Keywords: Model order reduction, domain decomposition, Kolmogorov n-width, freezing method, calibration, hyper-reduction, optimal control, stabilization, proper orthogonal decomposition, Dynamic Mode Decomposition

Cette section va être assez courte, non pas que je n'ai personne à remercier, mais parce qu'il faut que je prépare la soutenance ! Je remercie Prof. Chinesta et Prof. Haasdonk d'avoir accepté de rapporter la thèse. Je remercie Yvon de m'avoir permis d'effectuer cette thèse dans de bonnes conditions. Je le remercie aussi pour ses bonnes idées qui, distillées régulièrement, m'ont permis de progresser dans le sujet. Merci à (Dr) Roxana Crisovan pour la collaboration, et merci à Prof. Abgrall de l'avoir facilitée.

Je termine par le plus important assez sobrement. Merci à la famille , aux copains/copines du labo, aux copains/copines d'ailleurs pour les bons moments et pour la rigolade, indispensables pour finir une thèse. v

Quelques approches non linéaires en réduction de complexité

Résumé

Les méthodes de réduction de modèles offrent un cadre général permettant une réduction de coûts de calculs substantielle pour les simulations numériques. Dans cette thèse, nous proposons d'étendre le domaine d'application de ces méthodes. Le point commun des sujets discutés est la tentative de dépasser le cadre standard «bases réduites» linéaires, qui ne traite que les cas où les variétés solutions ont une petite épaisseur de Kolmogorov. Nous verrons comment tronquer, translater, tourner, étirer, comprimer etc. puis recombiner les solutions, peut parfois permettre de contourner le problème qui se pose lorsque cette épaisseur de Kolmogorov n'est pas petite. Nous évoquerons aussi le besoin de méthodes de stabilisation sur-mesure pour le cadre réduit.

Mots clés : Réduction de modèles, décomposition de domaine, épaisseur de Kolmogorov, méthode de freezing, calibration, hyper-reduction, control optimal, stabilité de schémas numériques pour la mécanique des fluides, décomposition orthogonale en modes propres ...

vi

A few non linear approaches in model order reduction Abstract

Model reduction methods provide a general framework for substantially reducing computational costs of numerical simulations. In this thesis, we propose to extend the scope of these methods. The common point of the topics discussed here is the attempt to go beyond the standard linear "reduced basis" framework, which only deals with cases where the solution manifold have a small Kolmogorov width. We shall see how truncate, translate, rotate, stretch, compress etc. and then recombine the solutions, can sometimes help to overcome the problem when this Kolmogorov width is not small. We will also discuss the need for tailor-made stabilisation methods for the reduced frame.

Quelques approches non linéaires en réduction de complexité

Nicolas CAGNIART 12 Second application of ORBEM: a dense collection of subdomains, following a pattern 2. [START_REF] Bardos | Derivatives and control in the presence of shocks[END_REF] The ORBEM method allows for the rotation of the local basis . . . . . . . . . . .

List of Figures

Préambule

Le chapitre introductif commence par une description du problème cible, l'objectif que nous nous sommes fixé en début de thèse. Après avoir décrit brièvement une méthode type, empirique, utilisée aujourd'hui pour résoudre ce problème cible, nous motivons le besoin pour une approche plus rigoureuse, qui tienne compte de la très grande complexité du problème due aux nombreuses échelles spatiales impliquées. Dans cette direction, nous décrivons la méthode de bases réduites, qui répond simultanément aux deux contraintes: un cadre théorique rigoureux ainsi que des coûts de calculs réduits. La fin du chapitre introductif insiste sur le fait que certaines questions liées à l'application de méthodes type «bases réduites» au problème cible n'ont pas encore été résolues. C'est à quelques unes de ces questions que nous tentons de répondre dans les autres chapitres.

Dans le chapitre 2, nous étudions la facette «variabilité géometrique» du problème cible, et nous le faisons indépendamment des autres difficultés soulevées dans le chapitre introductif. Il apparait assez clair que cette problématique est proche des méthodes de décomposition de domaine, et que notre méthode devra être adaptée au contexte qui est celui de la réduction de modèles. Après avoir rappelé les quelques approches de la littérature proches de nos besoins, nous concluons sur la nécessité du développement d'une nouvelle méthode, plus flexible et proposons un cahier des charges. Le reste du chapitre est consacré à la description et à l'analyse d'une méthode répondant aux besoins fixés.

Dans le chapitre 3, nous cherchons de nouveaux outils à ajouter à la réduction de modèles standard et qui permettent de traiter des problèmes pour lesquels les variétés solutions ont une grande épaisseur de Kolmogorov. Nous commençons par décrire la méthode de Freezing, disponible dans la litterature et qui est une réponse possible, mais pas entièrement satisfaisante à la problématique du chapitre. Le reste du chapitre est consacré à la description d'une méthode alternative, la calibration. Des tests numériques sur l'équation de Burgers visqueux dans le cas périodique, viennent confirmer la viabilité de l'approche.

Dans le chapitre 4, nous appliquons la méthode de calibration à un problème plus réaliste: un écoulement autour d'un profil NACA. L'application de la calibration à ce problème qui est un problème en dimension deux, hyperbolique, et non périodique, pose de nouvelles difficultés que nous tentons de résoudre. De nombreux ingrédients sont nécessaires pour la résolution complète de ce problème. Par conséquent, la section numérique se concentre sur quelques aspects, ceux qui nous paraissent les plus importants.

La calibration dans les chapitres 3 et 4 est utilisée pour réduire l'épaisseur de Kolmogorov de variétés solutions. Le chapitre 5 part du constat que la calibration amène une propriété supplémentaire: elle ajoute à la régularité des solutions, comme fonction des paramètres. Ceci est vrai quelque soit le problème étudié. Un exemple particulièrement parlant, et celui que nous

Preamble 1

The introductive chapter starts with a description of the target problem, the initial goal of the thesis. After briefly discussing existing empirical models, we motivate the need for an approach with a stronger theoretical background: we choose to use reduced order modeling. We give a quick overview of how ROM is usually performed, and insist on the properties a problem should satisfy in order to be a good ROM candidate. We then provide evidences showing that the target problem is not, at first glance, suitable for ROM, and that additional steps need to be performed.

In chapter 2, we study difficulties due to the geometry variability of the target problem. We work with a model problem, that helps isolating this one issue from the other mentioned in the introductive chapter. We then give a quick overview of the ROM methods available in the literature that are designed to handle geometry variations. We show that there is a need for a method more flexible than the existing ones. The remainder of the chapter is devoted to the development of such a method.

In chapter 3, we try to develop a method that allows for the use of ROM in the context of solution manifolds with large Kolmogorov n-widths. We describe the method of Freezing that is a interesting answer to this specific problem, but not entirely satisfactory in our opinion. The remainder of the chapter focuses on the description of an alternative method: the so-called calibration. Numerical experiments are performed on the periodic viscous Burgers equation, and tend to confirm the viability of the method.

In chapter 4, we apply the calibration method to a more realistic problem: the flow around a NACA airfoil. This leads to additional challenges, as the solutions present shocks, whose positions are strongly parameter dependent. Also, the domain is now two dimensional, and we are not in the favorable periodic setting. A complete numerical scheme is quite involved, and we have rather chosen to focus on a reduced number of issues, the ones we consider the most important.

The calibration is used in chapter 3 and 4 to diminish the Kolmogorov n-width of solution manifolds. Chapter 5 starts by noticing that calibration adds a bonus property: it improves the regularity of the dependency of the solutions with respect to the parameters. This is true whatever the problem studied. An especially enlightening example, and the one we have chosen to study in this chapter, is the one of hyperbolic problems, where the shock positions' sensitivity to parameter variations is high. We end the chapter by deriving a few preliminary results, towards the development of numerical schemes for the computation of parameter derivatives of the solutions.

The last two small chapters of this manuscript are a little different, but discuss connected topics. For the computation of proper orthogonal modes for time dependent problems and/or for parameter spaces with moderate dimensions, it is natural to apply a divide and conquer strategy. Chapter 6 gives bounds on the errors due to this kind of approach.

The starting point of chapter 7 was to notice that big data related ideas were receiving more and more attention in the reduced basis community. We start by mentioning some recent and promising results in that direction. We then discuss the DMD ( Dynamic Mode Decomposition ) method, and use its analysis to warn for inconsiderate use of such approaches. There are some cases, where brute force can not replace a good model, whatever the amount of data assimilated.

Chapter 1 Introduction

This first chapter is an extended introductory chapter. We start with a short description of the target problem, the original objective of this thesis. After briefly discussing existing empirical models, we motivate the need for an approach with a stronger theoretical background: we choose to use reduced order modeling. We give a quick overview on how ROM is usually performed, and insist on the properties a problem should satisfy in order to be a good ROM candidate. We then provide evidences showing that the target problem is not, at first glance, suitable for ROM, and that additional steps need to be performed. We finally detail how each of the chapters of this thesis is a new step towards the objective.

The first section states the starting point of the thesis. We briefly discuss the context in which this study takes place. The introductory section ends with a motivation for the use of Reduced Order Modeling (ROM). We then give an overview of the ROM framework. Instead of a general, heavy presentation, we have chosen to illustrate it on a simple heat equation. The end of this chapter is devoted to showing that there were (and still are) many ingredients missing to provide a complete ROM based solution to solve the initial objective. Following this remark, we have focused on simpler, more reasonable objectives. Some of them will seem far away from the initial goal. we consider them as steps, or elementary bricks, towards a sensible and robust solution to the initial objective.

Vertical axis wind turbine placement optimization

Wind farms have started catching attention for clean energy production for a few decades now. To get a grasp of their growing importance, one can for instance take a peak into the list of major European offshore wind farms, in [103]. The infatuations of the beginning have opened up to a more mature market where optimization of cost, production and maintenance hold a bigger place.

Whatever the precise objective function, these optimization problems have many layers of complexity. Indeed, for such problems, many length scales are involved, each having interacting influences. The scales identified a priori are: the scale of the boundary layer, the scale of the wing (shape, material etc.), the scale of the farm (for instance, the relative placement of the turbines), and finally the atmospheric scale (required for boundary conditions). The first two scales have already received some attention because of the many different applications among which aeronautics. It also benefits the fact that experiments in wind tunnels can easily be programmed and conducted. The last two scales are more difficult to study, from a theoretical stand point as well as numerically and experimentally.

In this thesis, the focus has been put on optimization at the scale of the farm. The analysis of the measurements of working wind farms have shown that the average power loss due to the influence of upstream turbines averages 10% to 20%. It is also shown that in the wake, one finds higher level of turbulence, and thus higher loads on the turbines, which means higher maintenance costs. Many different configurations have been and are currently being tested. We mention two of them: in [START_REF] Jane Barthelmie | Modelling and measurements of power losses and turbulence intensity in wind turbine wakes at middelgrunden offshore wind farm[END_REF], the authors describe a bow shaped windfarm. A more natural 'grid' type farm is studied in [START_REF] Sten Tronaes Frandsen | the shadow effect of large wind farms: measurements, data analysis and modelling[END_REF]. The question that has naturally risen is: is there a way to diminish the wake effect as well as the load imposed on the turbines by adjusting the placement of the turbines ?

The (very ambitious) objective of the thesis was to give an optimized wind farm layout given the shape (or model) of a turbine, some constraints on the positioning of turbines and some other physical parameters (such as boundary and initial conditions). We have restricted the study to offshore windfarms and vertical wind turbines. This provides with useful assumptions: we do not have to take into account the topography. Also, vertical wind turbines allow for the use of a cylindrical symmetry. The turbines are not sensible to the inflow direction, and the only parameters for a given turbine shape are the direction and speed of rotation.

We note right away that even if the optimization only takes place at the scale of the farm, the behavior at other scales aforementioned still need to be modeled. In other words, the computation for one specific configuration is already a challenging task. But the situation is even worse: the objective being optimization, we are in a many-query context. This means that a focus needs to be put on the computational cost of the method.

The majority of the literature on the computation of the power output of wind farms uses rough empirical models 1 . We start our journey by giving a very quick overview on how these models are constructed, and on the theoretical results they rely on. We then give the major drawbacks of these approaches. This serves as a justification for the central discussion of this thesis: how to use the theory of Partial Differential Equations (PDE) and Reduced Order Modeling (ROM) to solve the optimization problem at hand.

A complete state of the art of the engineering methods currently being used to estimate a priori the output of a wind turbine farm is out of the scope of this thesis. We refer to [START_REF] Crespo | Survey of modelling methods for wind turbine wakes and wind farms[END_REF][START_REF] Frandsen | Analytical modelling of wind speed deficit in large offshore wind farms[END_REF] for surveys, and have rather chosen to briefly present one 'generic engineering method'. We underline some of its flaws and thus motivate the need for more advanced models. We also take this opportunity to once again highlight the overall complexity of the task at hand.

The first component, common to all rough engineering models, is a wake deficit to power output relation. We describe one possible route. Let some cylindrical control volume around the rotor, with the longitudinal axis in the direction of the flow. The process is illustrated in Figure 1.1. We apply the conservation of momentum equation and suppose that the viscous term, the pressure term, basically all terms except the convective term and the force due to the turbine are negligible. Some hypotheses can be justified by physical considerations and we refer to the aforementioned articles for more details. We denote with u the velocity, with T a source term that represents the force acting on the turbine, with fl the density and with Ê the cylindrical control volume of interest. We end up with a relation such as:

⁄ Ê Ò•(flu ¢ u) ¥ T.
Denote u in and u out respectively the inflow and the outflow. We apply the divergence theorem We can see how, by placing such models end to end, one can derive a first naive way of estimating the power output of wind farms with aligned turbines.

To get more advanced models and results for more general configurations, one needs to add other components to try and account for other various physical phenomenon. We mention a few of them, but there are as many variants as there are paper on the topic. For instance, one needs to model the wake geometry, or equivalently a procedure to select the downstream turbines that are being impacted by a given wake. In [START_REF] Kusiak | Design of wind farm layout for maximum wind energy capture[END_REF] they try using a reasoning similar to the one that lead to equation (1.1) to account for the interaction between wakes. Other models have been constructed to estimate the power loss due to an increase in the turbulence intensities, see for instance [START_REF] Jane Barthelmie | Modelling and measurements of power losses and turbulence intensity in wind turbine wakes at middelgrunden offshore wind farm[END_REF]. This type of rough modeling is currently being used on real life data. The numerical results presented in the literature often compare the energy output of the model with experimental data. Thanks to the extra degrees of freedom given by parameters in the model, a reasonable match is (most of the time) found between the two.

The examples of more advanced modeling just mentioned illustrate something important: there are many effects that need to be taken into account. The propagation of the wake, the wake interactions, and the change in the nature of the flow (increasing turbulence levels) have important effects on the output. One can not conclude, a priori, on the effects that can be neglected. For instance, a model that accounts for wake interaction, without discussing turbulence levels may be dubious.

I draw two important conclusions from this short analysis. The first one is that there is a need for more advanced modeling, to better understand the different mechanisms and their influences. The second conclusion is that it will be hard to find intermediate steps between the rough calculations resulting in relations such as in (1.1), and the full solution of the problem. The objective of this thesis is to find such steps (that might look like sideway steps) towards a complete resolution of the problem, or at least towards a model with rigorously justified assumptions.

PDE models

As a more realistic approach, we have chosen to use the theory of Partial Differential Equations (PDE). We take advantage of this section to define some notations that will be used throughout this manuscript. Ω denotes a smooth domain in R d where d oe 1, 2, 3. The problems considered will often be time dependent PDEs and the model equation that we have chosen to work with is:

't oe [0,T],
ˆu ˆt + L(u)=f on Ω.

(1.2)

To complete the system, and have a chance to have a well posed problem, one needs to provide appropriate initial u(t = 0) = u 0 and boundary conditions for u or ˆu ˆn on ˆΩ. L will be throughout this manuscript a first or second order partial differential operator.

We make a constant use of the following functional spaces: L 2 (Ω) the space of square integrable functions over Ω;

H 1 (Ω) := Ó u oe L 2 (Ω), Òu oe ! L 2 (Ω) " d Ô
. Also, denote H 1 0 (Ω),t h e elements of H 1 (Ω) with zero trace. That is H 1 0 (Ω) :=

) u oe H 1 (Ω), and u =0a.e on ˆΩ* . Let Y be any real Hilbert space. We will denote by < •, • > Y and ΕΠY := Ô < •, • > Y the scalar product and the norm respectively. The dual space of Y will be denoted Y Õ . For instance, ! H 1 0 " Õ (Ω) = H ≠1 (Ω). For an introduction on Sobolev spaces, we refer to the reference book on the subject [START_REF] Robert | Sobolev spaces[END_REF].

In this thesis, our focus will be put mainly on three equations. A rigorous theoretical presentation of each of them is not in the scope of this thesis, nor is it its topic. We briefly state here some of their characteristics, and detail the chapters of the thesis and the particular contexts in which each of them appears. Burgers We will encounter in this manuscript both the viscous '>0 and inviscid cases ' =0 .

The viscous case is a one dimensional non linear but simplified model of the Navier-Stokes equations. It allows for simple numerical experiments, as will be conducted in chapter 3.

The inviscid case is a simple hyperbolic problem, and is often chosen as a first test case for hyperbolic solvers in the literature. We will throughout this manuscript focus on the viscosity, physically meaningful solutions. We will use it in chapter 5 as a model for Euler equation.

Euler We will study this equation in a two dimensional setting in chapters 4 and 5. It is, as the inviscid Burgers equation, an hyperbolic problem. Our main focus will be put on the development of shocks.

Navier-Stokes This is the target equation. Because of the issues that will be raised in section 1.5, we will actually not work with it a lot in this manuscript. We present in Figure 1.2 the model problem we work with in chapter 2. The wind turbines are modeled by cylinders, to focus on the optimization of the geometry of the farm, rather than on the realistic computation of a CFD flow around a turbine. Navier-Stokes simulations are also performed in the numerical experiments of section 3.9.

Numerical methods

A discussion about the (many) different numerical schemes used for CFD computations is not in the scope of this manuscript. This is the topic of many books in the literature, for instance [START_REF] David | Computational fluid dynamics[END_REF] or the more recent [START_REF] Tj | Computational fluid dynamics[END_REF]. Also, a more specific overview of PDE solvers' specificities for the computation of flows around wind turbines can be found in [START_REF] Vermeer | Wind turbine wake aerodynamics[END_REF]. We only briefly come back one aspect. Instead of considering realistic physical blade bodies, one can model the wind turbines with equivalent source terms. The most commonly used methods in that direction are the actuator disk and actuator line models. A presentation of both methods can be found in [START_REF] Luis A Martinez | A comparison of actuator disk and actuator line wind turbine models and best practices for their use[END_REF]. The form of the source term is chosen using basic physical considerations in the same vein as the ones that were used in section 1.1. The numerical results obtained for wind farm flow computations are promising. But as for the engineering models, they require parameter fitting. More precisely, one has to tweak the parameters of the modeled blade for each numerical simulation. One can argue that the quality of the results presented is mitigated by this parameter fitting. Nevertheless, this method can be seen as an intermediate method between the complete numerical solution, and the basic engineering models such as the one presented in section 1.1. This type of methods will not be further studied in this manuscript, and this for a very simple reason: we use even coarser approximations in the course of this thesis. In chapter 2, we use cylinders to model the windturbines. The chapters that follow use simpler shapes or even different equations, to focus on very specific issues.

We conclude this short section by stating one important assumption, that we will have to keep in mind all through this manuscript. It actually explains why we do not further discuss the standard numerical CFD methods. We assume that we have a fine solver that exactly captures the true physical solution. This fine solver can for instance be of Finite Elements (FE) type, of Finite Volumes (FV) type or a Discontinuous Galerkin (DG) scheme. To insist on the fact that the resulting solution cannot be distinguished from the continuous solution, it is often referred to as 'truth approximation'. The reasons for this point of view is easily understandable. Behavior/structures that can not be captured by fine schemes, are out of reach of the desired cheap approximate solvers. The best thing the latter can do is try and match the 'best known' solutions, which are the ones obtained with fine schemes.

Remark 1 Only a few references do not use this premise. In [START_REF] Maday | A parameterizedbackground data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics[END_REF][START_REF] Taddei | An adaptive parametrized-background data-weak approach to variational data assimilation[END_REF], the idea is to use data fitting in addition to some learned model. Instead of trying to match the fine solutions as best as possible, we allow for a bias in the model.

We denote from now on N the number of degrees of freedom of the chosen fine solver providing the truth solution. This gives us a reference computational cost. More precisely, any computational cost of order N will be considered a high cost. We present one more method before entering the core of this manuscript. It is intermediate between the engineering models of section 1.1 and the full reduced model framework of the remainder of this thesis. Let D be a parameter space and let J be a functional defined on D that we are trying to minimize. What we describe is a typical surface type method, an empirical answer to parametrized problems that can be easily implemented. We present a possible implementation in Algorithm 1 below.

A first try at optimization

Data: Parameter space: D Result: µ opt Define Ξ 0 some coarse sampling of the parameter space; Compute the solution u(µ) for each µ oe Ξ 0 ; repeat Find Ê k µDsuch that J(µ) is small for µ oe Ξ k fl Ê k ; Sample more finely D in w k : Ξ k+1 ; Compute the solution u(µ) for each µ oe Ξ k+1 ; k Ω k +1 ; until some accuracy/computational cost condition;

Algorithm 1: Surface type method An illustration of this algorithm for a parameter space embedded into R 2 is shown in Figure 1.3. The blue dots represent parameters for which the truth solution has been computed. This method requires the estimation of J in between sampled points. This is achieved using an interpolation procedure. The latter is an important component of the method and has a big impact on the overall accuracy. But as there is no theoretical results associated, it is done empirically. The direct consequence is that in order to achieve a decent accuracy, one needs to perform many fine scheme computations. It is obvious that this is not in accordance with the goal of computational cost reduction. Moreover, this issue is made even worse because of the curse of dimensionality. Indeed, as the parameter space is a subspace R dim(D) , for a fixed discretization step in the parameter space (and so equivalently for a fixed accuracy) the number of fine computations required grows exponentially with dimension of the parameter space. The model order reduction framework gives a theoretically more sound and computationally more reasonable answer to this kind of parametrized problem.

Model order reduction

We start with a macroscopic overview of the framework as well as its main requirements. We then detail the steps and key results. For a more thorough presentation, we refer to the two recent books on the subject, [START_REF] Jan S Hesthaven | Certified reduced basis methods for parametrized partial differential equations[END_REF] and [START_REF] Quarteroni | Reduced basis methods for partial differential equations[END_REF]. We denote from now on with µ a generic parameter and with D a generic parameter space.

The objective is to construct a method that allows for many queries of the type µ ae u(µ). Let µ oeD. The solutions u(µ) are typically solutions to some PDE:

Y _ _ ] _ _ [ ˆu(•,t;µ) ˆt + L(u(•,t; µ)) = f in Ω u(µ)(t = 0) = u 0 in Ω u or ˆu ˆn = g on ˆΩ.
(1.3)

ROM assumes no a priori parametric dependence. f , u 0 , g, Ω and L may depend on the parameter µ. We will always explicitly state the parametric dependency. The question that model order reduction tries to answer is: is there some regularity, whatever the precise meaning, of

; DaeX µ ' ae u(µ) (1.4)
and if so, can we take advantage of it to accelerate the computation of (1.4). Our motivation in this manuscript is optimization. Note that this can also be used to solve inverse problems. In the course of this thesis, the regularity of (1.4) will take various forms. It ranges from differentiable with an explicit derivative, as in section 1.4.1, to cases that require a preconditioning step, see section 1.5.1.

The fundamental notion for ROM that will be used throughout this manuscript, is the concept of solution manifold. Several definitions are possible depending on the context, see for instance the introduction of chapter 3. The most common one2 , that will be used unless specified, is given by:

M := {u(•,t; µ),µoeD,toe [0,T]} . (1.5)
The name manifold is not chosen loosely. The premise of ROM is to think of M as a smooth manifold, embedded in a well chosen Hilbert space X, even if no theoretical results regarding regularity are available for most real life problems. As we are aiming for the reduction of the computational cost of solutions in M, one sensible first step is to try to capture and compress the characteristics of this manifold. One adapted M u(t 0 ; µ 0 ) u(t 1 ; µ 1 ) Figure 1.4: Solid line: Graphical representation of the solution manifold M. Dotted line: A possible trial space theoretical tool is the notion of Kolmogorov n-width. This quantity measures the 'linear width' of any subset of normed spaces. More precisely, for any manifold M it is defined as:

d n (M,X):=inf En sup f oeM inf goeEn Îf ≠ gÎ X . (1.6)
The first infimum is taken over all linear spaces of dimension n embedded in X. A graphical example is presented in Figure 1.4. The Kolmogorov n-width returns the worst approximation, on the best linear space of dimension n. A few theoretical results are available to estimate a priori this n-width. In [START_REF] Melenk | On n-widths for elliptic problems[END_REF], they prove n-width estimates for solutions to elliptic problems where the parameter dependence is on the source term. The latter is taken in some compact of a high order Sobolev space. A more recent result gives estimates of the Kolmogorov n-width under holomorphic mappings [START_REF] Cohen | Kolmogorov widths under holomorphic mappings[END_REF]. More precisely, they show that the exponential decay of the n-width is conserved through the image of a Frechet differentiable function.

We conclude this overview of ROM by introducing the offline/online paradigm. The capture and modeling of the characteristics of the solution manifold M can be seen as a learning phase, referred to as offline in the ROM community 3 . It often amounts to creating a linear space of moderate dimension, that represents well M. The target error, for a fixed basis size, is given by the Kolmogorov n-width. This learning phase is expensive from a computational cost, as it involves the computation of a moderate number of fine solutions. The online phase uses this learning phase, and provides a way of computing approximations of members of M at a reduced cost4 . The analytical example of the next section will help understand this concept of offline/online stages.

Analytical example

As stated in the preamble, we have chosen to tackle the description of ROM by focusing on toy examples. The first one we study is a favorable case, as the regularity of the problem with respect to the parameter is explicit. Let the following heat equation:

; ≠Ò • (µÒu)=f in Ω u =0 on ˆΩ. (1.7)
We choose a parameter space D that satisfies:

I D is a compact subspace of L OE (Ω)
÷µ min oe R +ú , 'µ oeD,µ>µ min a.e in Ω.

How do the solutions u(µ) behave when we change the diffusion coefficients ? Existence and uniqueness of the solution in H 1 (Ω) are guaranteed by classic Lax-Milgram theory for coercive and continuous operators.

We study the limit of u(µ + h‹) ≠ u(µ) for a fixed direction ‹ contained in the unit ball of tangent space of D and h oe R. Let u h := u(µ + h‹) and let v h := u h ≠ u(µ). Using the linearity of the equation, the latter is solution of:

; ≠Ò • (µÒv h )=≠hÒ•(‹Òu h ) in Ω v h =0 on ˆΩ.
We can compute the standard a priori estimates for v h . For this, multiply the first equation by v h and integrate by parts:

⁄ Ω µÒv h •Òv h = h ⁄ Ω ‹Òu h Òv h .
We have used v h zero on ˆΩ. We use the hypotheses on D and on ΋ΠL OE to show the a priori estimate:

µ min |v h | H 1 AE h|u h | H 1 .
To conclude, we use the a priori estimates on u h :

µ min |u h | 2 H 1 AEÎf Î H ≠1 Îu h Î H 1 , with the Poincare inequality: ÷C oe R, s.t µ min |u h | H 1 AE CÎf Î H ≠1 ,
to finally obtain the following a priori estimates on v h :

|v h | H 1 AE 1 µ 2 min ChÎf Î H ≠1 . (1.8) Let w(µ, ‹) oe H 1 0 (Ω) be the solution to ; ≠Ò • (µÒw)=≠Ò • (‹Òu(µ)) in Ω w =0 on ˆΩ.
To prove Gateaux differentiability, we will prove estimates on:

÷ : ; D, D, R ae H 1 (Ω) (µ, ‹, h) ' ae u(µ) ≠ u(µ + h‹) ≠ hw(µ, ‹).
We know:

≠Ò • (µÒ÷(µ, ‹, h)) = ≠Ò • (µÒ (u(µ) ≠ u(µ + h‹) ≠ hw(µ, ‹)) = Ò•((h‹)u(µ + h‹)) + hÒ•(µw(µ, ‹)) = Ò•((h‹)u(µ + h‹)) + hÒ•(‹u(µ)) = Ò•((h‹)Ò(u(µ + h‹) ≠ u(µ)).
This gives a priori estimates on ÷:

µ min |÷(µ, ‹, h)| 2 H 1 AE h|÷(µ, ‹, h)| H 1 |u(µ + h‹) ≠ u(µ)| 2 H 1 .
We use the estimates on v h = u(µ + h‹) ≠ u(µ), see equation (1.8), and obtain:

µ min |÷(µ, ‹, h)| H 1 AE 1 µ 2 min Ch 2 Îf Î H ≠1 . (1.9)
This concludes on the Gateaux differentiability of the application, in the direction ‹, with derivative w(µ, ‹). Moreover, since for all µ oeD, w(µ, •):

; DaeH 1 0 ‹ ' ae w(µ, ‹)
is linear and continuous, we conclude on the Frechet differentiability of the parametrized problem over D.

From this analysis, we have one natural way of developing a method to approximate µ ae u(µ) at reduced cost. The first step is construct a good representation of the solution manifold M. The Frechet differentiability provides an easy solution, that we illustrate in Figure 1

.5. Define M u(µ 0 ) u(µ 1 )
Figure 1.5: One way of constructing a reduced basis: the Frechet differentiable case some threshold ', the maximum approximation error that we want on our solution manifold M. Sample the parameter space such that:

'µ oeD, ÷µ i oeD, s.t Îu(µ) ≠ u(µ i ) ≠ w µi (µ ≠ µ i )Î H 1 AE '.
The a priori estimates on ÷, see equation (1.9), allow us to compute a rigorous distribution of {µ i } i and the corresponding Frechet derivatives {w µi } i . This ends the offline phase as introduced in the previous section. The online phase is even easier in this setting. For any µ oeD,w ej u s t have to pick µ 0 in the pre computed set the closest to µ. We then have an explicit formula for an approximation of u(µ). The guaranteed error estimate is the chosen threshold '. Because of the Frechet differentiability, independent of the direction, we feel that we are starting to overcome the curse of dimensionality. We note the main differences with a realistic case. First of all, the smoothness with respect to the parameter does not often translate into an explicit formulation. Sometimes, the manifold even needs some preconditioning to enforce smoothness, see for instance chapter 5. Also, it feels like this type of construction neglects a lot of redundancy. For instance, we expect redundant pieces of information to be found in solutions for parameters far away in the parameter space. Nevertheless, the methodology developed on this analytic example is exactly the same as what is being done for ROM for real life problems, and is enlightening in that respect. The next section describes a typical offline phase.

ROM, the offline phase

As already stated, the first necessary step is to construct a basis that captures most of M. More precisely, we are looking for Ψ, a set of N functions in X the underlying Hilbert space, such that:

'u oeM, ÷u N oe span Ψ, Îu ≠ u N Î X small .
This first necessary step is the sampling of the parameter space. That is, we need to choose a representative set:

Ξ:={µ k ,k=[1...N]}µD.
It is easy to see that the size of Ξ influences the offline computational cost as well as the online accuracy of the method. The selection of the set Ξ is almost always5 done empirically. The reduced basis is then picked as a subspace of:

span {u(µ k ),µ k oe Ξ}µX.
The way the compression of the information contained in {u(µ),µoe Ξ} is done, varies among the many methods available. We mention here a few of them, the ones we feel the closest to our objective. A geometric approach close to the Centroidal Voronoi Tesselation (CVT) has been developed in [START_REF] Du | Centroidal voronoi tessellations: applications and algorithms[END_REF][START_REF] Burkardt | Centroidal voronoi tessellationbased reduced-order modeling of complex systems[END_REF]. The parameter space is splitted using a CVT algorithm. In solid mechanics, the most commonly used algorithm is the Proper Generalized Decomposition (PGD) [START_REF] Ladevèze | The latin multiscale computational method and the proper generalized decomposition[END_REF]. This greedy approach results in a separable approximation of the solution, along the different dimensions: time, space and parameter for instance. It is empirical for most applications. We mention some of the theoretical results are available. If the problem actually is a separable problem, then the PGD algorithm reduces to a POD algorithm (that we will detail below), see [START_REF] Nouy | A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations[END_REF]. Also, for the Poisson equation in the 2 dimensional unit square, it can be shown that each greedy iteration is well posed, and that the algorithm converges, see [START_REF] Le Bris | Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations[END_REF]. Finally, we mention the Balanced Truncation method [START_REF] Rowley | Model reduction for fluids, using balanced proper orthogonal decomposition[END_REF], which is built on a 'Linear Time Invariant' form of the system to solve.

We now give more details on the two methods that will be used in the rest of the manuscript. They are dominating in the ROM community, especially for CFD computations. We underline the up and downsides of both methods and conclude on the purposes they should be used for.

Reduced Basis (RB)

In this section, we give an overview of the reduced basis method. It relies heavily on the existence of a cheap, rigorous error estimator. That is, for each reduced basis Ψ: ={Â i }, it requires an application:

∆ Ψ : ; DaeR µ ' ae ∆ Ψ (µ),
that is an upper bound on the actual error made on the reduced basis approximation. More precisely, for u(µ) the truth approximation and u N (µ) the reduced basis Galerkin approximation, it should satisfy:

'µ oeD, Îu(µ) ≠ u N (µ)Î X AE ∆ Ψ (µ).
The construction of such an error estimator is described in chapter 3. The greedy algorithm follows naturally: Data: Fine sampling of the parameter space D: Ξ Threshold ' Maximum size of the basis N max Result:

Reduced basis Ψ ú k Ω 0; µ 0 Ω random parameter in D; Ψ 0 := {u(µ 0 )}; while k<N max and ∆ Â k >'do µ k+1 := argsup µ ∆ Ψ k (µ); Ψ k+1 := Ψ k fi u(µ k+1 ); k Ω k +1; end Ψ ú := Ψ k ;
Algorithm 2: The greedy RB method Recent results [START_REF] Buffa | A priori convergence of the greedy algorithm for the parametrized reduced basis method[END_REF] show that if the Kolmogorov n-width of the solution manifold M decays exponentially, i.e if:

÷c, C oe R 2 s.t 'nd n (M,X) AE ce ≠Ck , ( 1.10)
then the basis obtained using the greedy algorithm described above inherits this property:

÷(c Õ ,C Õ ) oe R 2 , depending on (c, C) s.t 'N, sup uoeM inf u N oeΨ N . . u ≠ u N . . X AE c Õ e ≠C Õ N (1.11)
Similar results are available if the argsup during the greedy algorithm is not done exactly [START_REF] Devore | Greedy algorithms for reduced bases in banach spaces[END_REF]. This method's main advantage is obviously its computational cost. Indeed, only a moderate number fine computations need to be performed, thanks to the error estimator. Another interesting property is that we have a guaranteed error bound on a fine sample of D. The major downside is that that for non linear problems, the error estimators available are not reliable, as the bounds are not tight. This reduces a lot the range of application of this method. It will be discussed in section 1.4.3.1 where we explain how to construct the error estimator. We will also discuss it in 3 when discussing a posteriori error estimations for the one dimensional viscous Burgers equation.

Proper Orthogonal Decomposition (POD)

In the RB algorithm, the objective is to mimic the search for the optimal space in the sense given by the Kolmogorov n-width. The only difference is the replacement of the true error by an error estimator. The Proper Orthogonal Decomposition (POD) uses a different objective function.

Let N be some prescribed size for the reduced basis and let J be the following functional:

J : I X N ae R ( 1 ,..., N ) ' ae q µj oeΞ Îu(µ j ) ≠ Πu(µ j )Î 2 X (1.12)
where Π is the orthogonal projection6 onto span {Â i ,i=[1...N]}. The objective of the POD method is to minimize J over all orthogonal basis of cardinality N in X.W ew i l lp r o v ei nt h e course of this section that J has a unique minimizer and that the resulting basis is in fact in span {u(µ j ),µ j oe Ξ} .

Remark 2 When using discretized solutions (i.e when X is finite dimensional), the POD reduces to a (correctly reweighted) Singular Value Decomposition (SVD), see for instance [START_REF] Gene | Singular value decomposition and least squares solutions[END_REF].

Remark 3

This problem has a solution even when considering continuous snapshots in X (instead of the sampled case described in this section). With our notation, we can replace the discrete q µj oeΞ by the continuous s D dµ, for an appropriate measured parameter space D. We start by deriving the first order optimality conditions, a set of necessary conditions on an hypothetical optimal basis Ψ: ={Â i ,i=[ 1 ,...,N]}. For this, we formulate the optimal problem as: min

ΨoeX N J(Â) s.t < i , j > X = " ij , 'i, j.
To avoid redundant constraints, define the following:

e j : ; X j ae R j ( k ) koe[1,...,j] ' ae (( k , j ) Y ≠ " kj ) .
For all k, denote ⁄ k oe R k the lagrange multiplier associated with the constraint e k and construct the Lagrangian L:

L : ; X N ◊ Π N k=1 R k ae R {Â i }, {⁄ k } ' ae J(Ψ) + q N k=1 <e k ,⁄ k > R k =0
We compute ˆL ˆÂi , for some i oe [1,...,N] in the direction "Â:

ˆL ˆÂi " = ≠ q µj oeΞ <"Â,u(µ j ) ><  i ,u(µ j ) > + q i≠1 p=1 (<"Â, p > Y )⁄ p i +2⁄ i i <"Â, i > ≠ q N k=i+1 (< k ," > Y )⁄ i k .
First order optimality conditions state that for the basis Ψ to be optimal, it needs to satisfy:

'i oe [1,...,N], 2 ÿ µj oeΞ < i ,u(µ j ) >u(µ j )= i≠1 ÿ p=1  p ⁄ p i +2⁄ i i  i ≠ N ÿ k=i+1  k ⁄ i k . (1.13)
Define the following functional on X: R :

; X ae X  ' ae q µj oeΞ <u(µ j ), > X u(µ j ).
We show that for each N , size of the basis, the first order optimality conditions, (1.13), are equivalent to:

{'i oe [1,...,N], i is an eigenfunction of R} .
The proof is done by induction on N . For N =1, equation (1.13) becomes:

ÿ µj oeΞ <Â 1 ,u(µ j ) >u(µ j )=⁄ 1 1 Â 1 ,
which concludes. Suppose it is true for some N . We know that the set of Âs satisfy:

'i oe [1,...,N + 1], 2R( i )= i≠1 ÿ p=1  p ⁄ p i +2⁄ i i  i ≠ N +1 ÿ k=i+1  k ⁄ i k (1.14)
Let i<N+1. We use the orthogonality of the basis to show that:

I 2 < R( i ), N +1 > X = ≠⁄ i N +1 2 < R( N +1 ), i > X = ⁄ i N +1 .
As R is symmetric, we easily conclude 'i<N+1 ,⁄ i N +1 =0 . The first order optimality conditions are thus equivalent to:

I 'i AE N, 2 R( i )= q i≠1 p=1  p ⁄ p i +2⁄ i i  i ≠ q N k=i+1  k ⁄ i k R( N +1 )=⁄ N +1 N +1  N +1
We conclude using the induction hypothesis.

Remark 4

In the literature, the inductive proof is not always conducted properly. It sometimes takes for granted that the optimal basis is hierarchical, i.e that:

argmin rank N +1 J(Â)= argmin rank N J(Â) fi  n+1
where  n+1 is the solution of some other minimization problem 7 . This is not trivial, and should be proved (using the same simple steps above).

To prove that this necessary condition is in fact sufficient requires more work. The first thing is to prove the existence of such eigenfunctions/eigenvectors. For finite dimensional scalar products, this is a simple consequence of the spectral theorem. The extension to X = L 2 (Ω) is presented in appendix and uses the Hilbert-Schmidt theorem. The last ingredient is to show that the basis Ψ constructed using the first N (when ordered with decreasing eigenvalues) eigenfunctions of R is effectively minimizing the functional J. The proof is done by a direct argument, i.e by showing that:

' Ψ orthogonal basis of cardinality N in X, J(Ψ) AE J( Ψ).

The proof can be found for instance in [START_REF] Volkwein | Proper orthogonal decomposition: Theory and reduced-order modelling[END_REF]. 7 The rank 1 POD of the projection of the original set onto the POD basis of rank N .

Remark 5

In the literature, it is not unusual to work with centered solution manifold in this context. More precisely, one can subtract the mean field or the stationary solution to the snapshot set before performing POD compression. A motivation for this additional procedure can be found in [START_REF] Wang | Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison[END_REF][START_REF] Holmes | Turbulence, coherent structures, dynamical systems and symmetry[END_REF].

The major downside of the POD method is its computational cost. We discuss this issue in a small chapter in this thesis see 6. There is one other downside. As we are optimizing the mean projection error, it is possible that at the end of the algorithm, there exists a subset of D for which the basis behaves very poorly. This issue is solved by the variant of the standard POD described in the next section.

Localized RB

To complete this small introduction on the construction of reduced basis, we mention adaptive RB methods. Adaptive is here to be understood in the following sense: instead of one single basis used on the whole parameter space D, we construct a small number of reduced bases, each of them with a domain of validity. The initial work in that direction was done in [START_REF] Fahl | Trust-region methods for flow control based on reduced order modelling[END_REF]. They define the notion of trust region in the parameter space. A more involved version has later been proposed in [START_REF] Maday | Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces[END_REF]. Their idea is to construct offline a metric in the parameter space. We present in Figure 1.6 a graphical illustration. Ideas related to hp can be implemented in the same spirit, see

M RB 1 RB 2 RB 3 Figure 1.6: Localized Reduced Basis
for instance [START_REF] Eftang | An" hp" certified reduced basis method for parametrized elliptic partial differential equations[END_REF]. We also mention a related method which uses interpolation between reduced basis [START_REF] Amsallem | Interpolation method for adapting reduced-order models and application to aeroelasticity[END_REF].

ROM, the online phase

In this section, we present how the basis constructed in the previous section are being used. This corresponds, using the ROM vocabulary, to the online phase. We once again work with the heat equation, see (1.7). We modify the setting compared to section 1.4.1, and put ourselves in a more classical ROM framework. The parameter dependency is now characterized by some function g over D:

g :

; DaeL OE µ ' ae g(•; µ).
The objective is to propose a method that allows for an efficient computation of an approximation of u(µ) oe X := H 1 (Ω), the solution to:

; ≠Ò • (g(•; µ)Òu)=f in Ω u =0 on ˆΩ (1.15)
when µ varies in D. As usual, we denote with C(µ) and -(µ) the continuity and coercivity constants of the bilinear form associated with the weak form of the equation. Suppose that we have managed, through POD or RB method for instance, to find an appropriate basis, that is, a linear space

X N := span {" i ,ioe [1 ...N]}µH 1 0 (Ω)
, that is almost as good as the optimal Kolmogorov n-width basis. This hypothesis can be formulated as:

'u oeM, ÷u N oe X N , Îu ≠ u N Î X ¥ d N (M,X).
The optimal representant is here the orthogonal projection of the true solution onto X N . As the true solution is not known, we need another way of choosing a good representant in X N .G o o d in the sense that the error should be controlled by the best projection error.

Remark 6

This question is the same as the one that appears when one looks for a finite element solution.

The choice is almost always the Galerkin method8 , as in the FE context. Pick u N (µ) oe X N that cancels the projection of the residual onto X N . In other words, pick u N (µ) in X N such that:

; ≠Ò • (g(•; µ)Òu N (µ)) = f in the dual space of X N u N (µ)=0 on ˆΩ. ( 1.16)
We can put (1.16) in variational form:

'v N oe X N , ⁄ Ω g(•; µ)Òu N (µ)Òv N = ⁄ Ω fv N
Cea's Lemma guarantees, for this elliptic coercive problem the following estimate on the Galerkin approximation:

'µ, Îu(µ) ≠ u N (µ)Î X AE C(µ) -(µ) inf v N oeX N Îu(µ) ≠ v N Î X .
That is, the error is controlled by the best approximation error.

Remark 7

More general cases such as saddle problems, objective oriented problems and problems with non compliant outputs can be treated. See for instance [START_REF] Martin | A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations[END_REF] for a review.

How do we implement the resolution of problem (1.16) ? We use the fact that both v N and u N (µ) lie on a finite dimensional space. The search for u N (µ) can thus be reduced to the search of a set

{-i } ioe[1...N ] oe R N , such that u N (µ)=
q N i=1i " i and thus:

'j oe [1,...,N], N ÿ i=1 -i ⁄ Ω g(•; µ)Ò" i Ò" j = ⁄ Ω f" j .
The desired set of coordinates is the solution to a system of small size, as problem (1.16) can be expressed as:

A(µ)-= F, where

I A i,j (µ): = s Ω g(•; µ)Ò" i Ò" j F j := s Ω f" j .
We now make explicit the offline/online paradigm introduced section 1.4 in this specific case. The objective is to pre compute as many quantities as possible in the offline phase. The hope is that during the online phase, whatever the parameter µ considered, no quantity with a complexity dependent on N the number of degrees of freedom of the truth solver has to be performed. In our example for instance, the vector F can be computed once and for all, as it is independent of the parameter µ. The cheap computation of µ ae A(µ) requires additional properties on µ ae g(•; µ). This will be discussed in section 1.5.2.1.

We end this quick overlook of reduced order modeling by mentioning a key feature, the a posteriori error estimators.

A posteriori error estimator

One of the problems with the engineering methods of section 1.1, and similarly with the method of section 1.3 is the fact that we have no way of certifying (or at least assessing) the answer given. Fortunately, this desired feature is available in the ROM framework. Let µ oeD. For simplicity we drop the µ dependency and denote u the continuous solution and u N its RB approximation. Define the following residual operator r in the dual space of X:

r : Y _ ] _ [ X ae R v ' ae s Ω g(•; µ)Òu N Òv ≠ s Ω fv ae s Ω g(•; µ)(Òu N ≠Òu)Òv.
Because we are using a Galerkin method, we know that the restriction of r onto X N cancels, but of course, r does not cancel over X. The objective is to give an estimation of the RB error Îu ≠ u N Î X as a function of the norm of the residual ÎrÎ X Õ . Using the coercivity of the initial problem, whose property is passed to both finite element discretization and reduced basis approximation, we have:

. . u ≠ u N . . 2 H 1 AE 1 - - - - - ⁄ Ω g(•; µ)(Òu ≠Òu N ) 2 - - - -= 1 - - -r(u N ≠ u) - -
This gives us directly the desired upper bound on the RB error:

. . u ≠ u N . . H 1 AE 1 - Îr Õ Î X Õ
The resulting question concerns the computational cost of this residual norm. It is often achieved by computing r's Riesz representant, i.e ê oe X, and by using the well known property: ÎêÎ X = ÎrÎ X Õ . A typical offline/online decomposition of such an error estimator will be presented in chapter 3.

Remark 8

The resulting error bound guarantees a maximum between the reduced solution and the truth solution. Again, the underlying assumption is that the truth solution and the continuous solution are indistinguishable.

Remark 9

The a posteriori error estimations heavily relies on the structure of the continuous partial differential operator. We have presented one simple case, but the formulation is well understood for a large class of problems. A review can be found in [START_REF] Quarteroni | Certified reduced basis approximation for parametrized partial differential equations and applications[END_REF].

Specificities of the problem at hand

The previous section could have ended the introductive chapter. The roadmap is clear: construct an adapted reduced basis offline. Then build a computationally efficient scheme that estimates the power output of a wind farm. Finally use a posteriori error estimators to certify the answer. Unfortunately, things are not so easy. The previous ROM framework does not apply directly to our problem, for several reasons. The geometric variability is the most visible issue. It makes the raw notion of Kolmogorov nwidth obsolete, and thus the rest of the analysis of section 1.4 unusable. This is the topic of chapter 2. Other, bigger, problems, inherent to ROM for fluid simulations, will be discussed in the remainder of this section. We start by presenting some toy examples leading to continuous solution manifolds with large Kolmogorov n-width. We then present examples where the continuous solution manifold is well behaved, but where standard numerical schemes involve terms with large n-widths. As the smallness of the n-width is a necessary premise of any ROM method, we end this section by discussing two classes of methods to deal with these issues.

Continuous solution manifold with large Kolmogorov n-width

One can easily construct examples inspired by realistic fluid dynamics problems, where the raw reduced order modeling presented in section 1.4 fails. We mention two of them.

Strong influence of the inflow direction

Let us take a wind turbine (or obstacle) fixed in Ω. Suppose that the main inflow direction varies, either in time or with respect to some parameter. We can propose a few reasons for this specific behavior. One can for instance think of the modification of the outside boundary conditions, or of the deviation of the flow due to the presence of an upstream turbine. In any case, a basis reproducing flow structures propagating in all directions will necessarily be of big cardinality. As a consequence, instead of the standard solution manifold, see equation (1.5), it seems more adapted to consider a 'transformed' solution manifold. A natural choice is:

M := {u(r ◊(t;µ) (x),t; µ),µoeD},
where r ◊ is the 2 dimensional rotation matrix. Intuitively, we expect that

'n, d n (M) ∫ d n ( M),
for a well chosen (t, µ) ' ae ◊(t; µ). It is clear that this choice of modified solution manifold causes many problems, among which the choice of ◊ during the simulation, the matching of this rotating domain with an outside solution. These two issues will be discussed respectively in chapter 2 and chapter 3.

A propagating front

Propagating fronts/shocks are known to be a limiting case for ROM, see for instance [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF]. In this section, we provide some quantitative evidences using a model example. We then try to give an explanation of why such strategies are still being used in the literature, and why the numerical results are not as poor as expected. This discussion is closely related to the one we will have in the next section about numerical stabilization. The specific toy example we discuss here is inspired by [START_REF] Taddei | Reduced basis techniques for nonlinear conservation laws[END_REF]. We consider the following continuous solution manifold: This can be thought of as the solution manifold of the transport of a step function. Its n-width can be explicitly computed. Let {µ j ,j=[1,...,N]} be a set of ordered parameters in D, and the corresponding snapshots: {u(•; µ j ),j=[1,...,N]}. Let X N be the space spanned by these snapshots. We start by computing the distance from X N to one member of M. Let µ oeD:

M = ) u(•; µ) oe L 2 (0,
inf voeX N Îu(•; µ) ≠ vÎ L 2 = Y _ _ ] _ _ [ 1 (µ≠µj )(µj+1≠µ) µj+1≠µj 2 1 2 on [µ j ,µ j+1 ] |µ ≠ µ 1 | 1 2 on [0,µ 1 ] |µ ≠ µ N | 1 2 on [µ N , 1]. (1.19)
We estimate the n-width as follows:

inf {µj }oeD max Q c c c c c a sup µoe[µj ,µj+1] 1 (µ≠µj )(µj+1≠µ) µj+1≠µj 2 1 2 sup µoe[0,µ1] |µ ≠ µ 1 | 1 2 sup µoe[µ N ,1] |µ ≠ µ N | 1 2 . R d d d d d b (1.20)
It is easy to see why the first infimum is obtained for a parameter distribution {µ j ,joe [

|µ ≠ µ 1 | 1 2 = sup µoe[µ N ,1] |µ ≠ µ N | 1 2 = sup µoe[µj ,µj+1] 1 (µ≠µj )(µj+1≠µ) µj+1≠µj 2 1 2
The second condition is equivalent to µ 1 = K 4 =(1≠ µ N ). With this new condition, we can compute explicitly the distance between the optimal parameters: K

= 1 N ≠ 1 2
, and the corresponding distance between X N and M:

dist (X N , M)= Û 1 N ≠ 1 2
This is obviously a bad convergence rate, and it jeopardizes the framework of section 1.4 9 . The real life situation is even worse, because the output of a numerical scheme is not as good as the optimal error given above. Also, it is known that compression algorithms (such as POD or RB) would result in oscillating basis, causing stability issues. Nevertheless, one can found in the literature such strategies being used, and they are associated with decent numerical results and convergence rates. I suggest one explanation. Suppose that instead of trying to reproduce the full solution manifold M given equation (1.18), you are trying to reproduce the discrete training set:

M train := ) u(•; µ k ),koe [1,...,N snap ] * .
We can see right away that the new n-width at hand is better behaved, as d n (M train )=0 for n>N snap , and independent of the dimension N of the underlying truth approximation space. One possibility to detect such behavior, without adding to much complexity to the implementation, is to modify the time step/parameter step being used in the online phase. In the case described above, one would keep in the online phase the parameter range of the offline phase [µ 0 ,µ N snap ], but with a different sampling strategy.

Numerical schemes involving large n-widths

The causes of the failure of ROM for the two previous examples are visible and easily understandable. This section is devoted to the presentation of another class of sets with less obvious non decreasing n-widths. Numerical schemes for CFD computations often involve some sort of numerical stabilizer. In this manuscript, the latter have two major origins: they are either related to the closure of Navier-Stokes equation or pure numerical tools that help enforcing physical properties on the solutions. Among the second class of methods, we will focus on methods that aim at preserving the Total Variation Diminishing (TVD) nature of entropic solutions to conservation laws. Even though these two classes have different natures, we can extract common properties:

• they are working at a different, smaller scale than the approximation error , i.e the n-width of the solution manifold measured in X norm. More precisely, the stabilization terms should be bounded by the desired consistency error

• most of them have a highly non linear dependence on the state variable.

We start by saying a few things on how non linearities are usually handled in the ROM framework. We then describe one example of TVD stabilization that is not a proper candidate for standard ROM. We then discuss the closure of Navier-Stokes equation. We compare our conclusions with some numerical examples in the literature. We end the introductive chapter by giving possible methods to deal with this specific issue.

Handling non linear terms in ROM

In this section, we discuss the computational costs incurred by non linear terms. This is one of the main topic of research in the ROM community at the moment. The most commonly used method is the Empirical Interpolation Method (EIM). It was originally developed in [START_REF] Barrault | An âĂŸempirical interpolation'method: application to efficient reduced-basis discretization of partial differential equations[END_REF] for a restricted set of applications among which the case where one of the physical parameters of the problem has a non trivial dependency on the parameter µ. We work once again with the model equation (1.15). In order to follow computational steps described in section 1.4, we need a good online/offline decomposition. A sufficient condition on g(x; µ) for this is its so called 'affine decomposability'. More precisely, we need a set of cardinality M of functions {g k ,k =1,...,M}oeX M , such that

'µ oeD, ÷{h k (µ),k =1,...,M}oeR M , s.t g(•; µ)= M ÿ k=1 g k (x)h k (µ).
The EIM method provides a way of constructing an approximate decomposition of such a type for situations where g is not rigorously affinely decomposable. It also provides a priori error estimates on the resulting approximation. Note that the strength of this method is that it does not solve the computational cost related problems by resorting to linearization or local linearization 10 . The simplicity of EIM have lead authors to widen its scope of application. It is now used when there is a non linear dependency on the solution itself. Denote R : X ae X some generic non linear11 function of u and {" i ,i=1 ,...,N} some well chosen reduced basis. A standard Galerkin method, see section 1.4, requires the efficient computation of terms such as:

<R( N ÿ j=1 -j " j )," i > X , 'i oe [1,...,N] for reasonable sets {-j ,j =1,...,N}.
How is this case being handled by EIM ? A sufficient condition is again the existence of an approximate affine decomposition of µ ae R(u(•; µ)), that is the existence of a set of cardinality M of functions {g k (x),k =1,...,M}oeX M such that:

'µ oeD, ÷{h k (µ),k =1,...,M}oeR M ,R (u(•; µ)) ¥ M ÿ k=1 g k (x)h k (µ).
(1.21)

We can extract two key ingredients that impact the overall accuracy of the method. The first one the is the existence of such a family {g k (x),k =1,...,M}oeX M . A necessary condition is for

R(M):={R (u(•; µ)) ,µoeD}
to have a small n-width, in the norm of the underlying Hilbert space, the X norm. As we will see, this property can be hard to satisfy for non linear stabilization mechanisms.

The second ingredient is less important but is still worth noting. It is related to the procedure that selects the representant on the EIM basis, i.e the set {h k (µ),k =1 ,...,M} in equation (1.21). In the EIM framework, the latter are chosen using pointwise estimates. As a consequence, the a priori estimates available are given in L OE norm. Moreover, the upper bound appearing in the error estimate, the Lebesgue constant, depends on the size of the EIM basis. This is less favorable than its counterpart in the standard Galerkin method 12 . We draw one important conclusion from the previous discussion for standard non linearities: in order not to deteriorate the overall accuracy, we either need R(M) to have a faster decaying Kolmogorov n-width than M or to work with an EIM basis bigger than the POD or RB basis.

Constructing a reduced TVD scheme

One needs to be careful when trying to apply an EIM type method to stabilization terms. As already mentioned above, these non linear terms act at the consistency error scale, and not at the approximation error scale. Denote R lim some operator acting on X and serving as stabilization mechanism for the truth numerical scheme. The following inequality will often hold:

ÎR lim Î X Õ π d N (M,X) .
Thus, when performing an EIM (or any other reduction) algorithm on such terms, taking as threshold some estimation of d N (M,X) does not guarantee a correct approximation R lim . Nevertheless, discussing the question of the n-width of d n (R lim (M)) is still relevant: we need to fit the computation of this term into the ROM framework. We make a few preliminary remarks:

• this term is inherently linked to the underlying mesh and corresponding truth approximation space. The n-width at hand is thus a "discrete" n-width

• its action is local

• the solutions we are trying to stabilize live in the projection of the solution manifold onto some reduced space M N := Π X N M and not in M

As a result, the natural n-width to estimate is the following:

d n (R lim (M N ),l OE ),
and for these terms to correctly enter the ROM framework, we need something like:

÷n πN, s.t d n ! R lim (M N ),l OE " π d 0 ! R lim (M N ),l OE " , ( 1.22)
or for some other right hand side that correctly characterizes the size of the limitation term. We provide now one example that illustrates the fact that for some standard discretization scheme, and their associated stabilization methods, a property such as equation (1.22) is not verified. We will deduce that trying to use such methods in a ROM framework is not a good strategy. Let a one dimensional transport equation. The parameter is chosen as the convection parameter:

u t + µu x =0.
Denote u(•,t; µ) the fine solution transported at speed µ, at time t. The initial condition is chosen to be smooth to ensure the smallness of the n-width of the continuous solution manifold M := {u(•,t; µ),t ,µoeD}. Let ' be some prescribed accuracy. One can find some N dimensional vector space X N such that:

'u oeM, ÷u N oe X N , s.t Îu ≠ u N ÎAE'.
The question that arises is how to construct a stable procedure u(•,t; µ) ' ae u(•,t+ "t; µ).T h e situation is illustrated in Figure 1.7. The solid line is the trajectory of the continuous (or equivalently stabilized truth) solution. The plane represents the low dimensional linear space associated to some RB or POD basis: X N . The dotted line is the best projection error. A well conducted offline phase guarantees that the error between dotted and solid line remains small, here bounded by the chosen '. The problem is that the solution of a numerical scheme without proper stabilization might diverge. This is illustrated by the dashed line and we have denoted u w the reduced solution when no proper stabilization is provided. We now describe an hypothetical situation, that will help us understand the issue at hand. Say that the truth numerical scheme we are using uses forward Euler time discretization and a space discretization operator denoted L. For each time step t n , we compute the truth solution u n+1 as:

u n+1 = u n +∆tL(u n ).
A standard way of guaranteeing the TVD nature of numerical approximations to conservation laws, is to use flux limiters. We roughly state the goal, and refer to specialized literature for 1.5. Specificities of the problem at hand

Πu 0 u 0 Πu(•,t k ) u(•,t k ) u w (•,t k )
Figure 1.7: Solid line: truth solution; Dotted line: Best projection onto some reduced basis; Dashed line: Output of a system with no proper stabilization a more precise description of this problem, see for instance [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF]. Godunov's theorem state that linear high order 13 schemes for the resolution of conservation laws are not monotone, and thus subject to stability issues. A reasonable choice is thus to use the high order scheme everywhere, except where it behaves poorly, i.e where solutions exhibit sharp gradients or reach a local extremum. At those locations, the flux limiters enforce the use of a first order scheme. For our current problem, as we are dealing with smooth solutions, we consider that our truth discretization differential operator L is composed of a sum of one linear component, the high order spatial discretization operator, denoted A truth , plus a non linear correction term, a local ingredient that only acts at local extrema, denoted R lim .W eh a v e :

u n+1 = u n +∆t ! A truth (u n )+R lim (u n ) " .
For simplicity, suppose that the solutions in M only posses one local maximum. The stabilization term is then given by:

'µ oeD, 't, R lim (u(•,t; µ)) ≥ -(u(•,t; µ)) " pmax(t;µ) ,
where p max (t; µ) is the index of the control volume where the maximum of the solution for parameter µ and time t is located and " is the standard Kronecker delta. Denote by N lim (D) the cardinality of {p max (t; µ),t ,µoeD}.W ee x p e c tN lim (D) to be of order N , the size of the underlying truth discretization space. Let us try to construct a reduced scheme mimicking this stabilization procedure. For each time step t n , we compute the reduced solution u N,n+1 as:

u N,n+1 = u N,n +∆tΠ X N L(u N,n ). Denote A RB =Π X N A truth .W eh a v e : u N,n+1 = u N,n +∆tA RB (u N,n )+∆tΠ X N ! R lim (u N,n ) " . (1.23)
Because of the structure of R lim , there is no hope in reducing its computational cost. Indeed, we can explicitly compute its n-width:

d n (R lim (X N ),l OE )= ; max t;µ {-(u(•,t; µ)} if n<N lim (D) 0 if n>N lim (D). (1.24)
We thus expect no decay before n ¥N.

One can argue that trying to reproduce the corrective term alone is a naive approach, and that a more reasonable one is to reproduce directly the limited term over Ω. That is, one can try and perform an EIM procedure on L(X N ). We provide a partial answer in what follows. The assumption of the form of the scheme is that:

L(X N )=A truth (X N )+R lim (X N ).
For all p, denote E p a generic linear space of dimension p embedded in X. The n-width of the manifold of intereset, is given by:

d n (L(X N ),l OE ):=inf En sup f oeL(X N ) inf goeEn Îf ≠ gÎ l OE ,
and it satisfies for all n:

d n (L(X N ),l OE ) Ø inf En sup f oeA truth (X N )+R lim (X N ) inf goeEn+A truth (X N ) Îf ≠ gÎ l OE Ø inf En sup f oeR lim (X N ) inf goeEn+A truth (X N ) Îf ≠ gÎ l OE Ø inf E n+N sup f oeR lim (X N ) inf goeE n+N Îf ≠ gÎ l OE = d n+N (R lim (X N ),l OE ).
The first inequality comes from the fact than E n µ E n + A truth (X N ). The last inequality comes from the fact that the infimum over all spaces of dimension (n + N ) is better than the infimum over all spaces of dimension n plus A truth (X N ). For our one dimensional transport equation, this means that the Kolmogorov n-width, measured in the l OE norm, of the limited flux is constant for n<N≠N . This is not satisfactory from a ROM point of view.

Remark 10

The previous analysis of course does not constitute a proof of the impossibility to create a stable, reduced scheme for this class of problems. For instance, it is very easy to taylor a reduced scheme that can handle the transport of smooth quantities. Nevertheless, we have provided evidences that tend to show that we can not guarantee the reconstruction of local, stabilization terms. This will be further discussed in section 1.5.5.

Instability for Navier-Stokes equation

In this section we discuss another common source of instability in CFD codes, the one associated with the numerical approximation of solutions to the Navier-Stokes equation. Let • be some filtering operation. It can be:

• a spatial filtering operation, as in the Large Eddy Simulation (LES) methods

• a time filtering operation as in the Reynolds Averaged Navier Stokes (RANS) simulations

• the orthogonal projection onto some well chosen reduced basis, still denoted X N , for RB type methods

For LES and RANS methods, the filtering operation can be done using some convolution kernel. More precisely, denote for all state variable w:

w := G ú w,
where G any convolution kernel and ú is the standard convolution operator in time (RANS) or in space (LES). Whatever the filtering operation chosen, the pair (ū, p) is solution to the following system:

I ˆū ˆt +ū •Òū ≠ µ∆ū + Òp =ū •Òū ≠ (u •Òu) Ò•ū =0 . (1.25)
In all three cases, (LES, RANS and RB), the system (1.25) is underdetermined. This is due to the filtering of the quadratic term u •Òu. The right hand side of the first equation is called the residual stress tensor. It represents the interaction between the resolved and unresolved scales, and needs to be modeled in order to close the system.

Closure for LES and RANS methods

One of the approaches that can be considered is to set the residual stress tensor to zero. For sufficiently fine meshes, this leads to stable schemes, as all scales are resolved: this is called Direct Numerical Simulation (DNS) in the literature. For reasonable (computational wise) mesh size, these schemes are often instable. Figure 1.7 is also a good illustration of this type of instability. The dashed line u w is in this context the output of a reduced scheme that uses zero residual stress tensor.

To my knowledge, there is consensus to interpret this instability, at least for LES type methods. The starting point is the existence of an energy cascade between the different length scales coexisting, see for instance [START_REF] Leonard | Energy cascade in large-eddy simulations of turbulent fluid flows[END_REF][START_REF] Robert H Kraichnan | Eddy viscosity in two and three dimensions[END_REF]. More precisely, the conjecture is that there is creation of energy at large scales and that the mechanism ensuring global stability is the dissipation occurring at small scales. Because of the filtering operation, the small scales are not resolved. The conclusion is that the methods with zero residual stress tensor lack a dissipation mechanism. To our knowledge, the numerical study of this cascade mechanism has received little attention. We mention here one of the results we are aware of. In [START_REF] Robert H Kraichnan | Eddy viscosity in two and three dimensions[END_REF], they study of the spectral decomposition of Navier-Stokes solutions and show that the energy transfers14 are local, in the wave number space.

Following these considerations, the most common option is to model the residual stress tensor with a viscous term. A profusion of closure models are available in the literature. We mention a few of them:

• for RANS, common choices are the k ≠ ' and k ≠ Ê models • for LES: Smagorinsky, Variational Multiscale (VMS), Dynamic Subgrid Scale etc.

For the most advanced LES models, the eddy viscosity is a highly non linear function of the state variables and needs to be recomputed at each time step and in each cell of the physical domain. In the Dynamic Subgrid Scale, it is even more complicated, since you are using two different filters corresponding to different cut off length scales, each of them with its own eddy viscosities model.

Theoretical results

We now mention another class of method, not directly applicable to Navier-Stokes equation, but that is backed by firm theoretical results. These have been developed studied in [START_REF] Temam | Infinite-dimensional dynamical systems in mechanics and physics[END_REF] and related references. We briefly present the results, but stay at a formal level since the rigorous proofs are quite involved and far from the scope of this thesis.

The class of problems considered is the class of dissipative partial differential equations. That is, the differential operator L, see (1.2), can be splitted into two contributions, one linear viscous term and one non linear term. We also require the linear part to be dominating, in some sense. Let L such a generic differential operator. Denote A the linear contribution and R the non linear part. Equation (1.2) becomes:

ˆu ˆt + A(u(t)) + R(u(t)) = F.
(1.26)

The starting point of the method is to consider the (complete) set of eigen functions of A in X. Denote {" i ,i=1,...,OE} this basis, ordered in decreasing eigenvalues. In the remainder of this section, we denote:

• P N the orthogonal projection onto span {" i ,i=1...N}

• Q N the projection onto span {" i ,i>N } • p = P N u and q = Q N u
With this notation, the objective appears clearly. We want to solve for the high energy modes components, denoted p. The other components, stored in q, are the so-called unresolved scales. We project (1.26) onto the two orthogonal spaces P N X and Q N X and get a system equivalent to the initial problem:

I P N ˆu ˆt + P N A(u(t)) + P N R(u(t)) = P N F Q N ˆu ˆt + Q N A(u(t)) + Q N R(u(t)) = Q N F.
We use the fact that the basis {" i } i is independent of time, and that the projection operators P N and Q N commute with A. We get a coupled system equivalent to (1.26):

I ˆp ˆt + A(p(t)) + P N R(p + q)=P N F ˆq ˆt + A(q(t)) + Q N R(p + q)=Q N F.
To get familiar with the notation, we can reformulate some standard methods in this framework:

• a zero residual stress tensor method uses

P N R(p + q)=P N R(p)
• eddy viscosity closure models assume that:

P N R(p + q)=P N R(p)+‹ eddy (p)A(p).
We now describe the ideal case. Suppose you are able to construct a function Φ:X ae X such that the system's dynamics are exactly captured by

ˆp ˆt + Ap + P N R(p +Φ(p)) = P N F. (1.27)
Then, the knowledge of Φ gives us a well posed problem on the high energy modes p. In [START_REF] Foias | Inertial manifolds for nonlinear evolutionary equations[END_REF], it is shown that when the diffusion operator A has a spectral gap, such a function Φ exists, at least asymptotically in time. The graph of Φ is in that situation called inertial manifold.

There is no spectral gap for the two dimensional Navier-Stokes equation, and there is yet no proof of any similar property. To my knowledge, the strongest results are the existence of determining forms, see for instance [START_REF] Foias | A unified approach to determining forms for the 2d navier-stokes equations-the general interpolants case[END_REF][START_REF] Foias | A determining form for the two-dimensional navier-stokes equations: The fourier modes case[END_REF] and the references therein. Again, these results are true asymptotically in time. To put it simply, they say that the state of the two dimensional Navier-Stokes system is completely determined by a finite (small) number of degrees of freedom. It is obvious that this is not as strong as (1.27), and not sufficient to construct self sufficient schemes.

Closure models for RB methods

The closure of system (1.25) is an open question for RB type methods, and no standard procedure has yet taken over. The first idea is to take a zero residual stress tensor, and hope that the situation is better than for LES. One reason for that would be the existence of an intrinsic stability due to the fact that the reduced solution manifold only contains physically relevant solutions/structures. The corresponding well behaved trial and test spaces should thus give a more stable numerical scheme than say a full Finite Element method. This is true to some extent. For instance, in one of the numerical experiments of chapter 3, we show that the CFL condition is much less stringent for the reduced scheme, than the one of the fine scheme. Unfortunately, this intrinsic stabilization mechanism does not seem to be enough. It is now well accepted in the ROM community that a scheme with zero residual stress tensor generally fails for two and three dimensional Navier-Stokes problems, see for instance [START_REF] Wang | Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison[END_REF].

The concept of energy cascade introduced for LES also applies in this context. In [START_REF] Couplet | Intermodal energy transfers in a proper orthogonal decomposition-galerkin representation of a turbulent separated flow[END_REF], they numerically study the energy transfers between POD modes for the backward facing step problem. They show that for integer pairs (n, m) such that |m ≠ n| > 25, the energy transfers are neglectable. The underlying idea is that high energy POD modes contain large structures, whereas low energy modes contain small structures, and so that the discussion about energy cascade for LES also applies here.

The specific form of the filtering operation in the RB case has lead to new interpretations of the instability. For instance, in [START_REF] Rempfer | On low-dimensional galerkin models for fluid flow[END_REF][START_REF] Lassila | Model order reduction in fluid dynamics: challenges and perspectives[END_REF] they construct an enlightening example that we have chosen to include here. It is a simple finite dimensional example that tries to give a geometrical interpretation of the instability 15 The high dimensional complete system considered here is three dimensional, and possess a stable limit cycle, captured by a low dimensional subspace (here two dimensional). We have plotted typical trajectories in Figure 1.8. This stable limit cycle is contained in the z =0plane and is represented as a thick line. The problem has been chosen such that the projected system onto the z =0subspace is unstable. We show in Figure 1.9 a graphical illustration of the mechanism at hand. The plane on which the scheme is projected can be seen as the space spanned by the high energy POD mods (or first RB modes). The projection of the scheme is unstable, as there is creation of energy. The physical stabilization mechanism, here the diffusion that occurs at small scales, is materialized by the z direction. This point of view have lead to the development of geometrical answers such as the one proposed in [START_REF] Ae Deane | Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders[END_REF]. They are interested in steady state solutions and they advise adding transient snapshots to the basis. This amounts to adding a neighborhood of the low dimensional subspace. We cite one more result related to this subject. It can not be applied to the resolution of Navier-Stokes equation, but is a theoretical result on a closely related issue. In [START_REF] Rathinam | A new look at proper orthogonal decomposition[END_REF], they give sufficient conditions on L so that the error of the projected system is controlled by the best projection error. Unfortunately, the constraints are far from any real life application. For instance, one of the requirements is the Lipschitz continuity of the system in the direction orthogonal to the space of projection.

We conclude this section with the most common methods. As already mentioned eddy viscosity methods are by far the most widely used approaches in LES and RANS methods. They have naturally been transposed to ROM. We mention a few references [START_REF] Wang | Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison[END_REF][START_REF] Bergmann | Enablers for robust pod models[END_REF][START_REF] Thomas | Large eddy simulation and the variational multiscale method[END_REF], but note that there is an extensive literature on the subject.

Study of reduced basis for CFD numerical results in the literature

In this section we present a sample of numerical results taken from the literature. These have been carefully chosen to illustrate some of the hypotheses/conclusions that we have drawn in the previous sections. The first class of problems that we have chosen to discuss are the ones for which the truth scheme uses zero residual stress tensor, and no additional numerical stabilization. This happens when the underlying mesh is fine enough to capture all scales. In this situation, we expect that a reduced scheme which also uses zero residual stress tensor and no additional stabilization mechanism to both give satisfactory results and be computationally efficient. Recent, preliminary results tend to confirm this conjecture, see [START_REF] Martini | Certified reduced basis approximation for the coupling of viscous and inviscid parametrized flow models[END_REF][START_REF] Stabile | Finite volume pod-galerkin stabilised reduced order methods for the parametrised incompressible navier-stokes equations[END_REF].

The next class of examples we have chosen are situations for which the truth schemes uses some sort of stabilization that correctly enters the ROM framework. One example can be found in a recent paper [START_REF] Tomás Chacón Rebollo | Reduced basis method for the smagorinsky model[END_REF]. They study the backward facing step for moderate Reynolds, 50 to 450, and use a Smagorinsky eddy viscosity model. This term is linked to the underlying discretization space. In each cell K, one adds an eddy viscosity ‹ K eddy such that:

‹ K eddy à h K |Òu| K where h K
is the characteristic size of mesh element K and |Ò • | K is the Frobenius norm of the gradient in element K. We are in the situation of equation (1.23) and we need to keep in mind that the Smagorinsky contribution is acting at the consistency error scale, and not at the approximation error scale. The Frobenius norm of the gradient of a state variable is rougher than the state variable itself. We thus expect its relative n-width to decay slower. Nevertheless, because of the specific problem studied (there are no complex structures propagating), the situation is not as bad as the one described in 1.5.2.2. The numerical results show that an EIM basis of size 73 is enough to guarantee a small relative error for the Smagorinsky term. By comparison, the RB needed to represent the solution manifold is of size 17. As the reduced scheme correctly reproduces all the contributions of the fine solver, the resulting reduced scheme gives good results.

The next step it to solve problems with more complex stabilization mechanisms. This has been done for instance in [START_REF] Carlberg | Galerkin v. least-squares petrovgalerkin projection in nonlinear model reduction[END_REF][START_REF] Baiges | Explicit reduced-order models for the stabilized finite element approximation of the incompressible navier-stokes equations[END_REF][START_REF] Carlberg | The gnat method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows[END_REF]. More complex can in this context mean the use of local/directional quantities such as upwinded flux, utilization of approximate Riemann solvers, directional gradient reconstruction (ENO or WENO schemes for instance) in a FV context, characteristics-Galerkin method (see [START_REF] Boukir | A high-order characteristics/finite element method for the incompressible navier-stokes equations[END_REF]) in the FE context. We once again use the notation introduced in 1.5.2.2. The first step in all the references mentioned is to compute the contribution L(u N,n ) exactly at each time step. Such a method has a computational complexity of the same order as the original fine scheme. This make the usage of the RB framework dubious. Nevertheless, the fact that the resulting schemes give good results guarantees that the cumulative approximation error is not too big of an issue, even for very challenging CFD problems. At this point, the problem seems to have narrowed down to the reduction of the computational cost of L(u N,n ) at each time step.

To study this issue, we focus on the final numerical experiment presented in [START_REF] Carlberg | The gnat method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows[END_REF], but similar considerations could be derived by looking at [START_REF] Carlberg | Galerkin v. least-squares petrovgalerkin projection in nonlinear model reduction[END_REF][START_REF] Baiges | Explicit reduced-order models for the stabilized finite element approximation of the incompressible navier-stokes equations[END_REF]. The objective is to compute the flow around the Ahmed body, for one fixed set of parameters. The fine mesh they used has several million nodes. The fine scheme used to produce the training set involves several local ingredients. For instance, it uses a Roe scheme to discretize convective fluxes. The unsteady simulation they are trying to reproduce has around 1200 snapshots. We denote by M the solution manifold {u n ,noe [1,...,1200]}. They show that a basis of size 283 captures 99.99% of the energy. In order to reduce the complexity of L(u N,n ) at each time step, they have applied a reduction algorithm (here gappy POD) directly. As L involves local ingredients, and following the discussion of 1.5.2.2, we expect the n-width of stabilization terms appearing in the numerical scheme to reach a plateau for n<N≠N , for N in the order of millions. They find that using a subsample of 1500 nodes in the gappy POD is enough to obtain a stable scheme. At first, this does not seem to match our conclusions.

Before going further, we repeat something that has already been been mentioned in section 1.5.1.2, when discussing solution manifolds with large n-widths. When an RB type method is used only to reconstruct the training set of small cardinality, the overall complexity is not in N the number of spatial degrees of freedom, but rather in N snap the number of snapshots in the training set. For instance, for the TVD example of section 1.5.2.2, this means that N lim (D) is of order N snap , and not of order N .. We thus expect the plateau behavior to stop at N snap ≠ N instead at N≠N . Back to the Ahmed body problem. My interpretation is the following: as the space {L(u n ),noe [1,...,1200]} is roughly of dimension 1200, one needs to find enough sampling points {x i } so that the matrix {u n (x i ), (i, n)} is of rank 1200. For this three dimensional challenging example, because of conditioning, the actual number of points needed for the gappy POD is 1500.T h e gappy POD can then be understood as the following application:

; {u n ,koe [1,...,1200]}ae X {u(x i ),ioe [1,...,1500]} ' ae L(u).
If this interpretation is correct, this is not satisfactory from a ROM point of view, as this method would not work for a multiparameter setting, and because this is very sensible to the time discretization.

A few methods to deal with the issues

We now propose two classes of methods that are partial answers to the problems mentioned in section 1.5. The first one uses the fact that the reduced scheme does not need to be the same as the fine scheme. The second class of methods is of a different nature. It handles manifolds with large n-widths by adding a pre-conditioning step.

A new class of stabilization mechanisms

One possible direction is to use in the online section a different scheme than the one used in the offline, learning phase. This constitutes a natural approach after the discussion of section 1.5.2.

One can for instance use, in the reduced scheme, a different, simpler, eddy viscosity model than the one used for the truth scheme. We mention a few references that perform such procedure [START_REF] Iliescu | Variational multiscale proper orthogonal decomposition: Navier-stokes equations[END_REF][START_REF] Wang | Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison[END_REF][START_REF] Östh | On the need for a nonlinear subscale turbulence term in pod models as exemplified for a high-reynoldsnumber flow over an ahmed body[END_REF][START_REF] Wells | An evolve-then-filter regularized reduced order model for convection-dominated flows[END_REF] and the references therein. In a recent paper [START_REF] Fick | A reduced basis technique for long-time unsteady turbulent flows[END_REF], another strategy is advised. They enforce stability by using a constrained Galerkin method. The constraints are chosen in the offline phase. The initial numerical results are promising, and additional tests need to be performed to get a deeper understanding of the mechanism at hand. Whatever the ROM stabilization mechanism chosen, one important ingredient was missing. It has been recently provided in [START_REF] Maday | An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems[END_REF]. It explicits the key hypothesis when using this kind of strategy. Their setting is the following: during the offline phase, they use a fine scheme with SUPG type stabilization, unfit for ROM. On the other hand, their reduced scheme uses a simpler stabilization, adapted to ROM. More precisely, they add viscosity increasingly with the mode number, a variant of the Spectral Vanishing Viscosity (SVV) method [128]. As expected, the solution obtained with the reduced scheme happens to be different from the one obtained with the fine discretization. They reconcile the two by stating that there is a one to one correspondence between the two stabilized solution manifolds. The truth solution is then retrieved using a so called rectifier, a one to one mapping between the SUPG solution manifold and the SVV solution manifold 16 . This rectifier is constructed during the offline phase.

In my opinion, one ingredient could be added to the offline stage, to give a more physically sound stabilization mechanism. Suppose that you know that you want/need to add diffusion at some prescribed scales. The following algorithm could be implemented:

Data: Set of snapshots {u(µ i )}, spatial scale ' Result: Adapted Reduced Basis Ψ ' Filter the snapshots: {u(µ i )}ae{ū ' (µ i )} ; Ψ' k := POD {ū ' (µ i )}; 'i, ũ' (µ i ):=u(µ i ) ≠ Π Ψ' u(µ i ); Ψ' k := POD {ũ ' (µ i )}; Ψ ' := Ψ' fi Ψ' ;
Algorithm 3: Construction of basis adapted to SVV The resulting basis Ψ ' is perfectly adapted to the Spectral Vanishing Viscosity method. Indeed, by playing with the parameter ', one can choose the scales that need to be damped and how they should be damped. It is obvious that Algorithm 3 can be extended to a version with multiple spatial scales using different damping coefficients. This method should add less numerical viscosity than a standard SVV to the reduced solutions, and should thus lead to a better behaved rectification step.

I mention one other approach that fits the ROM framework, and that could be numerically investigated. It follows the presentation of section 1.5.3.2, and we will use the notations defined there. Even if no 'inertial manifold type' results are available for the two dimensional Navier Stokes system, this sound framework can inspire new numerical stabilization algorithms. We propose one here, inspired by the one developed in [START_REF] Edriss | On approximate inertial manifolds to the navier-stokes equations[END_REF]. Let M>N . We replace Q N the projection onto all unresolved scales by the projection P M onto the largest unresolved scales. We are looking for Φ M :

Φ M : ; P N X ae (P M ≠ P N )X p ' ae Φ M (p),
that mimics the behavior of the application Φ in the inertial manifold case. We can for instance take it as the limit of a fixed point algorithm:

Φ M : p ae A ≠1 ! P M f ≠ P M R(p +Φ M (p)) " . (1.28)
This involves taking the projection of the residual onto the largest non resolved scales and inverting the dominant part of the operator. There are no major obstacles to fit those computations into a ROM framework. Also, it matches the energy cascade principle. Indeed, this method models the interaction between the resolved scales and the largest unresolved scales. The other unresolved scales (for n>M) should not impact the stability of the method. This method is not backed by any rigorous results and, of course, we have no guarantee that this would be more stable than for instance a raw Galerkin scheme using the first M modes. I nevertheless believe that it is worth some numerical investigation. A version of this algorithm was tested numerically on a one dimensional reaction diffusion equation in [START_REF] Foias | On the computation of inertial manifolds[END_REF].

Calibration

Another direction is to transform the original problem, in order to enforce the smallness of the n-width of the problematic terms. For discontinuous solutions, see section 1.5.1.2, this means forcing the position in the mesh of the singularity. The same idea can be applied for stabilization terms. For instance, for flux limitation, this means forcing that the position of the maximum, so that it remains always positioned in the same mesh element.

In this introductive chapter, we stick to a quick glance at the method, as it is actually the main contribution of this thesis, and will thus be extensively described in chapters 4 and 5. We consider the transport equation again. This time, say we are able to recenter the maximum at each time step. We denote M the resulting modified solution manifold. The corrective term that needs to be added to the scheme is now concentrated in the same cell p 0 , whatever the member ũ of M considered: R lim (ũ) à " p0 . The Kolmogorov n-width is (greatly) reduced, whatever the norm ΕΠconsidered:

d n (R lim ( M), ΕÎ)=0for n Ø 1.
This can be compared to the n-width in the original case, see equation (1.24). Moreover, we have a bonus property, R lim is now a linear function of u.

Remark 11

The linearity does not appear like a useful property for flux limitation. If you transpose this discussion to upwinding, or any directional procedure, then it becomes a valuable property. This will be discussed in chapter 5.

L 2 = ⁄ Ω 5⁄ Ω Â(y)r(x, y)dy 6 2 dx Cauchy-Schwarz's inequality in L 2 gives 'Â oe L 2 (Ω), ÎRÂÎ 2 L 2 AEÎÂÎ 2 L 2 ⁄ Ω ⁄ Ω r(x, y) 2 dydx
We then use the fact that r is separable, and the Cauchy Schwarz inequality in R N snap :

'Â oe L 2 (Ω), ÎRÂÎ L 2 AEÎÂÎ L 2 N snap ÿ k=1 Îu(µ k )Î 2 L 2
This concludes that R is a continuous function X ae X. Using Fubini, we get the self-adjoint property. Compactness is trivial, as the image of R is a finite dimensional subspace of X.

Chapter 2

Domain decomposition in ROM context, for configurations with varying structures

The objective of this chapter is to propose a ROM based method to solve problems with challenging geometry variations, in the context of CFD. Our target application has been discussed in the introduction of this thesis, chapter 1, and the corresponding illustration can be found in Figure 1.2. In this chapter, we start with a quick overview of the ROM methods available in the literature that are designed to handle geometry variations. We then show that there are important properties missing before being able to solve the target problem. This analysis leads us to propose a variation of the RBEM method, both more flexible and computationally more challenging than the original version. After discussing the a priori estimates, we conclude by discussing possible implementations and the overall computational complexity. .

Introduction

The objective of this chapter is to tackle one of the issues raised when discussing the initial goal of this thesis, see the introductive chapter 1. We show in Figure 2.1 the target problem that we will be focusing on. It can be seen as a model version of the original problem. The wind turbines have been replaced by cylinders. Also, we replace Navier-Stokes equation with the following model convection diffusion equation, with Dirichlet boundary conditions

; ≠-∆u + -•Òu =0 in Ω u = u D on ˆΩ.
(2.1)

These simplifications will allow us to focus on the unusual parameter dependency. It is the number N obs and position of the obstacles inside Ω:

D = {x k oe Ω,koe [1 ...N obs ]} .
The physical domain will be denoted from now on Ω(µ). The solutions to (2.1) onto Ω(µ) will be denoted u(µ). With this unusual parameter dependency comes a new set of questions. What is the correct reduced space ? Indeed, the standard solution manifold M defined as:

M := {u(µ),µ oeD}
is not embedded in any obvious Hilbert space. We can reduce this question to a simpler one: how do you compare solutions with different number of obstacles?

The intuition is that the correct space to look at is the manifold of the restriction of solutions on subdomains of Ω. One can for instance study M defined as:

M := {u(µ)| Ω k ,µ oeD, Ω k some subdomain of Ω}.
An illustration is depicted in Figure 2.2. Following this initial remark, we expect that solving the

x 1 x 2 x 3 x 4 x 5 x 6 Ω Figure 2.
2: An illustration of the concept of solution partitioning target problem will involve matching solutions defined on neighboring domains. This is exactly the purpose of domain decomposition methods.

The first section will be a discussion on existing domain decomposition methods. More specifically we will give a short review of the intersection between domain decomposition and reduced order modeling. We will explicit the reasons why none of the existing methods is adapted for our specific need, and conclude on specifications required to solve the target problem. The rest of the chapter will be devoted to the analysis of the proposed method, as well as computational considerations.

ROM and domain decomposition

This section will start as most talks and lectures on domain decomposition methods, by presenting their origin. Suppose that you know how to solve a PDE separately on elementary domains, given any boundary conditions. Can you solve the same PDE on a combination of the elementary domains ? The situation where there are two elementary domains, one circle and one square is presented in Figure 2.3. Many variations were developed around the method proposed initially

Ω 1 Ω 2 Figure 2.3:
The original Schwarz problem by Schwarz. As our purpose is not to review all existing methods, we will (very) briefly describe one of them, taken from [START_REF] Lions | On the schwarz alternating method. iii: a variant for nonoverlapping subdomains[END_REF]. Let u 0 1 and u 0 2 be some 'guessed solution' inside Ω 1 and Ω 2 . The following iterative algorithm

≠-∆u n+1 i + -•Òu n+1 =0 in Ω i u n+1 i = u n (i+1)%2 on Ω u ˆΩi u n+1 i = u D on ˆΩ u ˆΩi
converges towards the solution to the full problem (2.1) under conditions on the overlap Ω 1 fl Ω 2 . These ideas have found many applications and are widely used. They are the building block of the design of parallel PDE solvers, see for instance [START_REF] Victorita Dolean | An introduction to domain decomposition methods: algorithms, theory, and parallel implementation[END_REF] and the references therein. As we will see, we use them here in a different setting.

To continue exploring the specificities of the problem at hand, we present a version of the Schwarz problem, closer to the target problem of this chapter. It will be used in the next section to illustrate the fact that existing methods are not suited to solve our problem. Let a parametrized version of the problem depicted in Figure 2.3. The parameter is taken to be the distance between the barycenters of the elementary domains. We can put this into a ROM framework:

• denote µ the distance between the barycenter of the subdomains Ω1 and Ω2 . Denote Ω 1 := Ω1 and Ω 2 := Ω2 (•≠µ)

• denote Ω(µ) the global domain, for a parameter µ:

Ω(µ)=Ω 1 fi Ω 2
• denote D, the parameter space in which µ lives. It is some interval of R such that Ω 1 and

Ω 2 overlap
This problem resembles the one that lead to the first domain decomposition method by Schwarz. We know how the restriction of the solution to both subdomains Ω1 and Ω2 behave. Can we propose a ROM strategy to quickly solve the combined parametrized problem ?

Remark 12

The computational difficulties due to this specific parameter dependence already appear on this simple toy example. We expect this to be even worse when multiple domains are intersecting, as in the target problem. We will see in the numerical section how to mitigate these issues.

We now recall some of the methods used in the ROM community to solve problems with geometric variations. Some of them will seem far from the target problem, but they will help understand the context in which this work takes place. We discuss the method in section 2.2.1 because it is the most widely used ROM method for variation of geometries. We try to apply it to the toy problem (Figure 2.3) and show that it is not adapted. The reasons why we discuss the method in section 2.2.3 will become clear at the end of this chapter.

Domain Deformation

One option to deal with geometric variations is to use domain deformation. More precisely, one can build an equivalent problem for which the parameter dependency appears in the variational form. A complete description can be found for instance in [START_REF] Quarteroni | Numerical solution of parametrized navier-stokes equations by reduced basis methods[END_REF][START_REF] Deparis | Stabilized reduced basis approximation of incompressible three-dimensional navier-stokes equations in parametrized deformed domains[END_REF]. It assumes the existence of a reference domain Ω and of a family of smooth mappings {F µ } such that the physical domains in which we want to solve the problem Ω(µ) can be expressed as

Ω(µ)=F µ ( Ω).
The mappings can be:

• affine: x = G(µ)x + g where G is a R d◊d matrix. • non affine: x = T (µ, x) where T is a smooth mapping R d ae R d .
The geometric variability is then handled as follows: in an offline stage, perform the fine computation for several values of µ. Define the mapped solution manifold M as:

M := {u(µ) ¶ F µ ,µ }.
Use standard compression algorithm on M. In an online stage, modify the variational form to make the problems on Ω(µ),µ oeDequivalent to the problem on Ω.

Remark 13

When the mapping is non affine, the additional terms in the variational form have non affine parameter dependence. The EIM method [START_REF] Barrault | An âĂŸempirical interpolation'method: application to efficient reduced-basis discretization of partial differential equations[END_REF] can be used to reduce the computational cost.

Is this idea adapted to solve the toy problem depicted in Figure 2.3 ? We need to choose a reference domain Ω and smooth mappings F µ such that 'µ oeD, Ω(µ)=F µ ( Ω). Let μ be some parameter in D. We take Ω=Ω(μ). We take the affine by parts mapping that is the identity in some neighborhood of the circle and that linearly stretches the rectangle, away form the overlap. Let x0 be some abscissa, away from the overlap. Let x r (µ) be the abscissa of the right edge of the rectangle. This situation is illustrated in Figure 2.4. We take the following mapping x ae

I x for x<x 0 x0 +(x ≠ x0 ) xr(µ)≠x0 xr(μ)≠x0 for x>x 0 .
This method is far-fetched in this situation; it is not using the underlying structure of the x0

x r (μ) Figure 2.4: As possible reference mesh Ω problem. Also, the extension to the target problem, with multiple intersections is not reasonable.

Reduced basis element method

The reduced basis element method (RBEM) is using ingredients that better match the target problem. It follows the following recipe:

• decompose the domain into non overlapping elementary blocks

• eventually use domain deformation onto well chosen reference blocks

• perform standard reduced order modeling to get a reduced basis onto each reference block

• glue solutions together using domain decomposition methods. The matching constraints on the interfaces can for instance be enforced using reduced basis, coarse finite element spaces or low order spectral spaces This method was initially developed and tested on the heat equation [START_REF] Maday | The reduced basis element method: application to a thermal fin problem[END_REF]. Results for the two dimensional Navier-Stokes equation are presented in [START_REF] Emil Løvgren | A reduced basis element method for the steady stokes problem[END_REF]. We also mention the static condensation reduced basis element method [START_REF] Vallaghe | The static condensation reduced basis element method for a mixed-mean conjugate heat exchanger model[END_REF][START_REF] Jens | A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and a posteriori error estimation[END_REF] which is an 'industrial' version of the RBEM, as well as ArbiLoMod [START_REF] Buhr | Arbilomod, a simulation technique designed for arbitrary local modifications[END_REF], a recently developed variant that allows for online local basis enrichment. All of these methods require a structured geometric decomposition. A quick look at Figure 2.1 will convince the reader that the target problem lacks this property: we have no a priori idea of the geometry of the overlaps. The method we will propose is an extension of the RBEM, that deals with this issue. [START_REF] Tonn | Reduced-Basis Method (RBM) for Non-Affine Elliptic Parametrized PDEs:(Motivated by Optimization in Hydromechanics[END_REF] handles another geometry related problem. The stages of the method proposed are:

Rotating obstacle

• build local (reduced) basis around objects

• rotate the objects during the simulation

• match these local solutions to an 'outside solution'

The exact relation of this method to the target problem is not obvious. We refer to the discussion in the concluding section of this chapter for more details. in Figure 2.5, we present a simplified version of their problem. The global domain Ω is decomposed into three non overlapping subdomains, Ω 0 , outside, Ω int , a rotating domain and Ω tampon which makes the junction between the two other domains. Define the rotation operator

Ω 0 Ω tampon Ω int Figure 2.5: Method developed in [131] r ◊ as: '◊, r ◊ : Y ] [ Ω ae Ω (x, y) ' ae 3 cos(◊) ≠sin(◊) sin(◊) cos(◊) 43 x y 4
Let ◊ be some prescribed rotation parameter. They suggest using the following mapping:

x = F ◊ (x)= Y ] [ x if x oe Ω 0 r ◊ Õ (x) if x oe Ω tampon , where ◊ Õ = ◊ rext≠ÎxÎ2 rext≠rint r ◊ (x) if x oe Ω int
Let Ω be some reference mesh such as the one presented Figure 2.6. The effect of the mapping insures global continuity of the mapping. To get higher regularity, we could choose something more involved than linearly going from rotation of angle ◊ to identity.

Remark 14 Trying to build mappings with smooth transitions between neighboring domains is one of the topic discussed in chapter 4.

With our application to fluid dynamics simulation in mind, suppose that we have Karman vortices following a mean flow direction, taken here as the parameter. We model this situation by considering the following solution manifold:

M := {u 0 ¶ r ◊ ,◊ oeD}
where u 0 is some reference solution. A few snapshots taken from this manifold are presented in Figure 2.8. The inflow has a fi 4 amplitude variation, and the structures are modeled by 2 dimensional gaussian functions. We present in Figure 2.9 the snapshots mapped back onto the reference domain, that is {F ≠1 ◊ (u 0 ¶ r ◊ )}. Inspecting the results, we can see that the structures are distorted in the patching domain. The method does not resolve the complexity of the directional problem. It brings it out of the direct vicinity of the blade, to the patching domain. The underlying hope is that the complexity is much smaller away from the objects.

One last example

In [START_REF] Ganesh | A reduced basis method for multiple electromagnetic scattering in three dimensions[END_REF] is studied a electromagnetic field problem where the geometric variability resembles the one we are dealing with. What 'saves' them, is that they transform their problem into a surfacic one. The distance between the obstacles, which is the parameter, then appears explicitly in the modelisation of the interactions.

From this analysis of the literature, we can draw some conclusions:

• RBEM type methods require a structured geometry. We would like to have something more general.

• because of the applications we have in mind, we have to be careful on how we choose to impose the matching conditions, see section 2.2.3

• the number of basis functions in each domain is small. The number of matching constraints that we can impose on each overlap is thus reduced

We will propose two versions of an OverlappingRBEM (ORBEM) method. The first one is a conforming method that is close the partition of unity method introduced by Babuska et al [START_REF] Markus | The partition of unity finite element method: basic theory and applications[END_REF]. This method uses a priori knowledge of the solution inside a localized subdomain. For instance, you can pick as trial space some coarse finite element function on Ω, plus some sharp gradient where you know it will appear. The a priori error estimations show that the global error depends on:

• local errors: the error in each subdomain using the local approximation space

• terms that depend on the smoothness of the partition of unity functions We will see in subsection 2.4.1 that despite the fact that this approach matches some of the objectives of our target problem, it does not fit well into a reduced order modeling context.

The second one is a non conforming method, close the Arlequin method described in [START_REF] Ben | Approches locales-globales méthode arlequin[END_REF] and for which a mathematical analysis can be found in [START_REF] Hachmi | Analyse mathématique de la méthode arlequin mixte[END_REF]. It was developed in these references in a solid mechanics context. We add to this approach the reduced order modeling aspect, as well 2.3. The set up as a more rigorous mathematical analysis. The latter is also close to the analysis of the mortar with overlapping, see for instance [START_REF] Achdou | The mortar element method with overlapping subdomains[END_REF].

The set up

We are still solving the parametric PDE defined in equation (2.1), on domains such as the one displayed in Figure 2.1. We will denote by a the associated bilinear form. Suppose that we have identified offline

• a set of 'generic' subdomains denoted by Ωj ,j=1...J

• for each Ωj , a reduced basis Ó "j,l ,l oe [1 ...N red j ]
Ô of small cardinality.

For all µ, we assume that the physical domain Ω(µ) can be decomposed into an overlapping set

Ω(µ)=Ω 0 fi A N dom € k=1 Ω k B .
such that

• 'k>0, Ω k is the image through some linear transformation of a generic Ωj . One can for instance think of rotation or translation. In the case where only translation is considered, this means that we have a satisfactory reduced basis noted {" k,l }, given as the translated version of the generic reduced basis:

'k, ÷j s.t ," k,l = "j,l (•≠x k ),loe [1,...N red j ]
For the rest of this chapter, j(k) denotes the application that gives the index of the corresponding generic domain. x k is the translation parameter.

• there are no complex structures on Ω 0 . That is, u| Ω0 can be represented by a coarse FE space

• 'k>0, dist(ˆΩ k ,ˆΩ) > 0.
These assumptions can be used to define a new notion. Our parameterized initial problem is assumed to have a small 'local Kolmogorov n-width'. It is

• small in the Ω k ,k > 0 even if there are small scales represented. Indeed, we have found a good basis given by a translated version of { "j(k),l ,loe [1,...,N red j ]} • small in Ω 0 because we only need coarse scales.

We show in Figure 2.10 an illustration of possible generic subdomains. The different color here represents the fact there can exist different basis for the same shape. We have two applications in mind. Either a very sparse collection of interesting subdomains, but with no underlying regularity in the positioning. This situation is depicted in Figure 2.11. Or a more dense collection of interesting subdomains, but with some kind of underlying organization. A typical situation is presented in Figure 2.12. Most of the rest of the analysis will be common to these two type of applications. The only difference will be in the numerical section.

Both methods require the construction of a partition of unity. This family of functions defined on Ω will be denoted {‰ i } ioe[0..N dom ] . It needs to satisfy the following properties:

• 'i, ‰ i is smooth on Ω • q N dom i=0 ‰ i =1on Ω • 'i, ‰ ≠1 i (1) µ Ω is a closed domain, with non empty interior • supp ‰ i µ Ω i
The characteristics of the set {‰ i } will appear explicitly in the numerical analysis of the method. Until the end of the chapter, we will denote

• Ω int = t i=1...N dom supp (‰ i )
Figure 2.12: Second application of ORBEM: a dense collection of subdomains, following a pattern

• Ω 0 = supp (‰ 0 )

The ORBEM method

In both conforming and non conforming methods, we will follow the same steps:

• make sure that at the continuous level, the method is equivalent to the initial problem

• prove the well posedness of the discretized problem

• compute a priori error estimation of the discretized solutions

• give computational details

Conforming method

There is not much to say about the analysis of the continuous problem. We are simply solving problem (2.1) using a particular subspace of H 1 (Ω).

Numerical approximation

To approximate H 1 0 (Ω int ),w ew i l lu s eX int," defined as:

X int," = span {‰ k " k,l }µH 1 0 (Ω int ) (2.2)
where k oe [1 ...N dom ], l oe [1,...N red k ].O nΩ 0 , we take the following approximation space

X 0," = ‰ 0 span{h k }, (2.3)
where {h k } is a coarse finite element space on Ω 0 . With an abuse of notation, we still denote with X int," and X 0," the spaces of functions extended by zero to the whole domain Ω.T h e discretized space for the whole problem is then given by:

X " = X int," + X 0," . (2.4)
Wrapping things up, we are looking for an approximation of the solution u oe H 1 (Ω) as:

I u " = q k q l ‰ k -kl " k,l + q n ‰ 0 -0n h n u " = u D on ˆΩ (2.5)

Well posedness

As we are dealing with a conforming approximation of a coercive elliptic problem, the well posedness of the discretized problem is a direct consequence of the well posedness of the original problem.

Best approximation error

This is a conforming method for an elliptic coercive problem. Letbe the coercive constant and C the continuity constant of the bilinear form a. Let u be the truth solution to problem (2.1). Let u " be the solution obtained with the Galerkin method in X " . Cea's Lemma gives

Îu ≠ u " Î H 1 AE C - inf v " oeX " Îu ≠ v " Î H 1
We want to link this global error to local error estimates, that is we would like to have an inequality such as inf

v " oeX " Îu ≠ v " Î H 1 . ÿ k inf v " oe span {" k,l } Îu| Ω k ≠ v " Î H 1 (Ω k ) .
We follow the lines of [START_REF] Markus | The partition of unity finite element method: basic theory and applications[END_REF]. For v " oe X " , denote {v ",k } k the decomposition onto the basis. That is,

v " := ÿ k ‰ k v ",k .
We have

'v " oe X " , ÎÒ(u ≠ v " )Î 2 L 2 = . . . . . Ò A ÿ k ‰ k (u ≠ v ",k ) B. . . . . 2 L 2
As the ‰ k are assumed to be smooth, we have:

'v " oe X " , ÎÒ(u ≠ v " )Î 2 L 2 AE 2 . . . ÿ ‰ k Ò(u ≠ v ",k ) . . . 2 L 2 +2 . . . ÿ Ò‰ k (u ≠ v ",k ) . . .
Let n overlap be the maximum number of subdomains that contain any given point in Ω.W e have:

'v " oe X " , ÎÒ(u ≠ v " )Î 2 L 2 AE 2 n overlap A ÿ k Ή k Ò(u ≠ v ",k )Î 2 L 2 + ÿ k ÎÒ‰ k (u ≠ v ",k )Î 2 L 2 B (2.6)
We conclude that there exists a constant K,

function of C, -, n overlap , Ή k Î L OE , ÎÒ‰ k Î L OE ,such that: Îu ≠ u " Î H 1 AE K ÿ k inf v ",k Îu ≠ v ",k Î H 1 (Ω k )
Forgetting about computational errors, the overall error depends on

• the parameters of the original problem

• the smoothness of the partition of unity

• the local approximation errors

This is exactly the property that we wanted. Let's look at the computational side.

Computational cost

In practice, this conforming method reduces to findingkl and -0n such that

Y _ _ ] _ _ [ q Ω k q l -kl a (‰ k " k,l ,‰ p " p,q )+ q n -0n a(‰ 0 h n ,‰ p " p,q )=0 'p oe [1 ...N dom ], 'q oe [1 ...N red ] q Ω k q l -kl a (‰ k " k,l ,‰ 0 h m )+ q -0n a(‰ 0 h n ,‰ 0 h m )=0 'h m oe coarse Finite Element space u " = u D on ˆΩ (2.
7) This formulation requires the computation of coupled terms such as:

K k,l,p,q := ⁄ Ω Ò(‰ k " k,l )Ò(‰ p " p,q ).
We would like to get rid of the coupled terms, coming from K k,l,p,q when p is not equal to k. We will see that, in that respect, a non conforming approach makes more sense. We will insist on this property in the numerical section.

Non conforming method

Unlike the conforming method, we will work a little bit at the continuous level. For simplicity, we will assume throughout this section that u 0 is known, having in mind a fluid dynamics problem, where inflow and outflow would be taken as uniform flow. The case with u 0 in a coarse finite element space would be treated the same way.

We work in the following Hilbert space

X = I (u k ) k=1..N dom oe N dom Ÿ k=1 H 1 (Ω k ) J . (2.8)
X is induced with the following broken norm:

ÎuÎ 2 ú := ÿ k ÎuÎ 2 H 1 (Ω k ) .
(2.9)

The appropriate bilinear form is given by: a :

; X ◊ X ae R (u, v) ' ae q k s ‰ k Òu k Òv k + q k s ‰ k -•Òu k v k . (2.10)
The non conforming formulation is to look for u in X such that

Y ] [ ≠∆u + -•Òu =0 in each Ω k in the distributional sense < (u k ≠ u l ), Ψ > L 2 (Ω k flΩp) =0 'Ψ oe L 2 (Ω k fl Ω p ), <u k , Ψ > L 2 (Ω k flΩ0) = <u 0 , Ψ > L 2 (Ω k flΩ0) 'Ψ oe L 2 (Ω k fl Ω 0 ) (2.11) For k, p, denote M k,p := L 2 (Ω k fl Ω p ) and M k,0 := L 2 (Ω k fl Ω 0 )
. We define the following bilinear forms

• b k,p (u, Ψ) the set of 'internal' matching conditions, i.e for u oe X and Ψ oe M k,p .T h e s e correspond to the second equation in (2.11).

• b k,0 (u, Ψ) the set of matching conditions with Ω 0 , i.e for u oe X and Ψ oe M k,0 .T h e s e correspond to the third equation in (2.11).

For all {g k oe L 2 (Ω k fl Ω 0 )},d e fi n eV ({g k }):

V ({g k })= ; u oe X, s.t 'k, p, 'Â oe M k,p ,b k,p (u, Ψ) = 0 'k, 'Â oe M k,0 ,b k,0 (u, Ψ) = <g k , Ψ > L 2 (Ω k flΩ0) < (2.12)
By considering the isomorphism between V (0) and H 1 0 (Ω \ Ω 0 ), it is easy to see that our initial problem is equivalent to the following: find u oe V (u 0 ) such that

'v oe V (0),a (u, v)=0.
(2.13)

Discrete formulation/ well posedness

We need to pick a discretization space for X. It is natural to pick

'k, W k := span ) " k,l ,l=1...N red k * . (2.14)
The global discretization space is then given by:

X " = I (u k ) k=1..N dom oe N dom Ÿ k=1 W k J (2.15)
It is more tricky to choose a suitable discretization space for M k,p and M k,0 . We do not specify them yet, and denote with M ",k,p and M ",k,0 some generic discretization space of M k,p M k,0 , respectively. As in the continuous case, define, for

{g k oe L 2 (Ω k fl Ω 0 )}: V " ({g k })= ; u oe X " , s.t 'Ψ k,p oe M ",k,p ,b k,p (u, Ψ k,p )=0 'Ψ k,0 oe M ",k,0 ,b k,0 (u, Ψ k,0 )=<g k , Ψ k,0 > L 2 (Ω k flΩ0)
< .

(2.16) With this notation, the discretized problem becomes: find u " oe V " (u 0 ) such that:

'v " oe V " (0),a (u " ,v " )=0
In order to prove the well posedness of the resulting discrete problem, we need to prove the V " (0) ellipticity of a.

Remark 15 Unlike in the conforming case, V " (0) is not a subspace of V (0). The ellipticity is not trivial.

Let v in V " (0).W eh a v e : a(v, v)= N dom ÿ k=1 ⁄ Ω k ‰ k |Òv k | 2 + N dom ÿ k=1 ⁄ Ω k ‰ k -•Òv k v k
A rigorous proof will not be presented here, and we will stick to formal arguments. Let some domain Ω k . Away from the boundary ˆΩk , ‰ k is bounded from below. The problem occurs near the boundary as ‰ k |Òv k | 2 from the bilinear form can not compete with |Òv k | 2 of the broken norm. The hope comes from the fact that where ‰ k is small, there exist p oe [0,N dom ] such that ‰ p is big, and because of the matching conditions (v is in

V " (0)), v p is close to v k .
The ellipticity constant should thus be close to the ellipticity constant of the continuous non conforming case, if the matching is good enough. More quantitative arguments can be found in the next subsubsection.

A priori error estimations

We start with an application of Strang's 2nd lemma [START_REF] Strang | An analysis of the finite element method[END_REF], which is the standard tool for a priori estimation for non conforming methods. With an abuse of notation, we also denote by u the representant of the truth solution in X:

u := {u k ,koe [1 ...N dom ]} .
Let u " := {u ",k } k be the solution to the discretized problem. Strang's 2nd lemma states:

Îu ≠ u " Î ú AE C - A inf v " oeV " (u0) Îu ≠ v " Î ú + sup v " oeV " (0) a(u, v " ) Îv " Î ú B .
(2.17)

The best approximation error will depend on the offline phase and on the choice of the reduced basis. Our hypothesis in this whole chapter is that this error is small, see the definition of local Kolmogorov n-width section 2.3. We will thus focus on the consistency error. This error measures 'how far' we are from having a global H 1 solution.

Define the following linear form on X " :

K : I X " ae R v " ' ae q N dom k=1 s Ω k ‰ k Òu k Òv ",k + q N dom k=1 s Ω k ‰ k -•Òu k v ",k
As u is in H 1 (Ω) and thus consistent over overlapping subdomains, we have:

'v " oe X " ,K (v " )= ⁄ Ω Òu • A N dom ÿ k=1 ‰ k Òv ",k B + ⁄ Ω A N dom ÿ k=1 ‰ k v ",k B -•Òu We use q N dom k=1 ‰ k v ",k oe H 1 0 (Ω)
and that u is the solution to the initial problem, to show that:

⁄ Ω ÒuÒ A N dom ÿ k=1 ‰ k v ",k B + ⁄ Ω A N dom ÿ k=1 ‰ k v ",k B -•Òu =0.
Combining the two previous relations, we have:

'v " oe X " ,K (v " )= N dom ÿ k=1 ⁄ Ω Òu •Ò‰ k v ",k (2.18)
On domains that do not overlapp, the previous quantity is zero. Let's handle a situation with two subdmains overlapping. Let Ω k , Ω p these subdomains. Define K kp the restriction of K on this intersection, that is:

K kp : v " ae ⁄ Ω k flΩp ÒuÒ‰ p v ",p + ⁄ Ω k flΩp ÒuÒ‰ k v ",k = ⁄ Ω k flΩp ÒuÒ‰ p (v ",p ≠ v ",k )
In order to get an idea of the consistency error, we need to bound K k,p on the subspace V " (0). We first follow the lines of the RBEM method. As we will see, this path does not match all of the objectives set up for the method, but will help get a better understanding of the differences with RBEM. A natural choice for M ",k,p is to pick some reduced basis that represents well the space spanned by:

) Ò‰ k Ò" k,l ,l=1,...,N red k * € ) " k,l ,l=1,...,N red k * . (2.19) As v " oe V " (0),w eh a v e : v ",k ≠ v ",p oe M ‹ ",k,p .
We deduce the following estimate for K kp :

'v " oe V " (0),K kp (v " ) AE . . .Π ‹ M ",k,p (Ò‰ p •Òu) . . . L 2 (Ω k flΩp) . . .Π ‹ M ",k,p (v ",p ≠ v ",k ) . . . L 2 (Ω k flΩp)
As for each overlap, the set given in (2. [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF]) is of small n-width, we know that we can find a reduced space M ",k,p of small dimension that guarantees a satisfactory upper bound on K k,p , and thus on the consistency error. Nevertheless, this strategy does not entirely fulfill our objectives. Indeed, it requires the knowledge of the sets given in (2. [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF]) and thus the knowledge of the overlaps a priori. We will see in the next section an alternative, weaker but more flexible matching.

Implementation details

The offline section does not cause specific issues. The first step is to identify the generic subdomains. These can be given for instance by the geometry of the mesh (hole with a specific shape, object of complex shape etc.), or by localized singularities of some physical parameter.

We then restrict the solutions to these generic subdomains, and perform standard reduced order modeling. This involves some mesh interpolation, but nothing to dangerous. For the online section and unlike for the conforming method, we only need to compute the bilinear form a in each subdomain independently. Indeed, the quantities appearing during the online phase are given by:

'k, '(l, m), ⁄ Ω k ‰ k Ò" k,p Ò" k,m = ⁄ Ωj(k) ‰ k (• + x k )Ò "j(k),l Ò "j(k),m .
(2.20)

Two big issues remain:

• the partition of unity {‰ k (µ),k oe 1,...,N dom } carries all the geometric variability. Its construction is a challenging task. We will propose two ways of mitigating the associated online computational cost.

• the matching constraints are very involved numerically. Both the stability and accuracy of the method depend on the properties of the discretization spaces M ",k,p and M ",k,0 .

Partition of unity

We can follow two routes in order to solve the complexity implied by the geometric variability. We can either limit the geometric variability, or accept a high set up cost, at the beginning of the simulation.

High set up cost

Suppose that we are solving a time dependent PDE. There will be a set up process before the actual resolution of the problem. During this set up, we compute the set of {‰ i ,i=0,...,N dom ]} and the associated quantities, see (2.20). This will be computationally expensive, but will only be done once.

Structured problem

If we are dealing with a problem such as depicted in Figure 2.12, the pattern in the geometry will induce a pattern in the partition of unity functions {‰ i ,i =1,...,N dom }. That is, there will be a limited number of generic partition of unity functions { ‰k }, such that:

'i oe [1,...,N], ÷k(i), s.t ‰ i =‰ k(i) (•≠x i ).
This means a highly reduced online computational cost.

Start from a non overlapping decomposition

The idea of this section was taken from [START_REF] Oh | The piecewise polynomial partition of unity functions for the generalized finite element methods[END_REF]. This method starts with a non-overlapping decomposition of Ω: {Q k ,k=0...N dom }.

Remark 16

This premise restricts the number of situations that can be handled.

Different shapes can coexist in the family {Q k } k . The only requirement is that it covers Ω. Let Θ a 2 dimensional smooth 'window function', with a support included in a small neighborhood of the origin, and such that s Ω Θ=1 . We define our partition of unity functions {‰ k ,k=0...N dom }, as:

'k oe [1,...,N dom ],‰ k =Θú 1 Q k (2.21)
where the operator ú denotes the standard convolution operator. Such a family of function satisfy:

• 'x oe Ω, q N dom k=1 ‰ k (x)=1 • supp(‰ k )=Q k + supp(Θ)
• the resulting partition of unity functions are translated version of a generic function:

‰ k = ‰j(k) (•≠x k )
The first point is easily checked:

'x oe Ω, ÿ k ‰ k (x)= ÿ k ⁄ R 2 Θ(x ≠ y)1 Q k (y)dy = ⁄ Ω Θ(x ≠ y)dy =1 (2.22)
as long as 'k>0,d i s t (ˆΩ k ,ˆΩ) > characteristic size of the support of Θ.

With this choice, the partition of unity functions are independent of the non overlapping cover of Ω. It thus involves no additional online computational cost.

Matching

We will give a possible computationally reasonable choice for M ",k,p , subspace of M k,p = L 2 (Ω k fl Ω p ). As usual, we focus on one particular overlap, say Ω k fl Ω p .

One possible choice for constraints

As in the previous section, we use Θ some two dimensional 'window' function, of small support. We propose the following matching constraints:

'm oe [1 ...M], ⁄ Ω k flΩp ÷(v k )(•)Θ(•≠x m )= ⁄ Ω k flΩp ÷(v p )(•)Θ(•≠x m )
for some well chosen {x m ,moe [1,M]} in Ω k flΩ p , and for ÷ some linear operator H 1 (Ω k flΩ p ) ae L 2 (Ω k flΩ p ). For instance, when ÷ is the identity, this imposes that the average in some predefined neighborhood of x m of v k and v p match.

First question to answer is to see if it fits into the framework of section 2.4.2.1. Define Γ k,p as:

Γ k,p (x m ): ; H 1 (Ω k fl Ω p ) ae R v ' ae (v ú Θ)(x m )= s Ω k flΩp ÷(v)(•)Θ(•≠x m
) and take M ",k,p := span {Γ k,p (x m ),moe [1,...,M]}. The associated bilinear form b k,p is given by:

'v oe X " ,b k,p (v, Γ k,p (x m )) = ⁄ Ω k flΩp Θ(•≠x m )(÷(v k ) ≠ ÷(v p )).
With this particular choice of M ",k,p , what can we say about the consistency error? Recall that for two subdomains overlaping, it is given by: sup

v " oeV " (0) A 1 Îv " Î ú ⁄ Ω k flΩp ÒuÒ‰ p (v ",p ≠ v ",k ) B .
As one of our premises is that for all k, W k is of small n-width, we know that:

) w k,p := (v ",k ≠ v ",p )| Ω k flΩp ,v ",k oe W k ,v ",p oe W p *
also has a small n-width, whatever the particular overlap Ω k fl Ω p considered. Thus, if we manage to find enough independent matching conditions, this should be sufficient to guarantee the smallness of Îv ",p ≠ v ",k Î L 2 (Ω k flΩp) , and thus of the global consistency error. One can for instance think that by equating enough local averages of the solution and local averages of the vorticity, one can correctly match the solutions of a CFD problem, even if this is not done in an optimal fashion as in the RBEM version presented above. We note the difference with equation (2. [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF]). Where we were explicitly constructing a taylored reduced basis for a specific overlap, we now hope to find local (averaged) quantities that characterize the solution on any overlap. The precise quantities to consider and the size of the support of the window function Θ are of course problem dependent, and should be chosen in the offline section.

Offline, online decomposition

We show how to efficiently impose the matching condition proposed in the previous section. For this, we need a procedure to select a set of {x m } in the overlap as well as a way of efficiently computing the corresponding Γ k,p (x m )(v) defined in the previous section. For simplicity, we will use ÷ the identity mapping, but note that the same procedure could be performed for any linear operator acting on H 1 (Ω k fl Ω p ). Define Λ k,l as:

Λ k,l : I Ω k fl Ω p ae R x ' ae s Ω k flΩp " k,l (•)Θ(•≠x) Let v oe X " and {-k,l ,l=1...N red k } be the coordinates of v on W k . We know that 'x, Γ k,p (x)(v)= N red k ÿ l=1 -k,l Λ k,l (x) (2.23)
We have replaced the problem of estimating Γ k,p (x m )(v) by the estimation of Λ k,l (x m ),u s i n g linearity.

Let j(k) in J and Ωj(k) be the generic obstacle corresponding to Ω k . Select offline some points in Ωj(k) denoted {x j(k),n }. We pre compute the local quantities:

'j oe J, Λj,l,n := ⁄ Ωj "j,l Θ(•≠x j,n ).
We use the translation invariance to show that:

'x oe Ωj(k) , Λ k,l (x k +x)= ⁄ Ω k flΩp " k,l Θ(•≠(x k +x)) = ⁄ Ωj(k) "j(k),l Θ(•≠x)
and thus that:

Λ k,l (x k +x j(k),n )= Λj(k),l,n . Finally, Λ k,l is a function defined on Ω k , smooth, whose value is known at a discrete set of points, namely {x k +x j(k),n ,n }.
This ends the matching procedure. Indeed, let {x m }oeΩ k be a set of points chosen online. We use some interpolation method to approximate the Λ k,l (x m ). Then, with equation (2.23) we have an estimation of b k,p (v, Γ k,p ). The accuracy of the interpolation procedure needs to be assessed in the offline section.

Conclusion

We have proposed in this chapter an overlapping version of the RBEM method. This method has been constructed to handle situations with a high geometric variability. The main difference with the RBEM method is the way the matching constraints are imposed.

What has been presented is a preliminary work and a lot of questions still need to be answered. We have sketched a procedure to efficiently build partition of unity functions. This should be numerically investigated. Also, I have the feeling that the matching procedure could be improved. We have to keep its adaptability, but try to find an alternative resulting in a stronger matching.

We have one bonus property with the non conforming version of ORBEM. It matches the objective of the method presented in section 2.2.3. Each local basis can be modified independently online through non linear transformations. For instance, suppose that we are dealing with problems where the inflow direction varies in time. We would like the 'local' basis to rotate accordingly. This is illustrated in Figure 2.13. This can be handled simply by changing the

x 1 x 2 x 3 x 4 x 5 x 6
Figure 2.13: The ORBEM method allows for the rotation of the local basis mapping between the generic domains Ωj and the instances of these subdomains Ω k , as they will now involve a rotation. We do not need more evaluation of the bilinear form a, as it only involves local terms and is rotation invariant. Also, the matching process can be used as such. Summing up, for a prescribed rotation parameter on each local basis, the algorithm is exactly the same.

How to pick, online, a 'good' rotation parameter, such that the flow is best represented by the local basis is the topic of the next chapter 3.

Chapter 3

On the construction of a calibration procedure

The reduced basis method allows to propose accurate approximations for many parameter dependent partial differential equations, almost in real time, at least if the Kolmogorov n-width of the set of all solutions, under variation of the parameters, is rapidly decaying. The idea is that any solutions may be well approximated by the linear combination of some well chosen solutions that are computed offline once and for all (by another, more expensive, discretization) for some well chosen parameter values. In some cases however, such as problems with large convection effects, the linear representation is not sufficient and, as a consequence, the set of solutions needs to be transformed/twisted so that the combination of the proper twist and the appropriate linear combination recovers an accurate approximation. This chapter presents a simple approach towards this direction, preliminary simulations support this approach. There is an article version of this chapter available, see [START_REF] Cagniart | Model order reduction for problems with large convection effects[END_REF]. Note that we used Freefem++ [START_REF] Hecht | New development in freefem++[END_REF] for some of the numerical simulations used in this chapter. .

Introduction

Fast reliable solutions to many queries parametric Partial Differential Equations (PDE) have many applications among which real time systems, optimization problems and optimal control. Many different methods for reducing the complexity of the computations when such many queries are required have blossomed for answering this specific need. One of the approaches that have emerged is reduced order modeling (ROM). Methods in this category have been developed and are now well understood and set on firm grounds, both for steady cases or time dependent problems where time can be considered as another parameter.

The reduced basis method, which is the method that we focus on in this chapter, enters in this frame and consists in, i) defining a sequence of low dimensional spaces for the approximation of the whole set of the solutions to the parametric PDE when the parameters vary (called hereafter the solution manifold associated to our problem); ii) once such a sequence of low dimensional spaces (known as reduced basis spaces) is determined, an approximate solution is sought in such a chosen reduced space to the PDE for the values of the parameter we are interested in. The approximation is often based on a Galerkin formulation. For such reduced basis methods, both the variety of applications and the theory are now quite sound. For instance, reliable algorithms with a priori estimates and certified a posteriori errors have been developed for elliptic and parabolic problems, with or without so-called affine parameter dependence, see e.g. the two recent books on the subject [START_REF] Jan S Hesthaven | Certified reduced basis methods for parametrized partial differential equations[END_REF] and [START_REF] Quarteroni | Reduced basis methods for partial differential equations[END_REF] and, of course, the publications therein.

Reduced basis methods, classically, consider the solution manifold associated to the parametrized problem as outlined above and are appropriate if this manifold can be approximated accurately by a sequence of finite dimensional spaces. The mathematical frame for this is inherently linked to the notion of Kolmogorov width of solution manifolds, i.e. on how well the solution manifold can be approximated by a finite dimensional linear space. More precisely, let M be a manifold embedded in some normed linear space X. The Kolmogorov n-width of M is defined as:

d n (M,X)=inf En sup f oeM inf goeEn Îf ≠ gÎ X (3.1)
The first infimum being taken over all linear subspaces E n of dimension n embedded in X.

Even if, from the practical point of view, there are various ways for checking that M can be approximated by a series of reduced spaced with small dimension, the first natural mathematical question is to provide an estimation of the Kolmogorov n-width of M. Second, the question of an applied mathematician is if one can actually build an optimal, or close to optimal sequence of basis sets for these spaces?

Of course, in the vast majority of real cases, there is no analytical expression for this dimension but there are some papers giving bounds for some restricted classes of problems in the literature. For instance, in [START_REF] Melenk | On n-widths for elliptic problems[END_REF] bounds on d n are found for solution manifolds corresponding to regular elliptic problems and where the parameter dependence is on the forcing term. More general cases can be handled using the results in [START_REF] Cohen | Kolmogorov widths under holomorphic mappings[END_REF]. The hypothesis therein is on the regularity of the solution with respect to the parameter dependence, it is proven that, under analyticity assumption on the behavior of the parameters in the PDE, the small Kolmogorov n-width of the manifold of parameters D (AE cn ≠t , t>1) implies the smallness of the Kolmogorov n-width of the associated solutions manifold M D (AE cn ≠s , s<t≠ 1).

In practice, instead of the "optimal" linear subspace of dimension n in the sense described earlier, we build a "good" linear subspace. In the literature, the two most classical algorithms are the greedy method based on a certified (or at least fair enough) a posteriori estimator, and the Proper Orthogonal Decomposition (POD). We proceed assuming that the chosen algorithm has given a "good" basis "close" to the optimal one, that is, we assume that our reduced family of spaces {X n } n satisfies:

d n (M,X) ¥ sup f oeM inf goeXn Îf ≠ gÎ X (3.2)
A first paper on this subject is [START_REF] Maday | A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations[END_REF], where the authors derived error bounds on the error for the Reduced Basis Method (RBM) approximation in case of a single parameter dependent elliptic PDE. More general results have been obtained more recently for the greedy approach of the RBM [START_REF] Binev | Convergence rates for greedy algorithms in reduced basis methods[END_REF][START_REF] Devore | Greedy algorithms for reduced bases in banach spaces[END_REF]. The optimality considered in the case of POD is slightly different. The POD focuses on minimizing the average error (parameter wise), in some norm. More precisely, we have the well known relation

⁄ D Îu(µ) ≠ Π POD u(µ)Î 2 dµ = ÿ i>N POD ⁄ i (3.3)
where Π POD is the orthogonal projection onto the POD reduced space of dimension N POD and the ⁄ i are the eigenvalues of the associated correlation operator, in decreasing order. The faster the decay of the eigenvalues, the fewer modes are needed for a good (in average) reconstruction of the solution manifold.

Up to now, most of the literature on the subject, deals with problems where one can expect/check/prove/ or hope, that the solution manifold M D has a small Kolmogorov n-width. There are however cases where the plain approach does not work and some transformation of M D needs to be done. An example is for instance the use of the Piola transform in the processing of the velocity field when the PDE is the Stokes or Navier Stokes problem and the parameter includes the geometry of the computational problem (see e.g. [START_REF] Emil Løvgren | A reduced basis element method for the steady stokes problem[END_REF]). The choice of the Piola transform indeed provides better reduction than a simple change of variables.

The most classical and simple example illustrating limitations of reduced models due to large Kolmogorov n-width is the pure transport equation, with constant speed c>0. Formally, we consider the following parametric PDE over the domain

Ω=(a, b) µ IR Y ] [ u t (x, t)+cu x (x, t)= 0 , in Ω◊]0,T[ u(x, 0) = u 0 (x), in Ω c oeD:= [c min ,c max ]. (3.4)
The analytic solution is given by u(x, t; c)=u 0 (x ≠ ct).

(3.5)

We can consider two solution manifolds. Either the space time solution manifold

M x,t D = {u(•, •; c),coeD}, (3.6)
or a more natural solution manifold in our context is the snapshot solution manifold

M x D = {u(•,t; c),toe [0,T],coeD}. (3.7)
We will first give an illustrative idea of d n (M x c ), i.e for a fixed convection parameter. Thus, the only "parameter" left is time and

d n (M x c
) is, of course, smaller than d n (M x D ). Suppose now that our initial solution is compactly supported and let ¸denote the Lebesgue measure of its support. Let us assume in addition that its support is included in ]a, a + ¸[. Then, there are at least (b ≠ a)/¸snapshots {u(•,t k ; c)} k obtained for t k = k¸/c that are two by two orthogonal proving that a lower bound of the Kolmogorov n-width is (b ≠ a)/¸. For a given accuracy, reducing ¸, we can make the size of the reduced basis needed arbitrarily large. Another example of badly behaved manifold space can also be found in [START_REF] Taddei | Reduced basis techniques for nonlinear conservation laws[END_REF], or in chapter 1 of this manuscript.

The objective here is to give a proper framework and to introduce notations generalizing the following observation: apart from translation, the solution manifold for the whole time simulation can be represented by a unique basis. However, let us stress that this translation is not a linear process hence the Kolmogorov process cannot capture it. An additional ingredient to existing reduced order methods has thus to be added so as to capture this very simple problem structure.

Most of the literature in the reduced order modeling community on convection dominated problem focus on the stabilization issue, and not on the reduction of the Kolmogorov n-wdith. For instance, the authors in [START_REF] Dahmen | Double greedy algorithms: Reduced basis methods for transport dominated problemsâĹŮ[END_REF] have proven that using, as usual, the residual of the PDE as a surrogate for the true error, is not adapted if convection is dominating as the relative a posteriori estimator is not fair enough. Their method involves other norms than the natural ones, and increases the stability at each iteration by enriching the trial space. Once again, their method improves the stability of the construction of a reduced basis, but does not handle the fact that the solution manifold can have a large Kolmogorov n-width.

In the same direction let us quote the papers related to the so called GNAT approach [START_REF] Carlberg | Efficient non-linear model reduction via a least-squares petrov-galerkin projection and compressive tensor approximations[END_REF][START_REF] Carlberg | The gnat method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows[END_REF] where the authors propose also an alternative reduction approach for these type of problems.

In [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF], the authors address the stability issue in another direction. They give ideas and show numerical examples illustrating the fact that using L 1 -minimisation, instead of the -more classical -L 2 -minimisation (corresponding to a Galerkin scheme, which is natural in the reduced modeling context), does a better job for handling shocks (as appears in non linear convection problems) and provides more stable results. However this approach does not cure the problem that we have indicated above related to the large dimension of the solution manifold.

Let us also mention at this level, as an intermediate approach, the paper [START_REF] Carlberg | Adaptive h-refinement for reduced-order models[END_REF]. As standard reduced order modeling fails, the author chooses, in a preprocessing step, to "chop off" the reduced basis functions resulting in a kind of adaptive coarse enriched finite element method.

Very few papers tackle the n-width issue directly. In [START_REF] Taddei | Reduced basis techniques for nonlinear conservation laws[END_REF], the authors propose a method that is the first attempt to use shock fitting related ideas in the context of reduced order modeling. The idea is to decompose the spatial domain into zones separated by shocks. In each zone, classical reduced order modeling is performed, and the shocks dynamic is handled using another equation. For them, it is given by Rankine-Hugoniot conditions. This method, just as any other shock fitting method, is somehow limited to one dimensional problems.

In [START_REF] Gerbeau | Approximated lax pairs for the reduced order integration of nonlinear evolution equations[END_REF], the authors develop a method where the POD basis is reconstructed at each time step to follow the propagation of the phenomenon. More precisely, by referring to Lax-pairs, they choose as reduced basis the modes of the Schrödinger operator where the potential is taken as the solution at the previous time step. Even if no theoretical proof of this ansatz is presented, the numerical results presented in that paper illustrate the interest of the approach for selecting the reduced space and adding stability to the process without curing however the large increase of the dimension of the reduced space when the accuracy requirement increases.

The method presented in [START_REF] Iollo | Advection modes by optimal mass transfer[END_REF] is similar to our work in many aspects, in particular in looking for a change of variable for better representing the solution manifold. Their approach relies on the existence of a main mode u 0 that, by convection, represents most of the solution. The proper change of variable (written as a sum of advection modes) is fitted by evaluating Wasserstein distances between the snapshots in M x D , with modes being obtained by solving Monge-Kantorovich optimal transport problems w.r.t. the reference mode u 0 . Various numerical results illustrate the approach, however only in cases where the solution exhibits indeed such a main mode u 0 which is doubtful in nonlinear processes. We will come back on their ideas in the following sections.

In the first section, we introduce the notion of "calibration" of a solution manifold based on our knowledge of the process (differing here somehow from the optimal transport problem approach in [START_REF] Iollo | Advection modes by optimal mass transfer[END_REF]). We detail the steps of a procedure using the calibration manifold, the socalled freezing method [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF][START_REF] Ohlberger | Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing[END_REF]. As will quickly appear, the mathematical ingredients needed for a rigorous analysis of the freezing method are very involved. In the end of the section, we motivate the need for a lighter, more natural framework. The rest of this chapter is devoted to the development of this alternative method. We extensively describe its specificities on the one dimensional unsteady viscous Burger equation. We propose a self sufficient reduced scheme, with efficient offline/online decomposition. We then present some numerical simulations confirming the overall feasibility of the method. The last sections of this chapter describe possible extensions to the method.

Formal presentation

Let us consider a general time dependent parametric PDE in some physical space

Ω µ IR d , d =1, 2, 3 Y ] [ u t + L µ (u)= 0 in [0,T] ◊ Ω u(•,t = 0; µ)=u 0 (•,µ) in Ω B(u; µ)= 0 on ˆΩ (3.8)
where µ varies in some compact parameter space D. Our approach considers the corresponding snapshot solution manifold M x D as defined in (3.7), that we assume embedded in some Hilbert space X.

Let us assume that the solution manifold has a simple structure, not reflected though by the Kolmogorov n-width but hidden by a transformation of the solution manifold. As stated in the introduction, we can think of the transport equation as being the simplest example for which this is occurring. The objective is, through a "calibration" step, to recover the simple structure of the solution manifold.

Specifications

This first section will be formal. We will try to dress up a list of specifications that we would like our calibration process to satisfy. We are looking for a family of applications The sense in which this should be understood can vary depending on the framework considered. Namely, it depends on the way the family F acts on the solution manifold. It will be explicited for one example in the next section.

F := {" : X ae

the following application:

'" oeF,":

; X ae X u ' ae "(u)
is smooth. This property will be needed to derive the calibrated equation. It also naturally appears when studying the stability and accuracy of numerical schemes.

4. the application " ¶L µ should have an explicit form such as:

'µ oeD, '" oeF, ÷M µ," , L µ ("(v)) = " (M µ," (v))
where M µ," is some differential operator.

5. from the knowledge of the calibrated solution "(t; µ) ≠1 (u(•, •; µ)), and of the current calibration parameter "(t; µ), we need to be able to go back to the true, 'uncalibrated' solution. This means that the applications " oeF should be invertible.

Remark 17

The smoothness with respect to µ is never discussed here, as it is not required. We refer to the chapter 5 of this thesis, where we will try to adapt optimal control methods to this particular framekwork.

Following [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF], we call such a process a decomposition of a solution u(•,t; µ) into two components:

• phase: the calibration function "(t; µ) oeF • shape: the solution "calibrated" v := "(t; µ) ≠1 (u(•,t; µ)).

We will now present one framework that satisfies the previous specifications.

One possible framework

This section describes the freezing method, introduced in [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF]. The mathematical analysis is based on Lie group theory. We refer to [START_REF] Kirillov | An introduction to Lie groups and Lie algebras[END_REF] and the references therein for a rigorous presentation of Lie groups and Lie group actions, and restrict ourselves to formal arguments. From now on, let G be a Lie group and let us denote by a some group action on X: a : G ◊ X ae X (", u) ' ae a(", u).

(3.9)

If we compare to the previous section, that means that we parametrize our family of applications F by some group G. The Lie group setting makes sense for two reasons. First, the applications should be invertible. Second, the parameters should lie on a smooth manifold. The problem is now to find appropriate

• Lie group: G • group action: a • application ": [0,T] ◊D ae G (t, µ) ' ae "(t; µ) (3.10)
satisfying the list of specifications of the previous section. The most important one, and the sole reason of this whole chapter, is that the calibrated solution manifold M = {a("(t; µ) ≠1 ,u(•,t; µ))}µX (3.11) has a Kolmogorov n-width with good decay properties.

We restrict ourselves to group actions acting linearly on X. That is, we choose '" oe G, a(", •):

; X ae X v ' ae a(", v)
as a linear application. This solves right away the 3rd point in the specifications section. Other choices are in theory possible, but the analysis of the calibrated equations would require more involved ingredients than the one that will be discussed here. The 5th point of the specifications is also trivially satisfied in this Lie group framework, as the group action is invertible. Throughout this section, we explicit the different notions on one simple example. We consider the pure transport equation with constant speed:

; u t + cu x =0 on R u(t = 0) = u 0 (3.12)
where the parameter space is D =[ c min ,c max ]. Is natural to pick G = R d and the following group action:

a : ; R d ◊ X ae X (", u) ' ae u(• + ") (3.13)
It is clear that this group action, for fixed ", acts linearly on X. The only sensible choice for " is:

"(c, t):=ct. (3.14)
With these choices, we can check the remaining specifications:

1. the calibrated solution manifold has a (very) satisfactory n-width:

d n ( M)= ; 1 if n =0 0 else . (3.15)
That is, one shape function is enough to capture the whole simulation.

2. " is a smooth function of t 3. as L : u ae cu x , it is trivial to check '" oe G, 'u oe X, a(", L(u)) = L (a(", u)) .

So we have M " = L, for all ".

The question is now to find a calibrated equation, whose solution would be in the calibrated manifold, equivalent to the initial problem.

The freezing method

We go back to the general case. We start by a formal derivation of an equivalent formulation on the shape component. Let

{g(t; µ) oe G, (t, µ) oe ([0,T] ◊D)}
be some fixed, well chosen, phase component function. The corresponding calibrated solution manifold is:

M := ) v(•,t; µ):=a ! g(t; µ) ≠1 ,u(•,t; µ) " , (t; µ) oe [0,T] ◊D * (3.16)
The PDE satisfied by v is given by: (a(g(t; µ),v(•,t; µ))) t + L µ (a(g(t; µ),v(•,t; µ))) = 0.

(3.17)

We need some smoothness hypothesis on the group action. This corresponds to the second specification. We restrict ourselves to the group actions such that:

'v oe X, a(•,v): ; G ae X g ' ae a(g, v) (3.18)
is continuously differentiable. Denote a 1 its derivative.

'v oe X, " oe G, a 1 (", v):

;

T " G ae X ⁄ ' ae a 1 (", v)⁄ (3.19)
a 1 is supposed to be a continuous linear operator and T " G is the tangent space of G at ".

Remark 18

For details on Lie algebras, see [START_REF] Kirillov | An introduction to Lie groups and Lie algebras[END_REF].

In [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF], they assume that L is equivariant under the group action, that is:

'" oe G, a (", L µ (u)) = L µ a (", u) (3.20)
We rather use a weaker assumption. We suppose that there exists (a possibly different) differential operator M µ," such that

'" oe G, L µ (a (", u)) = a (", M µ," (u))
We want to insist on the fact that equivariance is not a core requirement for the method. This is discussed in section 3.8. The equation satisfied by the calibrated solution, equation (3.17) is equivalent to:

a (g(t; µ),v t (•,t; µ)) + a 1 (g(t; µ),v(•,t; µ))g t (t; µ)+a (g(t; µ), M µ," (v(•,t; µ))) = 0
Using the fact that the group action is invertible, we get:

v t (•,t; µ)+a ! g(t; µ) ≠1 , [(a 1 (g(t; µ),v(•,t; µ))g t (t; µ))] " + M µ," (v(•,t; µ)) = 0 (3.21)
To better grasp the new ingredients, we go back to our model translation problem. The tangent space can be identified with R d . What does the smoothness of the group action, equation (3.18), imply in this simple case ? Let ⁄ oe R d .

'h oe R,a (" + h⁄, v) ≠ a(", v)=v(• +(" + h⁄)) ≠ v(• + ") (3.22)
If we suppose that X is embedded in C 1 (R),t h e na is differentiable in the sense of (3.18) and the corresponding derivative a 1 is given by:

a 1 (", v)⁄ = a(", Òv • ⁄). (3.23)

Phase component

The previous section assumed no restriction on the choice of the phase function, apart from its smoothness. We now have to show how to add a constraint on the phase function that actually reduces the n-width. They propose in [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF] the following generic coupled problem:

; v t (•,t; µ)+a ! g(t; µ) ≠1 , (a 1 (g(t; µ),v(•,t; µ))g t (t; µ)) " + M µ v(•,t; µ)=0 Φ(v, g)=0 in A ú (3.24)
where A is the tangent space of G at g and A ú the dual space of A.

We give one example of constraint that enters this frameowkr. Let u ú be some reference solution. Suppose that the calibration is chosen so that solutions in the calibrated solution manifold are as close to u ú as possible, in X norm. That is, we choose g oe G as:

g : v ae arginf goeG Îa(g, u ú ) ≠ vÎ 2 X .
First of all, it is easy to see why this constraint goes towards a quicker decay of the n-width. First order optimality constraint gives:

'"g oeA,<a 1 (g, u ú )"g , a(g, u ú ) ≠ v> X =0,
where < •, • > X denotes the standard scalar product in X. We are thus interested in the zeros of the following application:

'v oe X, 'g oe G, Φ v,g : ; AaeR "g ' ae <a 1 (g, u ú )"g, a(g, u ú ) ≠ v> X .
This enters the framework of equation (3.24). The existence of solutions to the coupled system depends on the properties of the constraint equations. Given v, a necessary condition is for the constraint equation to be well posed, and to result in a smooth calibration parameter g. We refer to [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF] for more details.

Conclusions on the freezing method

This is it, we have a method to overcome the difficulties associated with the development of ROM for (a restricted class of) solution manifolds with large n-width. Corresponding numerical example for can be found in [START_REF] Ohlberger | Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing[END_REF]. The purpose of this section is to motivate the need for an alternative.

One disadvantage of the freezing method has appeared from the beginning: it is its complexity. The development of a theoretically sound framework combining the study of PDE's with Lie group and Lie algebra theory is not only (far) from the scope of this thesis, it is also (to my knowledge) not available in the literature. To my understanding, the strongest general result in [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF] concerns smooth solutions (say continuously differentiable), and is only local in time.

We now discuss another property of the freezing method. Let the inviscid Burgers equation:

u t + 3 u 2 2 4 x =0.
Taking translation as the calibration process, it is easy to see why the calibrated equation is given by:

v t +(v ≠ "(t))v x =0
for some function t ' ae "(t), that can for instance be the shock's speed. To numerially solve this equation on the shape function, one needs to construct a scheme taylored for this task. Using a scheme designed for convection dominated problems to solve for v is not a viable strategy. The fact that the calibration procedure has made us lost track of the physical intuition can make the construction of such a scheme a complex task. What we do in this chapter is less ambitious. We construct a method that whilst still using the calibrated solution manifold, manages to keep the properties of the original, physically meaningful equation.

Road Map

We are ready to set up the notations. They will be used until the end of the chapter. F denotes a family of applications:

F = ) F : Ω ae Ω * . (3.25)
Note right away that we have restricted the choice of F to family of smooth mappings. The elements of F will be parametrized by t and µ, that is, we consider:

[0,T] ◊DaeF (t, µ) ' ae F t;µ
Finally, we take the action group resulting in the following calibrated solution manifold:

M x F ,D := ) u(F ≠1 t;µ (•),t; µ),µoeD,toe [0,T] * (3.26)
As for the freezing method, the set F is based on a priori expertise on the behavior of the solution.

The offline section is also similar as we pick elements in F to make the n-width of M x F ,D as small as possible. The novelty appears on the way we use the calibrated solution manifold in the online stage.

Algorithm

For simplicity, let us assume that we are using an explicit Euler scheme for the time discretization. Extensions to implicit, higher order time discretization, or more involved conservative numerical scheme, is straightforward1 . Our semi-discretized PDE then becomes Y ]

[

u n+1 ≠u n dt + L(u n ; µ)=0 in Ω u(•,t = 0; µ)= u 0 (•,µ) in Ω B(u n ; µ)= 0 on ˆΩ (3.27)
Here, as is classical, dt denotes the time step, and u n an approximation for the solution to (3.8) at time ndt.

Assume that we have a basis {" i } such that span{" i } approaches the calibrated solution manifold M x F ,D defined in (3.26) to a given accuracy. Since M x F ,D is assumed to be of small Kolmogorov n-width, we expect that we can find such a basis of moderate size. At each time 3.3. Algorithm step, we look for coordinates (-n+1 i ) i on the reduced basis and an application F n+1 oeF such that u(•,t n+1 ; µ) is well approximated by:

u n+1 := M ÿ i=1 -n+1 i " i ¶ F n+1 .
(3.28)

In order to expect the search for F n+1 be computationally tractable, let us assume that our family F can be parametrized by a few parameters: that is

'F t;µ oeF, ÷(" j ) j , such that F t;µ = F [" 1 (t; µ),...," m (t; µ)] .
(3.29)

In the discrete setting, the search for F n+1 then reduces to the search for (" n+1 j

) j , and we set

F n+1 = F # " n+1 1 ,...," n+1 m $ .
We are thus simultaneously looking for a proper appropriate reduced space (defined as the span of the (" i ¶ F n+1 ) i ) and for coordinates on this reduced space. We have chosen to derive our solution from some minimization problem of the form:

(" n+1 j ,-n+1 i )=argmin ("j ,-i) . . . . . ÿ i -i " i ¶ F ([" j ] j ) ≠ u n + dtL(u n ; µ) . . . . . (3.30)
for some appropriate norm ΕΠon X.

Remark 19

It is interesting to note that our approach, in this context, may be presented as a shock fitting method, and thus one may fear that it will suffer from the classical drawback of this class of approach, especially the difficulty to generalize to multidimensional framework. One reassuring element is that the position of the fitting F n is not defined through the Rankine-Hugoniot conditions but through the minimization process (3.30), and the evolution in time can be chosen to follow any appropriate conservative numerical scheme.

Several choices are possible for the sense in which we will minimize this quantity. One example will be given in the next section. We propose the following generic algorithm.

Initialize -i and " j (-n+1,0 i ," n+1,0 j )=(-ini i ," ini j ) (3.31) -ini i
and " ini j will depend on the previous timesteps, namely on (k i ) i and (" k j ) j for k AE n. Then, assuming that (-n+1,q i ," n+1,q j ) are known for some internal iteration q Ø 0,w eproceed Fit thei given [" n+1,q j ] j Find (-n+1,q+1 i ) i that minimizes the following quantity (in some sense):

ÿ i -n+1,q+1 i " i ¶ F 1 [" n+1,q j ] j 2 ≠ u n + dtL(u n ; µ) (3.32)
Fit the " j given (-n+1,q+1 i ) i

Find (" n+1,q+1 j ) j that minimizes the following quantity (in some sense):

ÿ i -n+1,q+1 i " i ¶ F 1 [" n+1,q+1 j ] j 2 ≠ u n + dtL(u n ; µ) (3.33)
until convergence (for which, say q = q ú ). Then, we set

(-n+1 i ," n+1 j )=(-n+1,q ú +1 i ," n+1,q ú +1 j
).

(3.34)

Illustration on the viscous Burger's equation in one dimension

The viscous Burger's equation has already received some attention in the reduced modeling context. We mention [START_REF] Veroy | Reduced-basis approximation of the viscous burgers equation: rigorous a posteriori error bounds[END_REF] for the stationary case and when the solution manifolds can be well represented by a small finite dimensional linear space, without any calibration. We consider Ω=(≠1, 1), and solve for the time dependent viscous Burger equation with no forcing term and periodic boundary conditions (we will see later why these are important in our analysis):

Y ]

[

u t + ‹uu x ≠ 'u xx =0in [0,T] ◊ Ω u| t=0 = u 0 u periodic (3.35)
The parameters of this problem are the triplets: µ =( u 0 ,‹,'). We want to choose a parameter domain D in order that the problem is

• convection dominated so that the solution manifold has a large Kolmogorov n-width

• not too stiff so as not to be bothered by stabilization issues as mentioned in the introduction, hence, we shall only consider the cases ' Ø ' 0 > 0 (see the recent paper [START_REF] Maday | An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems[END_REF] that tackles this problem).

We have identified the following parameter range:

D = Y ] [ ⁄ oe [0.5, 1.3], ‹ oe [4., 6.], ' oe [0.04, 0.2].
(3.36)

Variational formulation and truth approximation

For the truth approximation to the solution of problem (3.35), let us consider a semi implicit scheme (so as not to be bothered by a two stringent stability constraint) with time step This semi-discretized problem is trivially well posed. In order to finalize the discretization, let us introduce an appropriate finite element discretization, the truth approximation space, X N . We pick it fine enough so that, with the chosen time step dt truth , it is able to represent well our solution manifold. From now on, we will consider that the exact solution u(•,t; µ) and the "truth" solution u N (•,t,µ) cannot be distinguished.

dt truth . Let X = H 1 per (Ω) (3.

Model order reduction -offline stage

As mentioned earlier, the first question we need to answer is: does our solution manifold M x D (in practice represented by M x,truth

D

) have a large Kolmogorov n-width? And if so, can we find better behaved "calibrated" manifold solution? Figure 3.1 shows some snapshots ) u(•,t k ; µ),koe 1 ...K * taken in M x D for some parameters. From basic expertise on the Burger's equation, we choose the following mapping family: F = {F t;µ },w h e r eF t;µ are defined as translation operators:

F t;µ :Ωae Ω x ' ae x ≠ "(t; µ) (3.40)
with "(t; µ) oe R. With this choice, our family of mappings is a one parameter family, i.e:

F = {F ("),"oe R} . (3.41)
Unlike in the pure translation problem of the introduction (3.4), our parameter " is not constant (it is a function of µ and time) and has no analytical expression. One possible calibration (and the most natural one) is presented in Figure 3.2, where we pick " manually so that all steepest points coincide. Our calibrated solution manifold is then

M x F ,D = {u(•≠"(t; µ),t; µ),tin [0,T],µ oeD}, (3.42)
that is represented in Figure 3.2 where we understand that the Kolmogorov n-width of M x F ,D is smaller than the original one represented in Figure 3 This is confirmed in Figure 3.3 which presents the decay of the POD eigenvalues in logarithmic scale for M x D and M x F ,D . As we could have expected, to achieve a fixed accuracy, the number of POD modes needed to represent the calibrated manifold is much smaller than the number of modes needed for the original solution set. To confirm this, we present in Figure 3.4 the 3rd and 6th POD modes of the calibrated and non calibrated simulations. As we can see, in the calibrated case, with just 3 modes, our L 2 -projection focuses on reproducing the shock, whereas in the non calibrated case, the modes desperately try to represent shocks centered anywhere in Ω. We mention again the fact that even in the calibrated case, our algorithm could be improved using L 1 -minimization. We present in Figure 3.5 the projection of one of the snapshot on the first three POD modes. With 10 POD modes in the uncalibrated case, the projection shown in Figure 3.5 exhibit the oscillatory behavior as described in [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF].

At this stage, we suppose that we have found a "calibrated" solution manifold, with nice Kolmogorov n-width decay. That is, we have calibrated an original dataset, and obtained a reduced orthonormal basis: span {" i ,i=1...M}µX (3.43) that approximates well the calibrated solution manifold M x F ,D . We now need to explicit the algorithm presented in the section 3.3. The biggest question is • left: 3 POD modes in the calibrated case

• center: 3 POD modes in the non calibrated case

• right: 10 POD modes in the non calibrated case how do we pick the F oeF at each time step?

Model order reduction -online stage

As was introduced in the previous section (see (3.27)), for the time semi-discretization of the RBM approach, we use a forward Euler discretization with a time step dt that may be different from dt truth , at each time step we are looking for the solution to the following elliptic problem2 :

u n+1 = u n ≠ dt‹u n u n x + dt'u n xx (3.44)
with periodic boundary conditions over (≠1, 1), which leads to the following variational formulation that will be used to provide the Galerkin formulation of the RBM: knowing u n , compute u n+1 oe X such that

'v oe X, Èu n+1 (µ),vÍ = Èu n (µ),vÍ≠dt‹c(u n (µ),u n (µ),v) ≠ dt'a(u n (µ),v). (3.45)
One could fear that a problem with this discretization is the stringent CFL condition on the time-step. Our reduced basis formulation will allow for very fast computation, which will mitigate this issue on which we shall dwell upon later. As said already, we could also consider an implicit Euler scheme. We refer to [START_REF] Veroy | Reduced-basis approximation of the viscous burgers equation: rigorous a posteriori error bounds[END_REF] (stationary) and [START_REF] Nguyen | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous burgers' equation[END_REF] (non stationary), for the development of reduced order model in that case.

The full RBM discretization starts from the knowledge of the (supposedly accurate) approximation of u n as an expansion

u n := M ÿ i=1 -n i " i ¶ F n , ( 3.46)
where the {" i } i are the reduced basis elements of the good approximation of the calibrated solution manifold that have been introduced in (3.43) as a result of the offline process. F n is here F (" n ) where " n is the current translation value. In order to deduce the next approximation,

u n+1 := M ÿ i=1 -n+1 i " i ¶ F n+1 , where F n+1 = F (" n+1 ) (3.47)
as described in the previous section, we iterate between the search for the reduced coordinates (-n+1 i ) i and for the mapping F n+1 i.e. for the translation parameter " n+1 , we initialize these entities as follows:

-

n+1,0 i = -n i " n+1,0 = " n + ! " n ≠ " n≠1 " . (3.48)
In the first part of the iterative step indexed by q, assuming we know ((-n+1,q i ) i ," n+1,q ) we fit thei for a fixed translation parameter ", i.e.we are looking for (-n+1,q+1 i

) i that satisfy {-n+1,q+1 i } = argmin (-i)ioeR N . . . . . ÿ i -i " i ¶ F (" n+1,q ) ≠ u n ≠ dt‹u n u n x + dt'u n xx . . . . . 2 2 (3.49)
The nice feature with the chosen norm is that we pick our reduced coordinates such that our residual is orthogonal to the translated reduced space, the space spanned by the {" i ¶F (" n+1,q )} i . Using u n 's expansion on its reduced basis, the coefficients {-n+1,q+1 i } i are given by the first-order optimality condition:

-n+1,q+1 i = q j -n j È" j ¶ F (" n )," i ¶ F (" n+1,q )Í ≠ dt‹ q j q p -n j -n p È" j ¶ F (" n )(" p ¶ F (" n )) x ," i ¶ F (" n+1,q )Í ≠ dt' q j -n j È(" j ¶ F (" n )) x , ! " i ¶ F (" n+1,q ) " x Í.
(3.50)

In order to evaluate this expression, we need to compute the following integrals:

Y _ _ ] _ _ [ 'i, j, s Ω " j ¶ F (" n )(x)" i ¶ F (" n+1,q )(x) 'i, j, p, s Ω " j ¶ F (" n )(x)(" p ¶ F (" n )) x (x)" i ¶ F (" n+1,q )(x) 'i, j, s Ω " j ¶ F (" n ) x (x)" i ¶ F (" n+1,q ) x (x) (3.51)
We will see in the next subsection how to achieve efficient offline/online decomposition for these quantities.

Once this is done, we fit the ". Let us define first the residual function r("):

r(")= . . . . . ÿ i -n+1,q+1 i " i ¶ F " ≠ u n ≠ dt‹u n u n x + dt'u n xx . . . . . 2 2 , ( 3.52)
then we choose " n+1,q+1 as the "best", i.e residual minimizing, translation parameter. It is given by: " n+1,q+1 = argmin . .

q i -n+1,q+1 i " i ¶ F (") . . . 2 2 + Îu n ≠ dt‹u n u n x + dt'u n xx Î 2 2 ≠2È q i -n+1,q+1 i " i ¶ F ("),u n ≠ dt‹u n u n x + dt'u n xx Í. (3.54)
The second term is independent of ". The first one, using periodicity, happens also to be independent of ". We can thus replace the minimization of r by the minimisation of the following quantity r: r(")=≠È

ÿ i -n+1,q+1 i " i ¶ F ("),u n ≠ dt‹u n u n x + dt'u n xx Í. (3.55)
Here again, we we need to evaluate the quantities

Y _ _ ] _ _ [ 'i, j, s Ω " j ¶ F (" n )(x)" i ¶ F (")(x) 'i, j, p, s Ω " j ¶ F (" n )(x)(" p ¶ F (" n )) x (x)" i ¶ F (")(x) 'i, j, s Ω " j ¶ F (" n ) x (x)" i ¶ F (") x (x) (3.56)
for various values of " in order to derive the value of " that minimizes r (or r).

Offline/Online decomposition of the expressions depending on γ.

In both the search for " (see (3.55)) and (i ) i (see (3.51)), we need to compute scalar products of the form:

È i ¶ F (" n ), j ¶ F (")Í (3.57)
where  can be one of the POD basis or one of its x-derivatives. " n and " can take any value in Ω. Our key ingredient here is that, due to translation invariance (because we are in a periodic settings), we can replace the previous terms by

È i ¶ F (" n ≠ "), j Í = È i ¶ F (∆"), j Í. (3.58)
We have plotted in Figure 3.6 these quantities (after rescaling) as a function of ∆" for some pairs of chosen Â's and we notice that, as can be expected because we are essentially using a primitive function of the integrant, these are regular functions of ∆". For a sufficiently small time step, we expect ∆" to be of order dt ú c where c is some local characteristic velocity. We have chosen the following method:

• precompute the scalar products for a predefined set of values of ∆"

• using some regularity hypothesis, use spline interpolation to get approximated values for all " in [≠dt ú c max ,dtú c max ],w h e r ec max is the maximum expected shock speed during the simulation.

Remark 20

For the optimization of r we have also tested to linearize our problem around " n which leads to a doable method but does not work better that the above.

Remark 21

A common comment about this method is about mesh interpolation, and the related numerical errors. In the offline part, we have indeed to interpolate between meshes. But, as the computational time is not much an issue, this can be done as precisely as required. During the online section, the only thing that is required is the interpolation between the discrete quantities computed in (3.58). This error can be quantified offline. See figure 3.6 for an idea of the quantities that we are interpolating. This point will be discussed in section 3.8 of this chapter.

Numerical results

About the CFL Condition

We represent in Figure 3.7 the value of the CFL condition of our reduced scheme using the space calibrated M x F ,D as a function of the dimension M of the discrete space expressed in Equation (3.46). Of course, the bigger the reduced basis, the smaller the time step required for stability. We remark that there is a plateau for large values of M that is above the CFL-condition for the truth solver. More importantly, for M =5 , we can use a discrete time step 3, 000 times bigger than the one of the fine (finite element) scheme (that was dt truth AE 10 ≠6 ).

Convergence Illustration

On the next Figure 3.8, we have plotted the L 2 -error of the solution of (3.30) in case of problem (3.35) as a function of time for different values of the reduced basis for dt =2 .5 10 ≠4 .T h e different colors represent various values of M used in (3.46) (Note that on the same figure, the plots close the x axis represent the projection errors (best approximation) of the solution onto the set M x F ,D with the exact value of the translation "). We see that our numerical scheme is convergent, as a function of M . The final accuracy is somehow difficult to grasp since it is a function of ∆t and the number of degrees of freedom used in the spatial direction (here M ) as for any discretization of an evolution problem. It is also a function of the way the value of " is found as each time step as a solution of the full minimization problem (3.30). These results are not entirely satisfactory, as there is an order of magnitude difference between error of the reduced scheme and the best approximation error. We will propose one explanation in the next section.

Numerical results

Interpretation

In order to explain the order of magnitude difference between the errors obtained with the reduced scheme and the best approximation errors, we compare the results for various parameters. We start by constructing a basis that reproduces well solutions for parameters in D, chosen in (3.36). The first step is to discretize D. We randomly pick the following: We compute the truth approximation solutions of each of these triplets. Because of the computational cost, we have chosen to use the two level POD presented in chapter 6. That is, we perform a POD on each of the 48 time simulations and then perform a weighted POD on these POD basis. The error incurred by this divide and conquer strategy is controlled. We present in Figure 3.9, the mean (in time) relative L 2 best projection error, using a basis of size 10, at each sampled parameter. As we can see, the mean projection error is slightly worse Figure 3.9: Relative L 2 best projection errors, on the training set for small viscosities. One argument is that lower the viscosity, the stiffer the propagating front. Since our calibration procedure is not exact, the calibrated solution manifold is more difficult to capture.

⁄ oe{ 1.3,
We now want to compare the previous plot with the errors on the reduced solutions. Their relative L 2 error are displayed in Figure 3.10. Similar to what we have seen on the convergence plot, there is an order of magnitude of difference between the best projection error and the output of the reduced scheme. More importantly, the scatter plot is much more spread out. The low viscosities behave much worse, compared to the other simulations. My interpretation is that the order of magnitude difference is due to the raw Galerkin scheme used in the online section. We can not reproduce the solutions of the truth scheme, that are obtained using an upwinding scheme. This could be solved either using rectification ideas [START_REF] Maday | An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems[END_REF], or using the ideas developed in chapter 5.

Extension to non periodic setting

The periodicity has held an important role in the analysis of the calibration method so far. We try in this short section to extend the latter to a non periodic setting. The model equation that we work with is:

Y ]

[

u n+1 ≠u n dt + L(u n ; µ)=0 in Ω u(•,t = 0; µ)= u 0 (•,µ) in Ω B(u n ; µ)= g n on ˆΩ.
(3.60)

We start by presenting a typical problem entering this framework. Suppose that there is a complex phenomenon/shock/front moving inside Ω. We want to use the ideas of the previous section, that is, we want to use the fact that only the relative positions of the phenomenon/shock/front between two successive time steps is relevant. The absolute position in Ω has not a big influence. How can we translate this into a reduced basis framework ?

The method

We suppose that there exists an 'interior' domain Ω int and a calibrating function ", " :

; [0,T] ◊C ae R (t; µ) ' ae "(t; µ),
such that the calibrated solution manifold M int defined as

M int := {u(•≠"(t; µ),t; µ)| Ωint ,toe [0 ...

T],µoeC}

has a small n-width. To put it in other words, because we have removed periodicity, we need to truncate around the phenomenon/shock/front.

We require one more assumption in order to make this method relevant. Let {" i } i be some well chosen reduced basis of M int , and Π Ψ the orthogonal projection onto any orthogonal basis Ψ embedded in X. We suppose that

M 0 := ) u(•≠"(t; µ),t; µ) ≠ Π "i ¶F ("(t;µ)) u(•≠"(t; µ),t; µ),toe [0 ...T],µoeC * (3.61)
has also a small n-width.

Remark 22

This resembles the notion of 'local Kolmogorov n-width' introduced in chapter 2.

We can rephrase this using the notations of chapter 2. We decompose Ω into two overlapping subdomains : one outside domain, that is supposed to handle the boundary conditions Ω 0 (") and one inside domain, supposed to handle the complex structure moving : Ω int (").T h ed i ff e r e n c e with chapter 2 is that these two subdomains are function of the translation parameter ".O n e possible setting is presented in Figure 3.11. We need a few assumptions on Ω 0 and Ω int :

'", Ω=Ω 0 (") fi Ω int (") '", ˆΩ µ Ω 0 (") ÷-, '", dist(Ω int ("),ˆΩ) >- ÷-, '", mes(Ω int (") fl Ω 0 (")) >-
Using assumption (3.61), we will now denote by {h k } a good reduced basis of M 0 . In the course

Ω Ω 0 Ω int Figure 3
.11: This method requires an overlapping decomposition of Ω of this section, it will be thought of as Fourier/polynomial, coarse Finite Element, and RB type. We take the following conforming approximation space of X :

X N ("):=span {" i ,i=1...M} ¶F (")+span{h k }.

We could also have used a non conforming method, such as the one presented in chapter 2. The reasons why the non conforming approach was necessary do not apply here.

Remark 23

We need to multiply our 'inside basis' " i by a smooth indicator function. This is necessary so that the 'outside' solution is smooth. It is also necessary so that we can 'safely' compute the scalar products between {" i } and {" i ¶ F (")}.W ew i l ln o wa s s u m et h a tt h eb a s i s functions {" i } i have zero trace on ˆΩint (").

Conditioning

One key element of such a method is, of course, the conditioning of the resulting mass/stiffness matrices. It is closely related to the linear dependency of the basis X N ("). For standard ROM methods, the conditioning can be controlled in the offline phase by using procedures such as Gram-Schmidt orthogonalization. Here, the approximation space depends continuously on the translation parameter ", and so does the condition number of the mass/stiffness matrix.

For polynomial or Fourier basis, one way of having a better conditioned system, is to subtract to the fine basis {" i } the projection on the first few modes {h k }. Indeed, these basis are invariant by translations. Thus, imposing

'i, 'k<N, <" i ,h k > Ωint small guarantees 'i, 'k<N, <" i ¶ F ("),h k > Ωint(") small .
Another option is to use constrained optimization in the online phase, as done in [START_REF] Jp Argaud | Stabilization of (g) eim in presence of measurement noise: application to nuclear reactor physics[END_REF].

Computational details

Just as for the periodic setting, at each time step, we are looking simultaneously for the coordinates on the reduced space X N (") and for the correct calibrating parameter ". We have chosen to use the same iterative process as the one described in section 3.3. The only difference is the discretization space.

Let " n be the calibrating parameter at time step n. We need to evaluate the following quantities, for " in a small neighborhood of " n :

Y _ _ ] _ _ [ 'i, j, s Ω " i ¶ F (" n ) " j ¶ F (") 'k, l, s Ω h k h l 'i, k, s Ω " i ¶ F (") h k (3.62)
As we have chosen the {" i } to have zero trace on ˆΩint , the first term will be handled just as in the periodic case as it only involves relative translations. The second term will not cause any problem, as the basis {h k } is independent of the translation parameter and is as a premise supposed to be of small cardinality. The tricky part is the matching term, as it does not only involve the relative translation. We detail in the next section a few options available.

Coarse Finite element space

We will present one possible solution, when {h k } is a coarse Finite Element space, on a regular grid. Let

• K = {k, supp(h k ) fl supp(" i ) not empty }.

• K(")={k, supp(h k ) fl supp(" i ) ¶ F (") not empty }.

• l the Lebesgue measure of the support of the " i

• ∆x the characteristic size of the coarse mesh.

Online, we need to compute

'k oeK("), ⁄ Ωint(") " i ¶ F (")h k
Let " 0 = Â " ∆x Ê and " = " ≠ " 0 . Using the fact that the mesh is uniform, it is clear that the previous quantity is equal to:

'k oeK, ⁄ Ωint " i h k ¶ F (").
One just needs to be careful while assembling the global system. An interpolation such as the one proposed in section 3.4.4 concludes: let {" p } p be some discretization of [0, ∆x] and compute offline

'p, 'k oeK, ⁄ Ωint " i h k ¶ F (" p ).
(3.63)

Interpolate for estimations of s Ωint " i h k ¶ F ("x) for "x AE ∆x.

Polynomial/Fourier basis

Suppose that we have a coarse basis, globally defined on Ω, that has an explicit expression through calibration. More precisely, suppose that

'", h k ¶ F (") oe span {h p (•)} p (3.64)
and that the coordinates on the basis have an explicit form. By a simple change of variable, we have:

⁄ Ωint(") " i (•≠")h k (•)= ⁄ Ωint " i (•)h k (• + "). (3.65)
This method works for instance for a global polynomial or Fourier basis.

At h i r di d e a

This method presents similarities with the adaptive empirical projection method presented in [START_REF] Smetana | Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques[END_REF]. We focus on the 'coarse solution' manifold : M 0 defined on equation (3.61). Construct the basis of the restrictions of the functions in M 0 to subdomains of measure mes (Ω int ) :

Ξ:={v ¶ F (")| Ωint ,voeM 0 ,"oe Ω}.
A natural discretization, is to take the "s as multiple of "x fine , the characteristic size of the fine mesh on Ω int . Ξ is then of cardinality card(M 0 ) ú mes(Ω)/"x fine . We use a compression algorithm on Ξ to get a reduced basis of small cardinality, {◊ j } j with support Ω int .

We then learn offline the following scalar products :

'i, j, < " i ,◊ j > Ωint (3.66)
Online, we need to estimate terms such as :

⁄ Ωint " i h k ¶ F (")
We can use an EIM type method to express h k ¶ F (") on the basis {◊ j } j .

Numerical results

What is presented here has to be taken as a proof of concept. More involved numerical simulations should be performed. We have chosen the following test case:

• a gaussian initial condition

• homogeneous Neumann boundary condition

• a small viscosity such that the classical ROM fails, but big enough so that our centered reduced scheme gives decent results, see section 3.4.

in Figure 3.12, we display the first offline steps of the method. The left Figure shows the original snapshots. We use an offline calibration process, such as the one described in section 3.4.2 to obtain the calibrated solution manifold displayed on the central picture. The choice for the Figure 3.12: Snapshot sets at different steps of the offline stage. From left to right: snapshots with no calibration; centered snapshots; truncated snapshots truncation window should be studied as it has a big influence on the condition number and overall feasibility of the method. Here, it has been done empirically. The resulting snapshots are presented on the right picture of Figure 3.12. Following remark 23, we taylor an indicator function, as presented in Figure 3.13. We have Figure 3.13: The indicator function, that depends strongly on the solution manifold at hand chosen this indicator function to be non symmetric, as our snapshot set is not symmetric. Taking it stiff on the right makes sense to capture the shock. Taking it smooth on the left makes sense, as we want the complement to be well represented by a low dimensional space. These complements are members of the manifold M 0 introduced above. Some snapshots are represented in Figure 3.14. We have reduced the propagation of a shock, in a non periodic setting into two parts: one stiff part that can be handled exactly as in the previous section. The other part is the transport of a smooth quantity, which should be doable using a coarse basis. Indeed, what is being transported is much more regular that what we had initially. We present numerical evidence backing this remark in Figure 3.15. It compares the projection onto a non calibrated basis of cardinality 15, with the projection onto the union of a calibrated This concludes this small section on an extension of calibration to non periodic settings. The most important thing left to do is to conduct numerical investigations to make sure that the conditioning problem discussed in section 3.6.2 can be overcome. The different offline-online decomposition strategies should also be assessed.

Calibrating step

In this section, we give some insight on how to chose the calibration parameters in the offline stage. More precisely, say we start with some raw solution manifold M, and suppose that some a priori knowledge of the problem has given us some family F of transformations. As usual, we are looking for:

F : ; [0,T] ◊D ae F t,
¶ F ≠ Π V k≠1 (u k ¶ F )Î ; F k Ω argmin F oeF I(F ) ; V k := span (V k≠1 + u k ¶ F k ) ; k Ω k +1 ; end
Algorithm 4: Finding offline calibration parameters

Optimal method

Here, we rather use ideas close to the optimal snapshot location developed in [START_REF] Kunisch | Optimal snapshot location for computing pod basis functions[END_REF]. Instead of a greedy algorithm, the resulting basis is optimal, in a sense that will be made clear later. As in the description of the POD method, in the introductive chapter 1, we start by defining the correlation operator:

R F : {F p } p oeF N snap ae ; X ae X  ' ae q N snap k=1 <u k ¶ F k , > u k ¶ F k .
This operator is well defined, and we denote {⁄ i ({F p } p )} i the eigenvalue set of R({F p } p ), for each {F p } p oeF Nsnap and by {" i ({F p } p )} i the corresponding eigen vector/ eigen function set. The existence of the previous quantities for general scalar products is discussed in chapter 1. We know that the set of ⁄s is related to the average approximation error. More precisely, we know that:

'{F p } p oeF Nsnap , 'M oe N, ÿ i>M ⁄ i ({F p } p )= Nsnap ÿ k=1 . . . . . u k ¶ F k ≠ M ÿ i=1 <u k ¶ F k ," i ({F p } p ) >" i ({F p } p ) . . . . . 2 .
We are starting get a sense in which we are looking for an optimal calibration: we look for the set of {F p } p that minimizes the average projection error.

Let M oe N. Let J M be the following functional:

J M : ; F N snap ◊ R N snap ◊ X N snap ae R {F p } p , {⁄ p } p , {" p } p ' ae q i>M ⁄ i (3.67)
We are trying to minimise J M , subject to:

I 'i, (R F ({F p } p ) ≠ ⁄ i )" i =0 'i, (1 ≠Î" i Î 2 )= 0 (3.68)
We can show the existence of a minimzer. We will only sketch the proof and refer to [START_REF] Kunisch | Optimal snapshot location for computing pod basis functions[END_REF] for a complete description. Let

! {F n p } p , {⁄ n p } p , {" n p } p " n be a minimizing sequence in F N snap ◊ R N snap ◊ X N snap .
Suppose that F is compact. This will be the case in all the examples we will be using. For instance, in the translation case in dimension d, F is a bounded domain in R d .W e can extract a converging subsequence in F N snap . For simplicity, we still denote ({F n p } p ) n this subsequence. Denote {F ú p } p the limit. We can conclude using arguments given in [START_REF] Kato | Perturbation theory for linear operators[END_REF]. The sequence of operators (R F ({F n p } p )) n are close in a sense which ensures convergence of eigenvalues and eigenvectors towards the eigenvalues and eigenvectors of the limit operator R F ({F ú p } p ). This concludes on the existence of a minimizer of R F in F N snap .

How do we approach this minimizer ? Define a modified objective equation JM :

; F Nsnap ae R {F p } p ' ae q i>M ⁄ i ({F p } p ) (3.69)
This problem is of course equivalent to problem (3.67), (3.68). We want to compute the Gateaux derivative of J M . Let j oe [1,N snap ], we start with a formal computation of ˆJM ˆFj . The arguments for the existence of these derivatives are given below. We have:

ˆJM ˆFj = ÿ i>M ˆ⁄i ({F p } p ) ˆFj .
We use the fact that the ⁄({F p } p ) are eigenvalues of the operator R F ({F p } p ) and the product rule:

'i, 'p,

3 ˆRF ˆFj ≠ ˆ⁄i ˆFj 4 " i ({F p } p )+(R F ({F p } p ) ≠ ⁄ i ({F p } p )) ˆ"i ˆFj =0
We take the scalar product of the previous quantity with " i ({F p } p ):

'i, 'p, < 3 ˆRF ˆFj ≠ ˆ⁄i ˆFj 4 " i ({F p } p )," i ({F p } p ) > + < (R F ({F p } p ) ≠ ⁄ i ({F p } p )) ˆ"i ˆFj ," i ({F p } p ) >=0.
As R F is an autoadjoint operator, and (" i ({F p } p ), ⁄ i ({F p } p )) is an associated eigenfunction/eigenvalue pair, the second term cancels. We thus have

'i, 'p, < 3 ˆRF ˆFj 4 " i ({F p } p )," i ({F p } p ) >= ˆ⁄i ˆFj (3.70)
We use the definition of R F : ˆRF ˆFj :

I X ae X Â ' ae < ˆ(uj ¶Fj ) ˆFj , Ψ >u j ¶ F + <u j ¶ F, Ψ > ˆ(uj
{G p } p Ω Algorithm 5 1 {" p 0 ({F p,k )} k } p 2 ; 'p, 'k, F ú p,k Ω G p ¶ F p,k ;
Algorithm 6: Finding offline calibration parameters An iterative version with several layers follows easily. We could also chose other type of representants of the 'batch' POD basis. Another computationally cheaper version of algorithm 5 is discussed in section 3.9.

Remark 24

This method can seem scary, as it involves many mesh interpolations at each iteration. The computational cost set aside, some may worry about the accuracy of the resulting method. The answer to this, is that we are trying to compress the information contained in M. That is, even if one specific calibration loses some detail, it may still be in the final reduced basis. Also, as we are in the offline setting, we can afford to interpolate as precisely as we want. A final argument is that we can, instead of minimizing J, we can accept less precise calibration that preserves more informations, without degrading the resulting calibrated manifold too much.

Of course, we do not have any theoretical convergence results on the quasi-newton algorithm for this specific functional. A possible issue is then to get stuck at a local minimum. This has happened to us when applying the optimization algorithm on the dataset used in section 3.4 above. The output of the algorithm is presented in Figure 3.16. We have reached a local minimum where the shock fronts have clustered at two distinct locations. Among the many ways Figure 3.16: On possible ouput of an offline calibration procedure such as algorithm 5 of handling this situation, we have chosen one coming from the machine learning community: a clustering algorithm, and more precisely a hierarchical clustering algorithm. This empiric approach is easy to implement and seems adapted for our particular need. The output for the dataset presented in Figure 3.16 is presented in Figure 3.17. We will briefly describe how to read this so-called dendogram and refer to [START_REF] Charu | Data clustering: algorithms and applications[END_REF] for more details on this method. This algorithm's purpose is to decompose sets into clusters. The starting point is some user-defined metric on the set. To obtain the previous Figure, we have simply chosen the norm of the underlying Hilbert Figure 3.17: The output of the hierarchical clustering algorithm, for the snapshot set presented Figure 3.16 space. Then, the lines' length represent the distance between two members of the set. Here, green lines correspond to the cluster of the snapshots of the end of the simulations. The associated distance are really small as the solution manifold is almost a constant propagating front. The beginning of the simulations, when the front has not yet formed, shows a bigger 'inner cluster' distance, in red in Figure 3.17. The distance between members of different clusters is much bigger than any 'inner-cluster' distance, and corresponds to the blue line in Figure 3.16. This algorithm should be able to spot any situations such as the one presented in Figure 3.16. One option is then to split the dataset into two disjoints sets: the outputed clusters and then to perform a compression algorithm on each subset. Finally, just as in algorithm 6, we can find a good 'transformation' between representative basis of the two subsets. The global offline calibration parameter can be taken as a composition of a global calibration (to make the clusters match) and of a relative calibration (inter cluster).

This method for offline calibration has potential. We now have a way of calibrating sets for which the calibration parameters are not obvious. We have raised concerns on the associated computational cost but have proposed small improvements to mitigate the issue. What is left to do is to try this method on more challenging numerical examples, to assess its performances.

Two dimensional example

We briefly present a two dimensional extension for the calibration method. This section does not introduce any major novelty compared to what has been done before. It serves two purposes. The first one is to illustrate the fact that the equivariant property, see section 3.2.3, which was stated as necessary requirement in [START_REF] Beyn | Freezing solutions of equivariant evolution equations[END_REF], can be replaced by a less stringent hypothesis. In other words, the calibrated differential operator can be different from the original one. The second purpose here is to interpret our calibration procedure as a reduced order modeling version of mesh adaptation.

Remark 25 For a true, challenging, two dimensional example, we refer to chapter 4.

We choose the following favorable setting: we are given some parabolic equation in some two dimensional domain Ω with periodic boundary conditions such that the solution manifold M D benefits from translation. F is thus chosen as a subspace of two dimensional translations. We skip the description of the offline section, and assume right away that we have a calibrated basis:

{" i ,i=1...N red } such that: 'u(•,t; µ) oeM D , ÷F t;µ oeF s.t u(•,t; µ) oe span {" i • F t;µ }.

On the equivariance of L µ

The objective of this section is to show that equivariance is not a necessary property for the application of the calibration procedure. For this, we have chosen a family of mapping F for which the parameter dependency is of the following form:

F t;µ : 3 x y 4 ae 3 x ≠ ◊ t;µ (y) y 4 . (3.72)
The underlying idea is to be able to handle transformations around the affine mapping:

F µ : 3 x y 4 ae 3 x ≠ µy y 4
which transforms the lines x = cte into x ≠ µy = cte. For this restricted class of mappings, the determinant of the Jacobian is constant over Ω, and equal to one. We also have an explicit form of the inverse mappings:

'µ oeD,F ≠1 t;µ :

3 u v 4 ae 3 u + ◊ t;µ (v) v 4 .
Suppose that the differential operator L µ involves a Laplace operator. It is clear, with our choice of mappings, that ∆(u ¶ F t;µ ) " =(∆u) ¶ F t;µ (say in the sense of distributions). Nevertheless, we will show that the calibration procedure can still be applied.

A standard online section requires the efficient computation of the variational form. We focus on two standard terms:

I s Ω " i (F t;µ (x, y)) " j (F t Õ ;µ (x, y))dxdy s Ω Ò(" i ¶ F µ )(x, y) •Ò(" j ¶ F t Õ ;µ )(x, y)dxdy.
L 2 scalar products are directly handled by a simple change of variables:

⁄ Ω " i (F t;µ (x, y)) " j (F t Õ ;µ (x, y))dxdy = ⁄ Ω " i (u + ◊ t Õ ;µ (v) ≠ ◊ t;µ (v),v) " j (u, v)dudv. (3.73)
As for all the examples in this chapter, we suppose the smoothness of t ' ae F t;µ .T h u s◊ t Õ ;µ (v) ≠ ◊ t;µ (v) is close to 0 and these terms can be estimated using an interpolation method, see section 3.4.4.

There is a little bit more work for the H 1 scalar products. For clarity, we will denote ˆ1 and ˆ2 the derivatives with respect to first and second variables. With our specific form of mapping, we know that:

ˆ(" i ¶ F t;µ ) ˆx = ˆ"i ˆ1 ¶ F t;µ and that: ˆ(" i ¶ F t;µ ) ˆy = ˆ"i ˆ2 ¶ F t;µ ≠ ◊ Õ t;µ ˆ"i ˆ1 ¶ F t;µ .
The evaluation of the H 1 scalar product between two consecutive time steps will involve the computation of terms such as:

s Ω ˆ("i ¶Ft;µ) ˆy (x, y) ˆ("j ¶F t Õ ;µ ) ˆy (x, y)= s Ω ˆ"i ˆ2 (x ≠ ◊ t;µ (y),y) ˆ"j ˆ2 (x ≠ ◊ t Õ ;µ (y),y) ≠ s Ω ◊ Õ t;µ (y) ˆ"i ˆ1 (x ≠ ◊ t;µ (y),y) ˆ"j ˆ2 (x ≠ ◊ t Õ ;µ (y),y) ≠ s Ω ˆ"i ˆ2 (x ≠ ◊ t;µ (y),y)◊ Õ t Õ ;µ (y) ˆ"j ˆ1 (x ≠ ◊ t Õ ;µ (y),y) + s Ω ◊ Õ t;µ (y) ˆ"i ˆ1 (x ≠ ◊ t;µ (y),y)◊ Õ t Õ ;µ (y) ˆ"j ˆ1 (x ≠ ◊ t Õ ;µ (
y),y) With the proper change of variables, and because of periodicity, we can express these quantities in terms of relative variations:

s Ω ˆ("i ¶Ft;µ) ˆy (x, y) ˆ("j ¶F t Õ ;µ ) ˆy (x, y)= s Ω ˆ"i ˆ2 (u + ◊ t Õ ;µ (v) ≠ ◊ t;µ (v),v) ˆ"j ˆ2 (u, v) ≠ s Ω ◊ Õ t;µ (v) ˆ"i ˆ1 (u + ◊ t Õ ;µ (v) ≠ ◊ t;µ (v),v) ˆ"j ˆ2 (u, v) ≠ s Ω ˆ"i ˆ2 (u + ◊ t Õ ;µ (v) ≠ ◊ t;µ (v),v)◊ Õ t Õ ;µ (v) ˆ"j ˆ1 (u, v) + s Ω ◊ Õ t;µ (v) ˆ"i ˆ1 (u + ◊ t Õ ;µ (v) ≠ ◊ t;µ (v),v)◊ Õ t Õ ;µ (v) ˆ"j ˆ1 (u, v)
Remark 26 For these terms to be properly defined, '(µ, t),y ' ae ◊ t;µ (y) needs to be smooth.

The offline/online decomposition of the global method depends on the offline/online decomposition of ◊ µ . A sufficient conditions is for ◊ µ to be affine (or close to) affine decomposable, see the description of the EIM section 1.5.2.1.

Calibration seen as mesh adaptation

The idea of this section could have been discussed earlier in this chapter, and is not specific to the two dimensional problem at hand. I have chosen to mention it here, as the current setting provides good and yet easily understandable illustrations. This interpretation of calibration in terms of mesh adaptation was not in the first version of this work. It is indeed not necessary, and it does not provide any additional results. I have nevertheless chosen to add it here because calibration, as presented so far, is not being seen by many as a reasonable option from a numerical point of view. On the other hand, mesh adaptation in finite volume and finite element communities, for similar problems, is considered standard/mandatory. We thus hope to convince skeptical readers of the validity of calibration.

We have chosen to illustrate this simple setting with a two dimensional version of Burgers equation. For simplicity, we work with a superposition of 1D problem. More precisely, we are solving for the scalar function u oe H 1 (Ω) such that:

u t + c( u 2 2 ) x ≠ 'u xx =0,
where forall µ, c(µ) is a smooth function over some two dimensional rectangle Ω. We plot in Figure 3.18 some snapshots for various times, and various parameters µ. The initial condition is the same as for the 1D problem, see section 3. We display in Figure 3.19 one reference mesh on Ω, on which the truth, calibrated, solutions M are defined. If the calibration procedure is successfully performed, the shock of such calibrated solutions should always be located in the refined portion of the mesh. Now, what happens in Figure 3.19: Reference mesh the online phase ? Say we are at time t n , with a well chosen calibrating function F n and the corresponding calibrated solution ûn . Equivalently, that means that we have a physical solution u n := ûn ¶ F n defined on an adapted mesh. Meshes for front such as the one presented in Figure 3.18 are presented in Figure 3.20. The iterative algorithm of section 3.3 can be seen as an adaptive mesh procedure, where we are looking for the best mesh to represent u n+1 . In FE or FV contexts, this adaptation is done using some inter cell computations, as no more information is available, see for instance [START_REF] Frey | Anisotropic mesh adaptation for cfd computations[END_REF]. In our reducing context, we do have some additional pieces of information. More precisely, the iterative algorithm selects a small variation around the identity, such that the modified basis (or mesh) best represents u n+1 . As already stated, because our calibration functions in F are easily invertible, all the computations can be made on the same reference mesh (for instance the one depicted Figure 3.19) whatever the parameter/time considered. For ideas on the numerical errors due to this process, we refer to remark 21. We want to insist on one Figure 3.20: Meshes on which the physical solutions u n =û n ¶ F n are defined point: all the critics made on the calibration method can be exactly transposed to the standard mesh interpolation methods. We even mitigate some of the issues:

• only a limited number of mesh interpolation have to be computed, and all the computations are done in the offline phase (and thus can be done as precisely as we want, with methods as costly as we want)

• the underlying interpolation error in the online phase is controlled: for this, we refer to Figure 3.7 and the associated discussion

• the counterpart to the error indicator used in the FE/FV context, is here perfectly adapted to the reduced framework, and can be assessed in an offline phase This will again be discussed in the concluding chapter of this manuscript. One last remark on a related subject. It seems like the calibration method could be applied to problems with moving boundaries or moving interfaces. Such problems have received some attention in the reducing community recently, see for instance [START_REF] Ohlberger | Approximation of skewed interfaces with tensorbased model reduction procedures: application to the reduced basis hierarchical model reduction approach[END_REF]. It seems like the raw solution manifolds for such problems would suffer the same n-width issue as propagating fronts. I believe that these problems could be treated as the two dimensional Burgers problem with varying front shapes. This idea has not been further investigated in this manuscript.

Rotating obstacle

The example we build in this section is a little different from the ones we have seen so far, since the calibration process is not a translation. It as already been mentioned in the introductive chapter, and in chapter 2. The setting is a rotating obstacle, fixed inside Ω.T h em e s hu s e d for the computation of the fine/truth solutions is presented in Figure 3.21. We solve the incompressible Navier-Stokes equation in Ω. For a certain range of Reynolds number, we know that this particular problem leads to the creation of a Karman vortex street. We also know that the direction of the vortex street is roughly the same as the main inflow direction. Following this remark, we try to construct an example on which standard ROM fails and that would benefit from calibration. We choose as parameter the inflow direction:

u in (•,t; µ)= A cos(◊(t; µ)) sin(◊(t; µ)) B .
For sufficiently big parameter range for ◊, we expect that the raw solution manifold M is not an adapted candidate to standard ROM. M 0 is far away from the obstacle, and for a 'proper' choice of Ω int , it should deal with less complex structures, and not be as heavily direction dependent, as the initial problem. This translates into a better behaved Kolmogorov n-width. We use the following calibration family for M int :

F t;µ : Y ]
[

Ω
M int,F ,D := ) u(F ≠1 t;µ (•,t; µ),toe [0,T],µoeD * .
We expect this manifold to have a smaller n-wdith that the one of the original problem. This concludes the preliminary analysis. We have the proper candidates for ROM procedure, M 0 and M int,F ,D .

As an introductive example, we have chosen to consider a problem where the only parameter is time. Also, ◊ is chosen as a linear function with ◊(0) = ≠ fi 4 and ◊(T )= fi 4 . As an illustration, we present in Figure 3.22 the horizontal velocity component for some t ¥ 0 and one with t ¥ T .B y inspecting these two snapshots, it appears clearly that standard ROM is not adapted. Indeed, the two solutions are close to orthogonal for all reasonable scalar products. Another way of putting this is that a compression algorithm such as POD or RB would not find much redundancy of information in such a solution manifold. The offline section continues smoothly. We start by truncating our solutions around the obstacles, on Ω int . The size of the latter is problem dependent, and should be investigated. Our choice of mesh for Ω int is presented in Figure 3.23. The restriction of the solutions of Figure We have to calibrate the snapshots offline. For each snapshot u(•,t n ), the parameters F n can be chosen, as for the other examples, using some a priori knowledge. Here, it is the known inflow direction at time t n . A more realistic possibility is to use the algorithm proposed in section 3.7. The computational cost issues that were raised are still valid. It is even worse as we are dealing with a two dimensional problem. In order to make the method of Algorithm 5 computationally tractable, we propose a small variation. It relies on the use of filtering. For instance, let ÷ be some smooth indicator function of some small neighborhood of 0. One can smooth any function defined on Ω,s a yu, by looking at ū defined as:

ū : x ae (u ú ÷)(x)= ⁄ Ω u(•)÷(x ≠•).
◊ 0 (t; µ) Ω Algorithm5(G 0 úM 0 ); M◊ 0 := ) u(t; µ) ¶ F ◊ 0 (t;µ) ,t , µ * ; repeat Let G k be a filter finer than G k≠1 ; ◊ k (t; µ) Ω Algorithm5 1 G k ú M◊ k≠1 2 ; M◊ k := ; u(t; µ) ¶ F q k p=1 ◊ p (t;µ) ,t , µ < ;
k Ω k +1; until some accuracy/computational cost condition; ◊ ú (t; µ):= q k p=0 ◊ p ; Algorithm 7: Finding offline calibration parameters The idea is that as k increases, the filter are becoming finer. At the same time, algorithm 5 is being run on roughly calibrated solutions. To put it in other words, the range in which to look for better calibration parameters ◊ in algorithm 5 diminishes when k increases.

To highlight the gain of this preconditioning process, we present on Figure 3.25 three POD modes (2nd, 5th and 10th) for both original (left) and rotated (right) sets. Just as in the 1D case, without calibration, the reduced basis cannot reproduce the small scales of the flow, but rather has to deal with the inflow direction. The latter plays the exact same role as the position of the propagating front, in the one dimensional Burgers case.

Online Phase

The complete construction of a reduced scheme is not in the scope of this small section. We refer to chapter 2 for details on how to glue the reduced solution in Ω int to the outside domain Ω 0 . We rather focus here on the novelty: finding a 'good' rotation parameter online. It is not easy to find a simple problem that asseses the search for a calibration parameter. The method we have come up with can seem far-fetched, but it will nevertheless help us draw some preliminary conclusions on the feasibility of the method. This method starts with a fine, truth approximation of the solution {u(•,t n )} n . We insist on the fact that in the remaining of this section, the full simulation is known. Let {" i ,i=1,...,N} i be the calibrated reduced space on which the reduced solutions in Ω int are living. Following the discussion in the offline section, it is assumed to be of small cardinality and to represent well M int,F ,D :

'u oeM int,D , ÷◊ oe [≠fi, fi], s.t u oe span{" i ¶ F (◊),i=1,...,N}.
We need another reduced basis, {Â j ,j=1 ,...,M} j , 'bigger' than {" i ,i=1 ,...,N}.W e have M>N . We choose it such that it represents well the calibrated solution manifold, with rotations around the identity

'û oeM int,F ,D , '|"◊|AE', û ¶ F ("◊) oe span{Â j ,j=1...,M}.
We reassure the worried reader. This is not suppose to be the description of a real online scheme. It is just an attempt to asses the reconstruction of calibration parameter.

As we know the truth solution u(•,t n+1 ), we can compute its coordinates on the big basis at time t n , i.e its coordinates on {Â j ¶ F (◊ n ),j=1,...,M}:

u(•,t n+1 ):= q M j=1 <u(•,t n+1 ), j ¶ F (◊ n ) > j ¶ F (◊ n ) = q M j=1 -n+1 j  j ¶ F (◊ n ). (3.74)
◊ n is not the optimal parameter to represent u(t n+1 ). As for Burgers, we assume smoothness of the calibration parameter and look for ◊ n+1 in a small neighborhood of ◊ n . This and the assumption on {Â j ,j=1,...,M} guarantees that the projection error remains small. In a way, they play the same role as the iterative algorithm of section 3.3. The second step is to look for a rotation angle denoted ◊ n+1 such that

u(•,t n+1 ) oe span {" i ¶ F (◊ n+1 ),i=1,...,N}. (3.75)
Just as in the translation case, we compute the norm of the orthogonal projection of u(•,t n+1 ) on rotated versions of the calibrated basis, trying to minimize the projection error. That is, we are trying to minimize J defined as:

J : ; [≠fi, fi] ae R ◊ ' ae . . u(•,t n+1 ) ≠ Π "i ¶F (◊) u(•,t n+1 ) . . 2 . (3.76)
Using the orthogonality of {" i ,i =1 ,...,N}, and thus the orthogonality of {" i ¶ F (◊),i= 1,...,N}, it is equivalent to maximizing J:

'◊, J(◊)= N ÿ i=1 ! <u(•,t n+1 )," i ¶ F (◊) > " 2 .
Using the rotation invariance, J can be expressed as:

'◊, J(◊)= N ÿ i=1 ! <u(t n+1 ) ¶ F (≠◊ n )," i ¶ F (◊ ≠ ◊ n ) > " 2
This is where we use the coordinates of u(t n+1 ) on the basis {Â j ¶ F (◊ n )}:

'◊, J(◊)= N ÿ i=1 Q a M ÿ j=1 -n+1 j <Â j ," i ¶ F (◊ ≠ ◊ n ) > R b 2
We once again use the assumption on the smoothness of t ae ◊(t) and look for ◊ n+1 in a small neighborhood of ◊ n . Let ("◊) max be the maximum rotation angle possible between two successive time steps. Let Ξ be some fine discretization of [≠"◊ max ,"◊ max ]. We pre compute the following quantities

'i oe [1,N] ,j oe [1,M], '"◊ oe Ξ,<Â j ," i ¶ F ("◊) > Ωint (3.77)
We find the 'optimal' new rotation parameter, at a cheap cost, using interpolation or polynomial fitting, see section 3.4.4. This specific setting was tested on our initial example with rotating inflow condition. The results are presented in Figure 3.26. In green is plotted the true inflow direction, that changes linearly with time. In blue is the 'cumulative' optimal angle.

With this simple numerical test, we have proven two things. First, that the optimization procedure proposed in (3.76), that was proven to give descent results for viscous Burgers, can extend to more challenging problems. We have also shown that the interpolations of terms such as (3.77) make sense. This is just a first result, and we do not pretend that it concludes on the feasibility of the whole method. It is nevertheless a necessary first step. A computationally viable matching method needs to be implemented before a full reduced scheme can be constructed.

A posteriori error estimation

Ap o s t e r i o r ie r r o re s t i m a t i o n

In this section, we show that the tools developed in the ROM community for a posteriori estimates directly apply to our calibration framework. We remind the reader that rigorous a posteriori estimates are functions:

∆: ; [0,T] ◊DaeR (t k ,µ) ' ae ∆(t k ; µ) (3.78)
that satisfy an inequality such as:

't k , 'µ oeD, Îu(t k ,µ) ≠ u k (µ)ÎAE∆(t k ; µ), (3.79)
where u is the truth approximation solution, and u k is the reduced solution at time t k . The common step, when trying to construct ∆, whatever the equation considered, is to find a relation between the norm of the error (that we want to estimate), and the norm of the residual. It has been done for a simple model elliptic equation in the introductive chapter, section 1.4.3.1. More generally, it is standard for coercive elliptic PDEs, see for instance [START_REF] Martin | A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations[END_REF]. It has also been done for the unsteady viscous Burgers in [START_REF] Nguyen | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous burgers' equation[END_REF]. We will thus focus here on the computation of the norm of the residual.

We work with a general time dependent equation, say (3.27). For simplicity, we choose an explicit time discretization. The residual in that setting is defined as: r :

; X ae R v ' ae < u n+1 ≠u n dt ,v > +L(u n ; µ)(v) (3.80)
Let ê be the Riesz representant in X of r,i . e

'v oe X, < ê, v >= r(v).
ê is such that ÎêÎ X = ÎrÎ X Õ . For all n,l e tg(u n ; µ) oe X such that

'v oe X< g (u n ; µ),v > X = L(u n ; µ)(v)
We know that:

ÎêÎ X = . . . . u n+1 ≠ u n dt + g(u n ; µ) . . . . X .
We can sum up what we have so far. To compute the norm of the residual, we need to compute Îu n Î X , Îu n+1 Î X , Îg(u n ; µ)Î X , <u n+1 ,g(u n ; µ) > X , <u n+1 ,u n > X and <u n ,g(u n ; µ) > X .T o make the error estimation useful, these terms need to be computed efficiently. We thus need to have a proper offline/online decomposition. For this, we write the decomposition of each term on their respective basis. For the first two terms, there is no problem. Indeed, as {" i ¶ F n } is an orthonormal basis in X,w eh a v e :

÷-i , Îu n Î 2 X = . . . . . ÿ i -i " i ¶ F n . . . . . 2 X = ÿ i -2 i Î" i ¶ F n Î 2 X = ÿ i -2 i
The offline/online decomposition of the terms involving g(u n ; µ) depends on the nature of the operator L. It can eventually be handled using an EIM type method. Anyway, it is not impacted by calibration, and will thus not be further discussed. The only novelties are the quantities involving scalar products of terms not defined on the same basis, i.e terms for which we have no orthogonality property. For instance,

<u n+1 ,u n > X = ÿ i ÿ j -n+1 i -n j <" i ¶ F n+1 ," j ¶ F n > X
Just as in the previous sections, the terms only involve the relative transformation (F n+1 ) ≠1 ¶F n . As usual, we pre compute a few terms for "F in the neighborhood of the identity in F, and interpolate.

Unfortunately, we do not present any numerical experiments for the test case of section 3.4. There are two major reasons why. First of all, the error estimator's tightness is strongly dependent on the relative influence between convective and diffusive terms [START_REF] Nguyen | Reduced basis approximation and a posteriori error estimation for the time-dependent viscous burgers' equation[END_REF] . Our convection dominated Burgers test case is not adapted. Also, as we are solving a time dependent problem, the error estimate grows exponentially (cumulative error). To mitigate this issue, a space-time formulation has recently been developed in [START_REF] Yano | A space-time hp-interpolationbased certified reduced basis method for burgers' equation[END_REF] for viscous Burgers.

We have shown in this subsection that the a posteriori estimates naturally extend to our calibration method. Nevertheless, our method inherits the flaws of the standard method: the a posteriori error estimators are not adapted to convection dominated problems and are tedious for time dependent ones.

Conclusion

In this chapter, we have started by showing that standard reduced basis method (or actually most model reduction methods) were not adapted for convection dominated phenomenon. We have then described the freezing method, which was designed to answer this specific problem. After highlighting its undesirable properties, we have proposed a simpler, alternative, method. Unlike the freezing method, it does not act at the continuous level, but works with a semidiscretized scheme. We have extensively discussed this method on one favorable example, the one dimensional periodic viscous Burgers equation. We have shown that the resulting self-sufficient reduced scheme could be efficiently implemented in the standard online/offline paradigm.

The last sections of this chapter have been devoted to presenting extensions to this generic algorithm. We have started by showing how to extend this method to non periodic problems. We have then proposed one method that would help for problems where the offline calibration parameters can not be picked using some a priori knowledge, but rather have to be numerically approximated. In section 3.8, we have given ideas for the extension to two dimensional problems.

The last 'bonus' section shows how to extend the a posteriori error estimators to the calibration framework.

The next chapter of this thesis will be devoted to using the calibration idea for a more involved problem: the two dimensional flow around a NACA airfoil.

Chapter 4

Calibration for a challenging two dimensional example

The objective of this chapter is to use the calibration procedure introduced in the previous chapter, to solve a more challenging problem: the two dimensional Euler equation around a NACA airfoil. We start by showing that standard ROM is not fitted to solve such problem, as the shock's position is parameter dependent and thus that the Kolmogorov n-width of the solution manifold is large. We then propose an adapted calibration procedure, that uses Gordon-Hall (or transfinite) mappings. We conclude the offline phase by showing that the resulting calibrated solution manifold is better behaved. We then derive an online phase, that follows the lines of the previous chapter, by making use of the fact that only the relative (between two successive time step) shock's position are relevant. The computational complexity is controlled thanks to hyper-reduction ideas. We conclude with preliminary numerical results, that are strong evidences that a complete reduced scheme is within reach. .

Introduction

The objective of this chapter is to apply the calibration idea developed in the previous chapter, to more realistic problems than the one dimensional Burgers equation. We have decided to focus on the steady two dimensional Euler equation around an airfoil. The precise setting will be discussed in Section 4.2. To motivate the calibration idea in this specific setting, we refer to the illustrative Figure 4.1. The coloured lines are going through the barycenters of the mesh elements in which the gradient of the solution is the largest, for various pair of parameters: Mach number and angle of attacks (AoA). Each black line is a fitted line through the position of these barycenters. It is obvious that this example suffers the same problem as the one dimensional Burgers case. Because of the moving shock, the Kolmogorov n-width of the raw data set M D will not have the good decay properties required for standard ROM. We thus need a preconditioning step, and will propose an appropriate calibration. The choice in this note is to follow the steps of chapter 3, for this 2D hyperbolic problem. That is, we want to:

• calibrate the offline computed solution, to get a reduced basis as small as possible;

• have an online scheme that builds a "calibrated problem", making use of the calibrated reduced basis.

We highlight the differences with chapter 3:

• we were solving Burgers' equation in 1D with one propagating front i.e, there was only one calibration parameter. The shock's position and shape might require more calibration parameters;

• we were using periodic boundary conditions, so there was no matching with any exterior domain. As a result, the calibrated problem was just a translated version of the initial problem;

• we were using a standard Galerkin scheme in the online phase. This scheme is easily put in a reduced framework and, although different from the one used in the offline phase, 4.2. Problem setting was stable for the parabolic problem studied. As we are now working in an hyperbolic setting, it seems like we will require some form of numerical stabilization, which can lead to difficulties in a reduced setting. This issue has already been introduced in chapter 1.

In the first section of this chapter, we completely describe the problem we want to solve. We then give details on the 'truth' scheme we are using. In the second section, we describe our choice of family of mappings F, as well as one possible choice for µ ae F µ . We use this to perform the 'offline phase'. We make sure that the calibration procedure leads to a better behaved solution manifold. In the third section, we propose a cheap 'online' algorithm. This is the central part of this chapter, as most related work simply perform the offline calibration, and do not propose any numerical scheme actually using the calibrated manifold M F ,D , see [START_REF] Iollo | Advection modes by optimal mass transfer[END_REF][START_REF] Welper | Interpolation of functions with parameter dependent jumps by transformed snapshots[END_REF][START_REF] Rim | Transport reversal for model reduction of hyperbolic partial differential equations[END_REF]. In the online phase, we propose a standard L 2 minimization algorithm and a L 1 extension, as was advised in [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF]. In order to make the overall method computationally efficient, we describe how one could adapt hyper-reduction ideas [START_REF] Ryckelynck | Hyper-reduction of mechanical models involving internal variables[END_REF]. The final section is devoted to numerical experiments. We present different mappings and we show the importance of the smoothness of the mappings in F. We conclude this chapter by presenting some ideas that could be further investigated and implemented.

Problem setting

Naca0012 test case

We have chosen to perform our calibration ideas on the following well documented external flow test-case: the two-dimensional, inviscid, transonic flow past the NACA 0012 airfoil. The explicit form of the wing is given as:

y = w(x):=0.6• 1 0.2969• Ô x≠0.1260•x≠0.3516•x 2 +0.2843•x 3 ≠0.1015•x 4 2 , for x oe [0, 1]. (4.1)
We are using subsonic boundary conditions on the outside boundary and slip boundary conditions on the wing. The latter is a Neumann type boundary condition that imposes that the velocity of the fluid is tangent to the wing. It is commonly known, that from a certain threshold of Mach number, a shock appears. Both the position and the form of the shock depend on many parameters among which the Mach number and the angle of attack (AoA), i.e the inflow mean direction.

2 dimensional Euler equation

We are interested in the numerical approximation of the two dimensional Euler equations. Let us denote by Ω some domain around the airfoil described in the previous section, W the state vector of conserved variables and f =(f x ,f y ) the flux:

W =(fl, flu, flv, E) T f x (W )=(flu, flu 2 + p, fluv, u(E + p)) T f y (W )=(flv, fluv, flv 2 + p, v(E + p)) T ,
fl is the density, u and v are the components of the velocity, E = fl' + 1 2 fl(u 2 + v 2 ) is the total energy and ' is the specific internal energy. The system is closed by the equation of state relating the pressure p to the conserved variables:

p =(" ≠ 1) ! E ≠ 1 2 fl(u 2 + v 2 ) " =(" ≠ 1)fl',
where the ratio of the specific heat " is constant, with " =1.4 in our applications. We are interested in the steady solutions. We will take them as the steady limit of the following evolution equation:

I ˆW ˆt + divf (W )=0 ,t > 0,x oe Ω W (x, 0) = W 0 (x),x oe Ω. (4.2)
This problem is supplemented with the boundary conditions specified in the previous subsection.

We will take a quick glance at the fine computational method we are using, the Residual Distribution (RD) method. It is a second order oscillation free method. A complete description of this method for steady problems can be found, for example, in [START_REF] Abgrall | Residual distribution schemes: current status and future trends[END_REF][START_REF] Deconinck | Residual distribution schemes: foundations and analysis[END_REF].

Residual distribution scheme

This short presentation of the RD scheme follows the lines of [START_REF] Deconinck | Residual distribution schemes: foundations and analysis[END_REF]. In order to approximate the solutions (4.2), we are using a conforming mesh with triangular elements. We will denote with T some generic element in the mesh, with N the number of elements in the mesh and by M a generic vertex. In the RD schemes, the data are stored at the vertices. W i will denote an approximation of W (M i ). The scheme also requires a continuous approximation of the flux f (W ) over elements. It will be denoted (f (W )) h . Definition 1 Let W i be some current state, and (f (W )) h the corresponding continuous approximation of the flux.

1. 'T oe [1,...,N ] compute the residual Equivalently, denoting by -T i the distribution coefficient of node M i :

Φ T := ⁄ T div((f (W )) h )dx = ⁄ ˆT (f (W )) h • n d x.
-T i = Φ T i Φ T (4.5) with ÿ MioeT -T i =1. (4.6)
3. Possibly add some numerical stabilization

'T oe [1,...,N ], 'M i oe T, -T,stab i := -T i + ' T i (4.7)
4. Assemble the contribution for all vertices M , and solve:

ÿ T s.t M oeT -T,stab M Φ T =0. (4.8)
Note that as we are dealing with a system, the previous equality is to be understood in R 4 .T h e resolution uses an iterative process (pseudo time-stepping) to get to the solution

{W i } i .
This is a very general formulation and many classical schemes can be formulated within this framework. First, one can modify the way the the residual of each triangle is distributed among nodes, that is, the choice of thei . For instance, distributing the residual evenly among nodes corresponds to a Lax-Friedrichs type of scheme. One can achieve upwindng by taking into account the transport direction when distributing the residual. Second, many stabilization mechanism can be implemented with the use of specific ' i in (4.7). We have chosen a Lax-Friedrichs type of scheme, with an SUPG stabilization. The consequences of these particular choices will be discussed in the online section.

Remark 27 Reduced Order Modeling does not necessarily require a deep understanding of the underlying truth solver. We give these details about the CFD code because we intend to use it as part of our online scheme.

The used fine CFD mesh has 4510 grid points which corresponds to a total of 18040 unknowns. Snapshots in this solution manifold can be visualized in Figure 4.2. We have identified a range of parameters, for which the sensitivity of the shock position to Mach and AoA is high :

D := ; Mach oe [0.81, 0.83] AoA oe [0.0 ¶ , 3.0 ¶ ].
The positions of the shock for sample parameters in D are depicted in Figure 4.1. This problem has been already studied in [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF] in the context of model reduction using L1-norm minimization. It was shown the existence of discrepancies in the reduced solution, for problems with shocks. This is actually the motivation of this work.

In the rest of this chapter, we will denote u a generic component of the state vector W . For instance, one component of the output of the CFD code for parameter µ will be denoted u(•; µ). This choice of notation is not made to confuse the reader, but rather to match the standard notation in the ROM community.

Offline phase

As we will use a POD method to construct a reduced basis, we first need to select a moderate but representative snapshot set inside M D . We have chosen the following set of cardinality 12: Mach oe{0.81, 0.82, 0.83} AoA oe{0.0 ¶ , 1.0 ¶ , 2.0 ¶ , 3.0 ¶ }.

These snapshots are presented in Figure 4.2. We plot a few basis resulting from the application of POD to this data set in Figure 4.4. One can observe that just as in the 1D Burgers' case, in order to take into account the variability of the shocks' position and shape, the reduced basis tend to oscillate. This behavior is even clearer when looking at the restriction of the POD basis at the wing, see Figure 4.3. The first objective of this section is to propose a calibration procedure to mitigate this issue. For this, we construct in the next section a family of mappings F as well as an application µ ae F µ .

Preliminary remarks

As mentioned in the introduction, calibration starts with some a priori knowledge of the solution manifold. By analogy with the first dimensional Burgers' case, we choose the following calibration: let Ω be some reference domain and x0 some abscissa in Ω. Construct F a family of mappings from Ω ae Ω such that

'µ oeD, ÷F µ oeF, Ó (x, ŷ) oe Ω s.t u ! F ≠1 µ (•); µ " is discontinuous Ô µ{(x 0 , ŷ)}
To put it in other words, with this choice of calibration, the solutions in the calibrated manifold

M F ,D := ) u ! F ≠1 µ (•); µ " ,µoeD *
have vertical shocks, at position x0 . Again, using the analogy with the one dimensional Burgers case, we expect that the POD representation of the calibrated manifold would be more representative of the shape of the solutions and not try to catch the moving discontinuity. How do we achieve this calibration ? The first task is to locate the position of the shock. We have chosen the following simple strategy: first find the boundary element (on the wing) where the quantity of interest has the highest gradient. Then look at neighboring elements and pick the one with the highest gradient. Iterate until the end of the shock (i.e some condition on the gradient) or until one reaches some predefined distance to the wing. One can use other methods in order to locate more precisely the shock. For instance, in [START_REF] Siddiqi | Geometric shock-capturing eno schemes for subpixel interpolation, computation, and curve evolution[END_REF], they use ENO related ideas to locate the inner-cell position of the shock.

We denote as x = s(y; µ),µ oeDthe true shape of the shock and we will make the following assumption :

÷k small , 'µ oeD, ÷P µ oeP k (R),s (y; µ)=P µ (y). (4.9)

That is, the shock can be represented by a low order polynomial. All numerical experiments presented in this chapter have been done using a polynomial of degree 1:

P µ (y)=a 0 (µ)+a 1 (µ) y. ( 4.10)
In Figure 4.1, the colored lines are the barycenters of the control volumes with the highest gradient. In black, is the fitted polynomial, characterized by two parameters, a 0 (µ) and a 1 (µ).

Second step now, we need to construct the family F. The global picture is presented in Figure 4.5. We decompose Ω into three subdomains • Ω 0 where we will use the identity mapping

Ω 0 Ω L Ω R
• Ω L and Ω R , where we will perform the calibration

We have chosen to use a Gordon-Hall (G-H) type mapping [START_REF] William | Transfinite element methods: blending-function interpolation over arbitrary curved element domains[END_REF]. Its properties have been studied in [START_REF] Emil Løvgren | Global c1 maps on general domains[END_REF]. Examples in fluid dynamics have been numerically studied in [START_REF] Emil Løvgren | A reduced basis element method for the steady stokes problem[END_REF]. There are multiple reasons for this choice. First, for the offline part, what is important is its simplicity and its flexibility. Second, we will give computational cost related arguments in the online section below. The rest of this section will detail the application of the Gordon-Hall method onto Ω L . Similar work is, of course, performed on the right subdomain Ω R .

The reference domain has to be a rectangle in the original G-H algorithm. This fits in our framework, as we want the calibrated shock to be a vertical line. The situation is depicted in Figure 4.6, where we have plotted one possible instance of F ≠1 µ ( Ω). Contrary to most examples using Gordon-Hall type method in the literature, our domain of interest is embedded in a bigger domain. The mapping thus needs to be (at least) continuous on ˆΩL , and ˆΩR . More precisely, we need

(x 1 ,y 1 )=(x 1 , ŷ1 ) (x 3 ,y 1 )=(x 3 , ŷ1 ) (x 3 ,y 2 )=(x 3 , ŷ2 ) (x 1 ,y 2 )=(x 1 , ŷ2 ) ΩR ΩL (x1, ŷ1) (x3, ŷ1) (x3, ŷ2) (x1, ŷ2) (x2, ŷ2) (x2, ŷ1) Ω R Ω L (x1,y1) (x3,y1) (x3,y2) (x1,y2) (s(y2; µ),y2) (s(y3; µ),y3)

The actual G-H method

The G-H method is conceptually easy to understand. We denote with Γ i the edges of Ω L .W e choose a clockwise numbering, starting from the left boundary. Their counterparts on ΩL are denoted Γi . The steps are the following:

• map each edge of ΩL onto its counterpart on Ω L . That is define f such that :

'i, f µ | Γi =Γ i
• define the weights functions " i :

ΩL ae [0, 1] (x, ŷ) ' ae " i satisfying the following necessary conditions :

'i oe [1,...,4], ; " i + " i+2 =1 " i | Γi =1
These functions represent the relative positioning between the opposing edges.

• define the projection functions fi i ;

ΩL ae [0, 1] (x, ŷ) ' ae fi i satisfying the following necessary condition :

'i oe [1,...,4], Y _ ] _ [ fi i | Γi+1 =1 fi i | Γi≠1 =0 fi i | Γi oe [0, 1].
These functions define a new coordinate system in ΩL .

• for any point (x, ŷ) on ΩL , compute the projection on each edge fi i (x, ŷ).T h e n ,u s e a weighted combination of the f µ (fi i (x, ŷ)). The weights are the " i (x, ŷ).

Remark 28

The conditions on the sets {" i } and {fi i } stated above are necessary conditions. We have no explicit sufficient conditions to ensure the bijectivity of the G-H mapping.

As a first easy step, we have chosen to linearly stretch/shrink the domain. That is, we choose the following parametrization of the Γ i :

f µ | Γ1 :(x 1 , ŷ) ae (x 1 , ŷ) f µ | Γ2 :(x, ŷ2 ) ae (x 1 +x • (s(ŷ 2 ; µ) ≠ x1 ), ŷ2 ) f µ | Γ3 :(x 2 , ŷ) ae (s(ŷ 2 +ŷ • (ŷ 3 ≠ ŷ2 ); µ), ŷ2 +ŷ • (ŷ 3 ≠ ŷ2 )) f µ | Γ4 :(x, ŷ1 ) ae (s(ŷ 3 ; µ)+x • (x 1 ≠ s(ŷ 3 ; µ)),w(s(ŷ 3 ; µ)+x • (x 1 ≠ s(ŷ 3 ; µ))), (4.11)
where w is defined in (4.1) and s is given in (4.9). For example, take the left edge of the reference domain Γ1 ,t h es e t Ó (x, ŷ) oe Ω, s.t ŷ oe [ŷ 1 , ŷ2 ] and x =x 1 Ô . The vector valued function f µ | Γ1 chosen above is one possible parametrization of Γ 1 .

We use, for now, the same weight and same projection functions as in the original G-H formulation:

" 1 (x, ŷ)= ŷ ≠ ŷ1 ŷ2 ≠ ŷ1 " 3 (x, ŷ)=1≠ ŷ ≠ ŷ1 ŷ2 ≠ ŷ1 " 2 (x, ŷ)= x ≠ x1 x2 ≠ x1 " 4 (x, ŷ)=1≠ x ≠ x1 x2 ≠ x1 . and fi 1 (x, ŷ)= ŷ ≠ ŷ1 ŷ2 ≠ ŷ1 fi 3 (x, ŷ)= ŷ ≠ ŷ2 ŷ3 ≠ ŷ2 fi 2 (x, ŷ)= x ≠ x2 x3 ≠ x2 fi 4 (x, ŷ)= x3 ≠ x x3 ≠ x1 .
The standard G-H mapping is given by :

GH(x, ŷ; µ)=" 1 (x, ŷ) • f µ (x 1 , ŷ)+" 2 (x, ŷ) • f µ (x, ŷ2 ) + " 3 (x, ŷ) • f µ (x 2 , ŷ)+" 4 (x, ŷ) • f µ (x, ŷ1 ) ≠ 4 ÿ i=1 " i (x, ŷ) • " i+1 (x, ŷ) • f i;µ , (4.12)
where f i;µ is the value of f µ in the corner between Γ i and Γ i+1 . Here, we have

f 1;µ =(x 1 ,y 1 ),f 2;µ =(x 1 ,y 2 )
f 3;µ =(x 3 ,y 2 ),f 4;µ =(x 3 ,y 1 ).
We will use, in the course of this chapter, the following notation :

R 2 aeF (a 0 ,a 1 ) ' ae G-H(•; a 0 ,a 1 ) (4.13)
This application takes as argument a shock position, and returns the corresponding G-H mapping in F.

Remark 29

It is important to know that the fi's, the "'s and f µ can be chosen independently from each other. This will be made clearer in Section 4.7.2 when we try to improve the method.

It is clear that this mapping suffers from major drawbacks :

• this mapping is continuous at the boundary, but has discontinuous derivatives;

• this mapping linearly stretches/shrinks the domain; this is not the best choice to diminish the Kolmogorov n-width;

• in x 1 and x 3 , the boundary ˆΩ is not C 1 .

These issues will be fixed in the numerical section 4.7.2. They are not a problem for the offline section. Thus, for simplicity, we will illustrate the usefulness of calibration using this rough mapping. We have computed separate POD basis on ΩL and ΩR . We present in Figure 4.7 the counterpart of Figure 4.3, that is, the x component of the velocity on the left part of the wing. As one can see, using calibration we got rid of the oscillations. We present in Figure As expected, the calibrated POD captures most of the information in the first 4 basis. The 5th basis only contains numerical noise. The first objective of this chapter has been solved, we know how to build a better behaved solution manifold.

We present in the next section a reduced scheme with a computational complexity independent of the size of the truth problem, based on the calibrated basis that we have just constructed.

Online phase

For the remaining of this section, we drop out the µ dependency, as we are focused on reducing one particular simulation. It will reappear in the offline/online decomposition section. Also, we use the following notation:

• t n is the discrete pseudo time;

• F n the mapping chosen at time step n. It maps Ω onto Ω. The inverse mapping will be denoted F ≠1 n ; • {" i } is some reduced basis on the reference mesh, of cardinality N red . The one constructed in section 4.3.

We denote by u n the solution at pseudo time step t n and ûn it's counterpart on the reference mesh. That is, we have ûn = u n ¶ F ≠1 n on Ω.

We see three paths, that we present with increasing difficulty:

• The easiest method we can think of is the following:

suppose we have some reduced solution at iteration n, ûn , defined on the reference domain Ω, and a "well chosen" mapping F n ;

map this reduced solution onto the real mesh, using F n ;

use the CFD code, on Ω,u s i n gû n ¶ F n as initial condition, to get u n+1 ;

map u n+1 back onto Ω. This implies finding a "good" (in some sense) mapping F n+1 , and the corresponding reduced coordinates.

• The second method is smarter, and more in the spirit of what has been done previously, in chapter 3

just as for the first method, suppose we have some reduced solution at iteration n, ûn , defined on the reference domain Ω, and a "well chosen" mapping F n ;

use a CFD code on Ω using ûn as initial condition. This implies of course the modification of flux and boundary conditions to make this "non physical problem" equivalent to the initial one Denote ũn+1 the output. By construction, we have:

ũn+1 ¥ u(•,t n+1 ) ¶ F ≠1 n ;
deduce a new "relative" mapping:

F n+1 ¶ F ≠1
n best suited to represent ũn+1 . From this, compute a better calibrated solution ûn+1 and the corresponding mapping

F n+1 such that u(•,t n+1 ) ¥ ûn+1 ¶ F n+1 ;
• a third approach is to construct a self sufficient reduced scheme. As mentioned in chapter 1, standard CFD codes often imply numerical stabilization unfit for the reduced setting. The self sufficient scheme will thus necessarily rely on new ingredients, such as the one introduced in section 1.

The first method will not be further discussed here, as the numerous mesh interpolations imply very high computational costs, as well as numerical errors. The third method is out of the scope of this chapter. We have chosen to prove the feasibility of the second method. It assumes the existence of a fully functioning CFD code. In the lines of what has been done [START_REF] Carlberg | The gnat method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows[END_REF], the idea is to keep the stability and accuracy properties of the existing code. The computational savings would be obtained using EIM/hyper reduction ideas.

The objective is to recast the original problem defined on Ω, onto an equivalent problem defined on Ω. This is a well studied problem in the elliptic and parabolic communities, see for instance [START_REF] Quarteroni | Numerical solution of parametrized navier-stokes equations by reduced basis methods[END_REF][START_REF] Løvgren | The reduced basis element method: Offlineonline decomposition in the nonconforming, nonaffine case[END_REF]. It relies on the variational form of the PDE at hand. A similar procedure for our hyperbolic problem could be performed on a non conservative formulation. There are two issues with this approach in our setting. The first one is that this derivation is not rigorous as some of the quantities appearing are not properly defined for discontinuous solutions. Also, this formulation is not suited for our purpose, as the resulting problem is no longer posed as a conservation law, and thus require some intrusion into the CFD code. The intent here is to find a mapping procedure fitted for conservation laws. We will see that it involves a modifications of both flux and boundary conditions.

We start with a step common to Finite Volume schemes and Residual Distribution schemes. Let {Ê i ,ioe [1,...,N ]} be the set of control volumes in Ω and let u be any state variable. The integration of the conservation law in space and time, in control volume i gives:

⁄ Êi u(w, t n+1 )dw ≠ ⁄ wi u(w, t n )dw + ⁄ Êi ⁄ t n+1 t n Ò•f (u)dtdw =0. (4.14)
It is known that equation (4.14) is equivalent to:

⁄ Êi û(ŵ, t n+1 )|J F ≠1 n |d ŵ ≠ ⁄ Êi û(ŵ, t n )|J F ≠1 n |d ŵ + ⁄ Êi ⁄ t n+1 t n Ò ŵ • (N T n f (û))dtd ŵ =0 (4.15)
where û := u ¶ F n , Êi = F ≠1 n Ê i and N T n f is the correct modified flux with

N T n = 5 (J F ≠1 n ) 22 ≠(J F ≠1 n ) 12 ≠(J F ≠1 n ) 21 (J F ≠1 n ) 11 6 n ,
where J F denotes the Jacobian of any mapping F . This equality is known as the Piola transform, which is usually used in a different context. For more details, we refer for instance to [START_REF] Emil Løvgren | A reduced basis element method for the steady stokes problem[END_REF]. We will make the assumption that the determinant of the Jacobian is sufficiently smooth and the mesh is fine enough so that we can consider N T n constant per element. The error due to this approximation will not be investigated in this chapter.

Remark 30 Some more rigorous approaches could be developed, but would lead to more intrusion into the CFD code. In [START_REF] Colella | High-order, finitevolume methods in mapped coordinates[END_REF] for instance, they choose to work with the average of û|J F ≠1 n | over control volumes, instead of û.

We arrive to the following equation in each control volume ŵi .

⁄ Êi û(ŵ, t n+1 )d ŵ ≠ ⁄ Êi û(ŵ, t n )d ŵ + 1 |J F ≠1 n | i ⁄ Êi ⁄ t n+1 t n Ò ŵ • (N T n f (û))dtd ŵ =0.
We have all the ingredients to feed the CFD code:

• am e s h : h e r ei ti st h er e f e r e n c em e s h ,o v e r Ω;

• the average of the solution over control volumes:

û i = 1 mes(ŵ i ) ⁄ ŵi û(ŵ, t n )
• a flux, in a closed form: with the Piola transform, here it just amounts to N T n f where the N T term will depend on the time step and is not constant over Ω.W ew i l ls e ei n Section 4.6.3 that using G-H type mapping allows for a proper offline/online decomposition • boundary conditions: we do not need to worry about the outside boundary conditions, as they are not be affected by the mapping. The slip boundary conditions for the original problem are given by u • n =0on the wing. In our case, these are imposed as follows: treat the boundary nodes as any other node, and add the correct quantity to impose the slip boundary condition. More precisely, let n =(n 1 ,n 2 ) be the norm at the boundary . The flux at nodes on the boundary are given by:

(f x ,f y ) • n = Q c c c c c c a fl 1 (u, v) • n 2 flu 1 (u, v) • n 2 + pn 1 flv 1 (u, v) • n 2 + pn 2 1 (u, v) • n 2 (E + p) R d d d d d d b

Finding the coordinates, for a fixed mapping

We enforce the slip boundary condition by subtracting the following quantity:

( fx , fy ) • n = Q c c c c c c a fl 1 (u, v) • n 2 flu 1 (u, v) • n 2 flv 1 (u, v) • n 2 1 (u, v) • n 2 (E + p) R d d d d d d b
We can use the Piola transform again for these terms. The subtracted quantity formulated in terms of the reference variables is simply given by

⁄ ˆK ( fx (û), fy (û)) • (N T • n).
The conclusion from this analysis is that under the assumption that the determinant of the Jacobian is constant per element, changing the normals in the CFD code is enough to compute the total residual in each triangle. This is the first part of the RD scheme, see section 4.2.3. What follows is the distribution of the residual among nodes, in each element. As mentioned in the offline section, the CFD code is of Lax-Friedriechs type. The residual is evenly distributed. This procedure is independent of the mesh and of the solution. There is no additional work. For an upwinding scheme, this is a much more difficult problem to tackle, not in the scope of this chapter.

As mentioned in section 4.2.3, the truth scheme uses SUPG type stabilization. We have not studied in this chapter how to modify this term in order to have an equivalent stabilization procedure on û. We will discuss this approximation in the numerical experiment section.

We now assume that we have performed the n+1th iteration with the CFD code. The output is denoted ũn+1 and by construction, ũn+1 ¶ F n ¥ u n+1 . As F n is not, a priori, the right mapping for u n+1 , we are looking simultaneously for:

• a better suited mapping F n+1

• the corresponding ûn+1 expressed in terms of the reduced basis defined on Ω.

Following the lines of chapter 3, define the following objective function, for p oe{1, 2}:

J p : Y ] [ F◊R N red ae R F, {-k } k ' ae . . .ũ n+1 ¶ F n ≠ q N red k=1 -k " k ¶ F . . . L p (Ω)
. (4.16)

Finding the coordinates, for a fixed mapping

In this section, we are working for fixed ũn+1 and F n . We first propose an optimization procedure when the mapping F in (4. [START_REF] Hachmi | Analyse mathématique de la méthode arlequin mixte[END_REF]) is assumed to be known. Fix F oeF and define J p F as:

J p F : Y ] [ R N red ae R {-k } k ' ae . . .ũ n+1 ¶ F n ≠ q N red k=1 -k " k ¶ F . . . L p (Ω) (4.17)
We are going to discuss 2 particular cases. First the p =2case, standard in ROM and then an extension to p =1minimization, which was advised in [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF].

L 2 minimization, standard Galerkin projection

The objective functional is thus given by:

J 2 F : {-k ,k oe [1,...,N red ]}ae . . . . . . ũn+1 ¶ F n ≠ N red ÿ k=1 -k " k ¶ F . . . . . . L 2 (Ω)
First order optimality condition gives us the -s. One needs to take into account the fact that the basis {" k ¶ F } k will most probably not be an orthogonal basis.

Q c c c a -1 -2 . . . -N red R d d d b = A Q c c c a < ũn+1 ¶ F n ," 1 ¶ F> L 2 (Ω) < ũn+1 ¶ F n ," 2 ¶ F> L 2 (Ω) . . . < ũn+1 ¶ F n ," N red ¶ F> L 2 (Ω) R d d d b
where A i,j :=<" i ¶ F, " j ¶ F> X is a symmetric invertible square matrix of size N red .D e fi n e

" F := F ¶ F ≠1 n .W eh a v e 'i, < ũn+1 ¶ F n ," i ¶ F> L 2 (Ω) = s Ω ũn+1 " i ¶ " F |J F ≠1 n | 'i, j, < " i ¶ F, " j ¶ F> L 2 (Ω) = s Ω " i " j |J F ≠1 |
As in the previous chapter, we have replaced the expensive problem involving the absolute mapping by a problem where mappings are close to the identity. We will see in section 4.6.3 how to achieve efficient offline/online decomposition.

Finding the mapping

One important remark, similar to the one made in chapter 3 is that the shock's position evolves smoothly in time. This is rigorously justified by Rankine-Hugoniot conditions. Let Â0 and Â1 be the maximum absolute values for the variation between two successive pseudo time steps of respectively position and slope of the shock. These are roughly given by: 'i oe{0, 1}, Âi ¥ W 1,OE (maximum shock speed).

We use these values to define the following neighborhood of the identity in F:

F rel := Ó G-H(â 0 , â1 ), |â i |AE Âi Ô ,
where the application G-H has been defined in equation (4.13).

Alternative differentiable objective function

Let û oeM F ,D . It is clear that for solutions with shocks, the following application is not smooth:

F rel ae X " F ' ae û(" F (•)).
More precisely, the derivative in the sense of distributions has a Dirac mass at the shock. We give here a formal proof, and refer to [START_REF] Bardos | Derivatives and control in the presence of shocks[END_REF][START_REF] Allahverdi | Numerical aspects of large-time optimal control of burgers equation[END_REF] for a rigorous one. Denote

Σ 0 := Ó (x, ŷ) oe Ω, s.t û is discontinuous Ô and v ae [v] the standard jump operator.
By construction, Σ 0 is independent of û oeM F ,D . Each solution in the calibrated solution manifold can be decomposed into a smooth component and one discontinuity:

'û oeM F ,D , ÷û smooth and ûj , s.t û =û smooth +[û j ]| Σ0 û ≠ û ¶ " F =û smooth ≠ ûsmooth ¶ "F +[û j ]| Σ0 ≠ [û j ¶ "F ]| "F ≠1 (Σ0)
The derivative in the sense of distributions has thus also two components:

û ≠ û ¶ " F ¥ ˆû smooth + "| Σ0 ˆΣ
where "| Σ0 is the Dirac mass at Σ 0 . We propose one option to circumvent this issue, an alternative and differentiable objective function. For ũn+1 the output of one iteration of the CFD code, denote

Σ(ũ n+1 ):= Ó (x, ŷ) oe Ω s.t, ũn+1 is discontinuous Ô
As already mentioned, because of R-H condition, Σ(ũ n+1 ) will be close to Σ 0 .W eu s et h ep =1 notation, but the following approach can be directly transposed to the p =2case. It is easy to see why for xi,j sufficiently far from the shock so that '" F oeF rel ," F (x i,j ) is on the same side of Σ 0 as xi,j

the following application is differentiable:

; F rel ae R " F ' ae ũn+1 (" F (x i,j )).
Following this remark, we denote Ωd the subdomain of Ω where we have removed some neighborhood of the shock. More precisely, let

Ωd := € Êi , for i oe [1,...,N ], s.t 'j oe [1,...,N quad i ],
xi,j satisfies condition (4.20) .

We denote Ω d it's counterpart in the physical domain.

Remark 31

For the L 1 norm, the overall problem as presented is not differentiable. This can be solved using Huber type minimization instead of the raw L 1 [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF].

Following the previous discussion, we define smaller objective functions. For every Ω sub subdomains of Ω, define the following J Ω sub :

J p Ω sub ,F : {-k } k ae . . . . . . ũn+1 ¶ F n ≠ N red ÿ k=1 -k " k ¶ F . . . . . . L p (Ω sub )
.

With this new notation, we replace the original problem J p F with the differentiable objective function J p Ω d ,F . We can now perform standard optimization algorithm to get the desired mapping

" F , as [≠ Â0 , Â0 ] ◊ [≠ Â1 , Â1 ] ◊ R N red ae R â0 , â1 , {-k } ' ae J Ω d ,G-H(â0,â1) (-) (4.21)
is a smooth application.

One possible algorithm

We now present one way of performing in practice the optimization of the quantity defined in (4.21).

• discretize the set [≠ Â0 , Â0 ] and [≠ Â1 , Â1 ]:

{â k 0 ,koe [1,...,N 0 ]} and {â k 1 ,koe [1,...,N 1 ]}
• denote Ψ F rel the following sample of F rel :

Ψ F rel := {G-H(â k 0 , âp 1 ),koe [1,...,N 0 ],poe [1,...,N 1 ]}.
• compute the coordinates for all mappings in Ψ F rel using section 4.5 and deduce the corresponding value of the objective function:

'" F oe Ψ F rel , compute inf -oeR N red J Ω d ," F (-).
• interpolate the previously computed quantities to get an estimate of inf

-oeR N red J Ω d ," F (-) over F rel .
Deduce the value of the optimal coefficients âopt 0 and âopt 1 , as in chapter 3

• deduce the reduced coordinates for the corresponding mapping G-H(â opt 0 , âopt 1 ) using section 4.5

Remark 32 Other ideas to find F n+1 can be implemented. They are however less natural in our framework.

• Shock fitting: close to what has been described in the offline section. Find the control volumes such that ũn+1 has highest gradient and fit a polynomial. This is made computationally efficient because we do not need to look for highest gradient all over Ω: Σ(ũ n+1 ) is close to Σ 0 .

• RH condition: update the shock's position and slope using the shock's speed's explicit form given by Rankine-Hugoniot.

Online/offline decomposition

We have not yet discussed the computational complexity of our full algorithm. For now, at each time step, we need to run the full CFD code to get ũn+1 over Ω. Until we manage to build a self sufficient reduced scheme, see the third method described in section 4.4 , this computational time is not easily reducible. The only ideas available in the literature are hyper reduction [START_REF] Ryckelynck | Hyper-reduction of mechanical models involving internal variables[END_REF].

In the previous section, we have restricted the problem from Ω to Ω d because of differentiability. Here, we replace Ω d by an even smaller, denoted generically Ω sub because of computational cost. Of course, we will look for Ω sub subsets of Ω d to keep the differentiability property. The hyper-reduction method is an empirical procedure that aims at selecting a "good" Ω sub .

We present here a version of the hyper-reduction procedure that uses a different objective function than J p Ω sub ,F defined in the previous section. Note that many different variants around the algorithm we propose here are possible. For û oeM F ,D ,d e fi n e

I p Ωsub (û): ) -k ,koe [1,...,N red ] * ' ae . . . . . . û ≠ N red ÿ k=1 -k " k . . . . . . L p ( Ωsub )
.

During the hyper-reduction procedure, we try to find Ωsub such that:

'û oeM F ,D , arginf {-k }oeR N red I p Ωsub (û) !) -k ,koe [1,...,N red ] *" ¥ arginf {-k }oeR N red I p Ωd (û) !) -k ,koe [1,...,N red ] *" .
That is, we want the optimization not to be affected too much by the reduction of the size of the problem. Of course, we do not know the continuous set M F ,D . Let us denote Ξ M F ,D a representative set of the continuous manifold, and let ' be some threshold. We perform the following greedy algorithm. N hyper := card( Ωhyper ); Algorithm 8: One possible algorithm to select Ωsub The idea of hyper reduction is that on the solution manifold there is a one to one correspondence between the restriction of the solution on Ωhyper and the full solution. For the problem at hand, we expect that the solutions in the solution manifold are characterized by their behavior in the vicinity of the shock. Because of calibration, the knowledge of the solutions in a reduced number of control volumes around Σ 0 , independent of µ, should thus be enough to completely characterize the solution.

Data: Ξ M F ,D ,{" k ,

Implementation details

For our choice of online implementation, the computation of the N T terms is not a pressing issue, as these are only required in a moderate number of cells, denoted by N hyper .W ew i l ln e v e r t h e l e s s emphasize that these terms, because of the choice of Gordon-Hall type mapping, would not be a computational problem even with no hyper reduction. By inspecting the structure of the G-H mapping, see equation (4.12), we can see that the weights and the projection functions are not parameter dependent. Also, in our problem, µ ae  i (•; µ) for i oe{ 1, 2, 3} are linear function of µ. Most of the terms appearing in equation (4.12) are thus trivially affinely decomposable. The fact that the computation of terms involving  4 also fall into the offline/online decomposition paradigm requires more work. We do not enter the details, but one could show it by using a variation of the G-H method, see section 4.7.2, and the fact that away from Γ 1 , the wing can be approximated by a polynomial.

Numerical Experiments

The framework presented in this chapter present many similarities with the method described in chapter 3. The choice in this chapter has thus been to focus the numerical effort on the real novelty: the resolution of an equivalent calibrated problem on a reference mesh using the Piola transform, see section 4.4. Of course, the overall performance of such an approach relies on the ability to construct a smooth family of mappings F. This has been challenging and is a big part of the numerical experiments presented below.

Mapping on a flat domain

The first experiment we discuss is a preliminary, alpha test: we try to reproduce one snapshot, using the Piola transform and a reference mesh. We are running the CFD code for Mach =0.81 and AoA =3.0 ¶ . The truth solution that we are trying to recover is presented in Figure 4.9. We first perform a 'control sample' test. We run the original CFD code on the reference mesh of section 4.3. The output solution is presented in Figure 4.10. As expected, it is not comparable with the truth solution: u(µ). Indeed, the problem solved is not equivalent to the original one. We need to modify fluxes and boundary conditions, as presented in section 4.4. As the steady Note that with this preliminary approach, the N T term is not updated at each pseudo time step. Figure 4.11 shows the resulting solution, that we denote û(µ). One can observe that the general behavior is correct. The shock is more or less located at the correct position and it has been straightened. In other words, quantitatively we have u(µ) ¶ GH (a 0 (µ),a 1 (µ)) ¥ û(µ) on Ω. This preliminary result is a first answer to the viability of using the Piola transform to construct equivalent problems on reference meshes. Nevertheless, we can see that we have some non physical behavior close to the wing. This could have been anticipated, as the mapping constructed in Section 4.3 suffers major flaws. The biggest problem seems to be at the wing and a consequence of the high gradient at the bottom left corner of the domain of ΩL . We conclude

Mapping on a curved domain

The choice of a flat wing of the previous sections was intentional in order to remind that the reference domain on which we are solving the problem is not the physical one. Nevertheless, because of the lack of smoothness of the resulting mapping, we have decided to use a more advanced mapping than the raw Gordon-Hall. Our starting point is the method developed in [START_REF] Emil Løvgren | Global c1 maps on general domains[END_REF], applied to the domains depicted in Figures 4. [START_REF] Ballarin | Reduced-order models for patient-specific haemodynamics of coronary artery bypass grafts[END_REF]

 i ¶ fi i | Γi : I Γi ae Γ i (x, ŷ) ' ae (x, y)
The mapping is then taken as a weighted combination of these mapped boundaries :

GH(x, ŷ)= 4 ÿ i=1 [" i (x, ŷ)Â i (fi i (x, ŷ),µ) ≠ " i (x, ŷ)" i+1 (x, ŷ)Â i (1,µ)]. (4.22)
where " i and fi i are respectively the weight and projection functions associated to Γi , see section 4.3.2. The linear weights and projection functions are not an option any more, as the reference domain Ω is not a rectangle.

We will first present the choice of weights and projections proposed in the original version [START_REF] Emil Løvgren | Global c1 maps on general domains[END_REF]. This was done in a very general case, and the focus was put on the smoothness of the overall mapping. The weights functions are taken as the solutions of the following Laplace problems :

'i oe [1,...,4], Y _ _ ] _ _ [ ≠∆" i =0 in Ω " i =1 on Γi " i =0 on Γi+2
ˆ"i ˆn =0 on Γi≠1 fi Γi+1 . The projection functions are also chosen as solutions to a Laplace problem : Remark 28 on the bijectivity of the resulting mapping still holds in this extended version of the Gordon-Hall method.

'i oe [1,...,4], Y _ _ _ _ _ ] _ _ _ _ _ [ ≠∆fi i =0 in Ω fi i = t

Additional ingredients

We will now deal with the issues mentioned in Section 4.3 one by one. The smoothness of ˆΩ is solved, with the new choice of Ω. Also, we had noticed in our flawed flat approximation that missing to take into account the curvature of the wing represents a too rough approximation. We thus need to chose fi 4 and  4 accordingly. The proper way of dealing with this curved boundary is to use the standard arclength definition. For instance, the projection function on the wing fi 4 is chosen as :

fi 4 | Γ4 :(x, ŷ) ae ⁄ x 0 Û 1+ 3 ˆw ˆx 4 2 .
The same holds for  4 .

We also need to be closer to the identity mapping on the left boundary. In the original formulation, homogeneous Neumann boundary conditions are imposed on neighboring edges when computing the projection function, see (4.24). This choice is not the right one for our particular problem. We present in Figure 4.14 on the left, the projection function fi 3 in the transfinite version of [START_REF] Emil Løvgren | Global c1 maps on general domains[END_REF]. Remember, fi 3 is the projection onto the edge Γ3 . It is clear that this particular choice deforms the coordinate system. This is one of the causes of the lack of smoothness of the mapping on the left edge Γ1 . The right picture in Figure 4.14 presents fi 3 for a better suited boundary condition. In order to enforce this, one necessary step is to modify  2 and  4 from the original version. Denote with H(x) some smoothed Heaviside step function. We write it for  2 , but note that the same can be done for  4 . We pick the following :

Ẫ2 (x, ŷ, µ)=fi 2 (x, ŷ)• x3 ≠ x2 s(y 2 ; µ) ≠ x 2 •(1≠H(fi 2 (x, ŷ)))+ 3 1+(fi 2 (x, ŷ) ≠ 1) • x3 ≠ x2 s(y 2 ; µ) ≠ x 2 4 •H(fi 2 (x, ŷ)).
That is, we want no stretching for fi 2 (x, ŷ) ¥ 0 or 1. A graphical illustration is presented on the left picture of We will modify one more ingredient. We take steeper weight functions for boundaries 1 and 3. For instance, we can pick: "1 (x, ŷ)=H (" 1 (x, ŷ)) ,

where the " 1 is the solution the the Laplace problem, see (4.23). This is presented on the right picture of Figure 4.15. What this achieves is that close to left boundary, the exact shape of the right physical boundary has no influence, and the converse. Finally, we choose the following set of {Â i ,ioe [1,...,4]}:

 1 (fi 1 (x, ŷ),µ):= 1 x 1 + fi 1 (x, ŷ) • (x 2 ≠ x 1 ),y 1 + fi 1 (x, ŷ) • (y 2 ≠ y 1 ) 2  2 (fi 2 (x, ŷ),µ):= 1 x 2 + fi 2 (x, ŷ) • (s(y 2 ; µ) ≠ x 2 ),y 2 2  3 (fi 3 (x, ŷ),µ):= 1 s(y 2 + fi 3 (x, ŷ) • (y 3 ≠ y 2 ); µ),y 2 + fi 3 (x, ŷ) • (y 3 ≠ y 2 ) 2  4 (fi 4 (x, ŷ),µ):= 1 arclen ≠1 (fi 4 (x, ŷ)),y 3 2

Remark 33

The offline/online decomposition of the global method will strongly depend on the way we pick the set of " i 's, fi i 's and  i 's.

Remark 34 This smarter choice of functions not only makes the G-H mapping smoother but it also makes

DaeF µ ' ae F µ smoother. This can be an interesting property in a optimal control context (see chapter 5).

To assess the gain of this more advanced mapping, we perform the same test as in subsection 4.7.1, this time with the new and improved G-H mapping, given by equation (4.22). The output solution û(µ) is presented in Figure 4.16. As for the results obtained with the original G-H, the overall behaviour is correct as û(µ) has a similar shape as u(µ) ¶ GH(a 0 (µ),a 1 (µ)).T h en o v e l t y is that we have managed to remove the non physical behavior at the boundary that we had in the raw G-H scenario, Figure 4.11. Before a more involved test run, we present yet another improvement. This goes one step further in building a smooth mapping at the boundaries. One recent development on transfinite maps is defined in [START_REF] Jäggli | An improvement on geometrical parameterizations by transfinite maps[END_REF] and is called boundary displacement dependent transfinite map (BDD TM). The idea is not to construct the whole mapping, but to construct a relative displacement with respect to the identity. Most of the method is the same, the only difference is that instead of  i function, which represent the position on the physical domain, a new function d i :[0, 1] ◊D ae R is introduced and it will represent the displacement:

d i (t, µ)=Â i (t, µ) ≠ Âi (t).
Each of the boundaries in the reference domain is parametrized by Âi :[ 0 , 1] ae R. Like this, the mapping will take into account the original positions of the points in the reference domain Ω and will move them by weighting only the difference between the original boundaries and the deformed ones. Let (x, ŷ) be a point in the reference domain Ω, the idea of BDD TM is to displace it through the quantity (x, ŷ)+ q n i=1 " i (x, ŷ)d i (fi i (x, ŷ),µ).I nt h ee n d ,t h eB D D transfinite mapping is defined as:

GH BDDT M (x, ŷ)=(x, ŷ)+ n ÿ i=1 1 " i (x, ŷ)d i (fi i (x, ŷ),µ) ≠ " i (x, ŷ)" i+1 (x, ŷ)d i (1,µ) 2 (4.25)
This has one major effect, on the left boundary for instance, where we want zero displacement. The resulting mapping restricted to a neighborhood of this boundary will be the identity, which guarantees overall smoothness.

Remark 36

The improvements on "'s and Â's presented for the TM method still apply to the BDD TM.

After this long preamble, we are ready to illustrate numerically the gain obtained from the methods just described. We present in Figure 4.17 the comparaison between the original method, section 4.7.2.1 and the pimped one, taylored for our specific application. We show one of the entries of N T . We have picked the one varying the most i.e ˆx ˆx . It is obvious that the preivous improvements to G-H have helped for the smoothness between neighboring domains.

Final experiment

What is presented in this section does not correspond to any actual step of the online section. The purpose is to provide a more quantitative result on the utilization of the Piola transform for resolution of a problem on a reference mesh. For this, we have chosen to perform the following test:

• pick a small number of pairs {(a 0 ,a 1 )} and construct the corresponding G-H mappings:

{G-H(a 0 ,a 1 )}

• as in the previous subsection, launch the CFD code, using the modified flux and boundary conditions. Denote the output û(a 0 ,a 1 ) for each mapping G-H(a 0 ,a 1 ). Once again, the mapping is not updated at each pseudo time step

• compare the output with the mapped 'truth' solution, i.e compare u ¶ (G-H(a 0 ,a 1 )) with û(a 0 ,a 1 )

We have chosen a simple comparison criteria: the position and slope of the shock. We present the results for two pairs (a 0 ,a 1 ) in Figure 4.18. Blue represents the shock of the truth solution mapped onto the reference domain, u ¶ GH(a 0 ,a 1 ). Red is the shock of û(a 0 ,a 1 ).G r e e ni st h e position of the shock of u, and has been plotted for control purposes. We have fitted one degree polynomials through each shock. The discrepancy on the left picture, between the output of the • numerical errors on the computation of the N T terms;

• the SUPG stabilization has not been touched, to avoid too much intrusion in the code. This means that we are not using the same stabilization procedure as the truth scheme. We refer to [START_REF] Maday | An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems[END_REF] for a study of this situation. They advise using an a posteriori procedure, called rectification;

• our method to locate the shock is basic. We would need something more involved to quantify the error

Conclusion

The purpose of this chapter was to propose a complete calibration procedure to make standard ROM methods fitted for solving the two dimensional Euler equation around an airfoil. We have proposed an offline calibration procedure, have shown that it leads to better behaved, non oscillatory basis. We have then proposed a fully functioning reduced scheme. The computational complexity and the optimization procedures have been theoretically studied. We have finally developed numerical experiments that serve as a proof of concept for the global method. Most of the stages in this chapter can be further investigated. Future work could involve :

• a deeper study of the offline calibration and its effect on the Kolmogorov n-width. We have proposed online some advanced mappings, where we impose no stretching in the vicinity of the shock. A numerical investigation has to be conducted

• the construction of fully reduced scheme, with the procedure advised in Section 4.4. This could lead to an interesting comparison between L 1 and L 2 minimization. Also, the optimization procedure can be studied. The differentiability issues can be also tested.

• the hyper reduction procedure can be numerically investigated. The conjectures made on the resulting Ωhyper , namely that the interesting control volumes are close to the shock, which is fixed in Ω can be tested

• the smoothness of the entries of the N T matrix has to be studied. All the smoothing ingredients proposed in section 4.7 could be further investigated.

• a more long term objective could be to use this method to try new airfoils shapes. As mentioned, the G-H is a very flexible algorithm. We could use the NACA 00012 as reference domain, but use fantasist choices for physical domains. This is a well suited framework for optimal control (for more details, see chapter 5)

Chapter 5

Calibration for optimal control problems: illustration on solutions to hyperbolic problems with shocks

In this chapter, we have chosen to use the calibration ideas introduced in chapter 3 to help solving optimal control problems. The starting point was to notice that calibration was increasing in some sense the regularity of the solution's dependency to the parameter at hand. We have chosen to illustrate this idea by studying the optimal control of an hyperbolic equation, for solutions with shocks. We start this chapter by showing how optimal control is usually performed. Of course, this procedure depends on the regularity of the solution with respect to the parameter, and is dubious for solutions with shocks. We thus propose an optimal control method that uses calibration, and illustrate it on the one dimensional inviscid Burgers equation. We conclude by sketching a procedure to solve the optimal design of a NACA airfoil. .

The title of this chapter should not mislead the reader. The ambition of this chapter on the optimal control side is modest. The point of view adopted here is rather to try and use the calibration ideas developed in the previous chapters, as well as reduced order modeling, in an optimal control context. As already mentioned several times in this manuscript, optimal control is a natural application of ROM and there is an extensive literature on the subject, see for instance [START_REF] Kunisch | Proper orthogonal decomposition for optimality systems[END_REF][START_REF] Qian | A certified trust region reduced basis approach to pde-constrained optimization[END_REF]. In this chapter, we add a calibration process to the resolution of an optimal control problem, for an hyperbolic equation with solutions with shocks. In that setting, it serves two purposes. The first one has has already been discussed in the introductive chapter 1. The calibration helps enforce regularity to both the solution and some numerical schemes with respect to the parameter, see section 1.5.2. The second one is that the domains where the calibrated solution remains smooth are constant in time and constant parameter wise. We can thus build reduced basis on each domain and perform the classical model order reduction.

We start by introducing optimal control on a simple elliptic example. We insist on all the properties required to get to a well posed problem. We then follow these steps for an hyperbolic problem. We show the major differences caused by the existence of discontinuities whose position is function of the parameter. We finally use calibration and show that it is an adapted framework as it enforces smoothness with respect to the parameter in a way that can be handled by ROM.

Most of the ideas are illustrated using the inviscid one dimensional Burgers equation with periodic boundary conditions. We end this section by roughly stating a way of using this idea to perform optimal design of an airfoil.

What is optimal control

Let U ad be some admissible parameter set. We consider the following time dependent parametrized PDE ˆu(•,t; µ)

ˆt + L(u(•,t; µ)) = f (µ), (5.1)
supplemented with proper boundary and initial condition. Suppose also that we are given some objective functional ; X ◊U ad ae R (u, µ) ' ae J(u, µ).

We are interested in the solution of an optimization problem of the form:

inf µoeU ad J(u(µ),µ)
where u(µ) is the solution to equation (5.1). What are the formal steps ?

• make sure that equation (5.1) has good properties. The problem has to be well posed and there needs to be some form of regularity on µ ae u(µ)

• pick some minimizing sequence µ n . Use hypothesis on U ad to show that µ n converges, in some sense, towards a limit µ ú oeU ad

• use the regularity µ ae u(µ) to prove that u(µ n ) converges to u(µ ú ) in some sense

• use hypothesis on J to conclude on the existence of minimizers:

J(u(µ ú ),µ ú )= inf µoeU ad

J(u(µ),µ).

A bonus property is the uniqueness of the minimizer

• look for (one of) the minimizers. This often is done using some gradient descent algorithm, and involves differentiating µ ae J(u(µ),µ)

In this chapter, we focus solely on the last step.

Introductive elliptic case

To give a better understanding of these formal steps, we apply them to a simple example. Let Ω be some domain and Γ: =ˆΩ. We are interested by the solutions of the following elliptic problem:

I ≠∆u + u = f (µ) in Ω ˆu ˆn = g(µ) on Γ.
The objective functional is chosen as

J(u, µ)= 1 2 ⁄ Ω (u ≠ u d ) 2 + Ÿ 2 ⁄ Γ g(µ) 2 .

What is optimal control

Suppose that U ad is embedded in some normed linear space, that 'µ oeU ad ,f (µ) oe L 2 (Ω) 'µ oeU ad ,g (µ) oe L 2 (Γ)

and that f and g have continuous dependence on µ. From Lax-Milgram, we have:

U ad ae H 1 (Ω) µ ' ae u(µ)
is well defined. The hypothesis on µ ae f (µ) and µ ae g(µ), and the linearity are sufficient to show the continuity of this application.

Let {µ n } be a minimizing sequence in U ad . We suppose that U ad is chosen such that there is a subsequence converging to µ ú . From the continuity, we know that u(µ n ) ae u(µ ú ) in H 1 (Ω). To conclude that µ ú is a minimizer of J,w en e e d : J(u(µ n ),µ n ) ae J(u(µ ú ),µ ú ). We develop:

'n, J(u(µ n ),µ n ) ≠ J(u(µ ú ),µ ú )= 1 2 s Ω (u(µ n ) ≠ u(µ ú ))(u(µ n )+u(µ ú ) ≠ 2u d ) + Ÿ 2 s Γ (g(µ n ) ≠ g(µ ú ))(g(µ n )+g(µ ú ))
This concludes the existence of a minimizer µ ú of J. The strict convexity of u ae J(u) guarantees the uniqueness.

The next topic on the line is the computation of ˆJ ˆµ . Formally,

ˆJ ˆµ = ⁄ Ω (u ≠ u d ) ˆu ˆµ + Ÿ ⁄ Γ ˆg ˆµg.
Can we give a meaning to ˆu ˆµ ? This requires new assumptions on the regularity of µ ae f (µ) and µ ae g(µ). Namely, we assume that f and g are Frechet differentiable and denote A µ and B µ their derivative. Denote ˆµu the application from U ad ae H 1 (Ω) given by: ˆµu : "µ ae the solution to

; ≠∆w + w = A µ "µ ˆw ˆn = B µ "µ
We can easily show that ˆµu is the Frechet derivative of u. We have now a rigorous definition of the Frechet derivative ˆµJ: ˆµJ :

; U ad ae R " µ ' ae s Ω (u ≠ u d )ˆµu"µ + Ÿ s Γ gB µ "µ
The standard way of looking for the minimizer, is to express ˆµJ as a function of the solution, denoted p, to the dual equation. This avoids having to solve for ˆµu directly. The dual equation is here given by:

; ≠∆p + p = u ≠ u d in Ω ˆp ˆn =0 on ˆΩ.
We start with the first term of ˆµJ:

s Ω (u ≠ u d )ˆµu"µ = s Ω (≠∆p + p)ˆµu"µ = s Ω p(≠∆ˆµu"µ + ˆµu"µ)+ s Γ pB µ "µ = s Ω pA µ "µ + s Γ pB µ "µ.
Wrapping everything up, we get the following expression for ˆµJ:

ˆµJ(" µ )= ⁄ Ω pA µ "µ + ⁄ Γ (p + Ÿg)B µ "µ
This gives a direction for a standard gradient descent algorithm. For instance, taking A µ "µ to be 'close' to p over Ω is a natural strategy. As we will see in the next sections, some of the steps above are not as easy for more challenging problems, and different parameter dependencies. Next section is devoted to the study of some hyperbolic equations in the presence of shocks. In that case, the derivative of µ ae u(µ) in the sense of distributions has a dirac mass. Of course, this has consequence on the smoothness of µ ae J(µ). Lastly, we have to be more careful with the dual equation.

Burgers equation

We first focus on one favorable case: Burgers equation with periodic boundary conditions. We have chosen to restrict the study to a parameter dependency on the initial condition. That is, we are solving Y ]

[

ˆtu + ˆx( u 2 2 )=0 on Ω ◊ [0,T] u(t =0, •)= u 0 (µ) on Ω u periodic (5.2)
for µ varying in some parameter space once again denoted U ad .

At the core of both optimal control and ROM, there is some form of regularity on µ ae u(•, •; µ). It will not be as obvious as in the elliptic case. We assume that:

• U ad and µ ae u 0 (µ) are chosen so that the solutions are L 2 (Ω)

• µ ae u 0 (µ) is smooth

• for all µ and all times, the solutions considered have at most one discontinuity. Its position for time t and parameter µ will be denoted "(t; µ)

• (µ, t) ae "(t; µ) is smooth. We will see in a later section that this property is guaranteed by the Rankine Hugoniot condition and by the smoothness of the solution away from the shock.

Objective function

We work with the following objective functional:

J : ; U ad ae R µ ' ae 1 2 s Ω - -u(x, T ; µ) ≠ u d (x) - -2 dx (5.3)
where u(•, •; µ) is the entropic solution of (5.2) with initial condition u 0 (µ).

Remark 37

The precise form of the objective function will have a big importance in the following section. The new ingredient that will be introduced in section 5. We give the steps of the proof of the existence of a minimizer in that setting, and refer to [START_REF] Castro | An alternating descent method for the optimal control of the inviscid burgers equation in the presence of shocks[END_REF] for a complete description. The ingredients are quite different from the elliptic case, as the solutions are not in the same functional spaces. Let {µ n } be a minimizing sequence, and {u(•, •; µ n )} the corresponding entropic solutions,

• suppose U ad is chosen such that we can extract a subsequence that converges towards a µ ú , in some sense.

• we use Oleinik one sided Lipschitz condition:

'µ, 't, u(x, t; µ) ≠ u(y, t; µ) x ≠ y AE 1 t to extract from {u(x, t; µ n )} n a subsequence that converges in L 2 (Ω)

• proving that the limit is u(•, •; µ ú ) concludes the proof.

The focus of this chapter is put on studying the differentiability of J. For simplicity, we restrict ourselves to the case where U ad is a compact subset of R M for some M . That is, we suppose that the variations on u 0 can be described by a moderate number of parameters. More general parameter dependence would not impact the method proposed here, but would just complicate the notation. The generic ˆµJ that will be used in the rest of the chapter has to be understood as the derivative with respect to one of the M parameters.

Formally differentiating the objective function, see equation (5.3), gives:

ˆµJ = ⁄ Ω ˆµu(x, T ; µ) ! u(x, T ; µ) ≠ u d (x) " dx.
Questions arise:

• can we give a meaning to ˆµu away from the shock ?

• as u(•,T; µ) is discontinuous, we can anticipate that the derivative of u in the sense of distributions will involve a dirac mass. How does this influence ˆµJ ?

• how do we compute ˆµJ efficiently ? is the dual method described in the simple elliptic equation still relevant ?

Smoothness away from the shock

We show in this section that the derivative of the solution with respect to the parameter is smooth, away from the shock. This is an important property and will be used constantly in this chapter. We use the following notation:

's, t, 'x, Z µ s,t (x) is the position at time t of the characteristics that passes through (x, s). For instance, for fixed x oe Ω, Z µ 0,t (x) is the position at time t of the characteristics that started at time 0 at x. For classical solutions to Burgers equation, i.e for solutions with no shocks, it is well known that the characteristics are straight lines: 'µ oeU ad , 't oe [0,T],Z µ 0,t :

;

Ω ae Ω x ' ae x + tu 0 (µ)(x).
For simplicity, we start by restricting the analysis to the set of classic solutions. Fix some (x, t) oe Ω ◊ [0,T]. The first step is to prove that µ ae Z µ t,0 (x) is smooth. This function returns the position of the foot of the characteristics that gets to (x, t) when the parameter µ varies. The situation is illustrated in Figure 5 Let "µ oe R. As we are only considering classical solutions, we know:

; x = Z µ t,0 (x)+tu 0,µ (Z µ t,0 (x)) x = Z µ+"µ t,0 (x)+tu 0,µ+"µ (Z µ+"µ t,0 (x)). So we have Z µ+"µ t,0 ≠ Z µ t,0 = tu 0,µ (Z µ t,0 ) ≠ tu 0,µ+"µ (Z µ+"µ t,0
).

(5.4)

Decompose the right hand side of (5.4) into two quantities:

• the foot of the characteristics are not the same:

tu 0,µ+"µ (Z µ t,0 ) ≠ tu 0,µ+"µ (Z µ+"µ t,0
)

• the velocities of the characteristics are not the same:

u 0,µ (Z µ t,0 ) ≠ u 0,µ+"µ (Z µ t,0 )="µˆµu µ 0 (Z µ t,0 )+O("µ) 2 Equation (5.4) becomes Z µ+"µ t,0 ≠ Z µ t,0 = tu 0,µ+"µ (Z µ t,0 ) ≠ tu 0,µ+"µ (Z µ+"µ t,0
)+t"µˆµu 0,µ (Z µ t,0 )+O("µ) 2 . This gives:

Z µ+"µ t,0 ≠Z µ t,0 = ≠t(Z µ+"µ t,0 ≠Z µ t,0 )ˆxu 0,µ (Z µ t,0 )+O(Z µ+"µ t,0
≠Z µ t,0 ) 2 +t"µˆµu 0,µ (Z µ t,0 )+O("µ) 2 . (5.5) We need one more argument before we can conclude on the differentiability of µ ae Z µ t,0 (x), and it will be given by the fact that no shock is created in the vicinity of the characteristics considered. This is illustrated in Figure 5.3. The thick line is the characteristic for u(•, •; µ) that passes through (x, t). The foot of the characteristic is thus Z µ t,0 (x). The dashed line is the characteristic of u(•, •; µ + "µ), emitted from Z µ t,0 (x). The dotted line is the parallel of this characteristic that passes through (x, t). What can we say about Z µ+"µ t,0 (x) ? It cannot be one the left of the dashed line. Otherwise, two characteristics of u(•, •; µ + "µ) would intersect, thus leading to the formation of a shock. A similar arugment prevents it from beeing on the right of the dotted line. The foot of the dotted line Z dotted is given by:

x = Z dotted + tu 0;µ+"µ (Z µ t,0 (x)).
With this, we can bound Z µ+"µ t,0

≠ Z µ t,0 : |Z µ+"µ t,0 (x) ≠ Z µ t,0 (x)|AE| Z µ t,0 (x) ≠ Z dotted | AE - -tu 0;µ (Z µ t,0 (x)) ≠ tu 0;µ+"µ (Z µ t,0 (x)) - - AE T |"µ ˆµu 0;µ | + O("µ) 2 .
This hypothesis that no shock is created is necessary in order to prove the differentiability of µ ' ae Z µ t,0 (x).

t =0 x t (x, t) Z dotted Z µ t,0 (x) Figure 5.3: Size of - - -Z µ+"µ t,0 ≠ Z µ t,0 - - -
We can now go back to equation (5.5). For fixed x oe Ω and t oe [0,T],w eha v e: µ ae Z µ t,0 (x) oe C 1 (U ad ) and its derivative is given by: One graphical illustration is presented in Figure 5.4. As we are dealing with discontinuous solutions, we will use:

ˆµZ µ t,0 (x)= tˆµu 0,µ (Z µ t,0 (x)) 1+tˆxu 0,µ (Z µ t,0 (x)) . ( 5
• [u] σ = u + | σ ≠ u ≠ | σ the standard jump operator • ūσ = 1 2 (u + | σ + u ≠ | σ )
the average operator For fixed µ, the normals to the smooth curve ‡(µ) in Ω ◊ [0,T] will be denoted:

‹ := (‹ x ,‹ t )= 1  1+(ˆt") 2
(1, ≠ˆt").

We want to find the derivative in the sense of distributions of ˆµu(x, t; µ). Let Q =Ω◊[0,T]◊U ad and w oeD(Q):

⁄ Q u(x, t; µ)ˆµw = ≠ ⁄ Q + ˆµu(x, t; µ)w ≠ ⁄ Q ≠ ˆµu(x
I = ⁄ Q uˆµˆtw + ⁄ Q u 2 2 ˆµˆxw + ⁄ Ω◊U ad u 0 (µ)ˆµw(x, 0,µ).
When integrating by parts, one must take into account the fact that all these quantities are discontinuous on Σ. We just cite the result here, and refer to [START_REF] Bardos | Derivatives and control in the presence of shocks[END_REF] for the full derivation. ˆµu is the solution of: Y ]

[

ˆtˆµu p + ˆx(uˆµu p )= 0 on Q ≠ t Q + ˆµu(x, 0) = ˆµu 0 (x) (ˆt +ūˆx)(ˆµ"(t; µ)[u] φ(t;µ) )=‹ x [uˆµu p ] φ(t;µ) + ‹ t [ˆµu p ] φ(t;µ) on ‡(µ) (5.8)

Remark 39

The proof is given in the calibrated case in the appendix.

First equation of (5.8) is a linear equation on ˆµu p . It needs to be solved pointwise away from the shock. Unlike for Burgers equation, we have no physical Rankine Hugoniot condition at the shock. The third equation of (5.8) is the equation solved by ˆµˆt". It determines the measure component in ˆµu. We can see another interesting feature. The characteristics for ˆµu are the same as the characteristics for u. We will see how to take advantage of that numerically in a ROM context.

Computation of ˆµJ

Using the same kind of calculations as in the previous subsection, and assuming for now that u d is smooth at "(T ; µ), we get:

ˆµJ = ⁄ Ω ˆµu p (x, T ; µ) ! u(x, T ; µ) ≠ u d (x) " dx ≠ ˆµ"(T ; µ)[u] "(T ;µ) (ū "(T ;µ) ≠ u d ).

Remark 40

The case where the shocks of u d and u(•,T; µ) are aligned needs a different treatment, as it involves the product of a dirac mass and a discontinuous function.

Fix µ oeU ad . In [START_REF] Bardos | Derivatives and control in the presence of shocks[END_REF], they propose one possible approach to compute ˆµJ,i n s p i r e db yt h e elliptic case. Let p the solution of the dual equation:

I ˆp ˆt + u(•, •; µ) ˆp ˆx =0 p(•,T)= p T (•).
(5.9)

The idea is to re write the objective function's derivative in terms of the solution to the dual equation, just as in the introductive elliptic case in section 5.1.1. This implies finding a set of boundary and initial conditions for the dual equation such that ˆµJ can be easily expressed in terms of p. Inspecting the form of (5.9), it appears that the characteristics are the same as for u. Thus, solving this backward equation will involve going back up the characteristics. For solutions with shocks, some characteristics enter the shock before t = T .W et h u se x p e c tt oh a v et od e fi n e a boundary condition for p on ‡(µ). Figure 5.5 illustrates this process. The dotted lines are the characteristics from Ω ≠ and Ω + subdomains entering the shock at t = T . The value of p inside the dotted lines depends on the boundary condition on ‡(µ). Keeping in mind these preliminary remarks, we follow the steps of the elliptic case. We start with the equation satisfied by p, see (5.9). Multiply by ˆµu p and integrate over Ω and [0,T]:

⁄ Ω◊[0,T ] ˆµu p (x, t; µ) 3 ˆp ˆt + u ˆp ˆx4 =0.
We integrate by parts and get:

s Ω◊[0,T ] ˆµu p (x, t; µ) ˆp ˆt = ≠ s Ω◊[0,T ] ˆt (ˆµu p (x, t; µ)) p + s σ(µ) ‹ t [ˆµu p (x, t; µ)p(x, t)] φ(t;µ) + s Ω ˆµu p (x, T ; µ)p(x, T ) ≠ s Ω ˆµu p (x, 0; µ)p(x, 0) s Ω◊[0,T ] ˆµu p (x, t; µ)u ∂p ∂x = ≠ s Ω◊[0,T ] ˆx (uˆµu p (x, t; µ)) p + s σ(µ) ‹ x [uˆµu p (x
⁄ σ(µ) (ˆt+ūˆx)(ˆµ"(t; µ)[u] φ(t;µ) )p = ˆµ"(T ; µ)[u] φ(T ;µ) p("(T ; µ),T)≠ˆµ"(0; µ)[u] φ(0;µ) p("(0; µ), 0). Finally: 0= s Ω p(x, T )ˆµu p (x, T ; µ) ≠ s Ω p(x, 0)ˆµu 0 +ˆµ"(T ; µ)[u] φ(T ;µ) p("(T ; µ),T) ≠ ˆµ"(0; µ)[u] φ(0;µ) p("(0; µ), 0)
The inspection of ˆµJ suggests choosing p(x, T ):=u(x, T ; µ) ≠ u d (x), and p| σ(µ) =ū φ(T ;µ) ≠ u d . This choice fits the objective, as the derivative of the objective functional can now be computed using the solution of the dual equation: .10) This requires the value of p on the t =0line. Let Ω s as depicted in Figure 5.5:

ˆµJ = ⁄ Ω p(x, 0)ˆµu p 0 + ˆµ"(0; µ)[u] φ(0;µ) ! ūφ(T ;µ) ≠ u d " . ( 5
Ω s := ) x oe Ω, s.t t ae Z µ 0,t (x) enters the shock before T * .
Following the discussion on the characteristics, we know that the value of p(x, 0) outside Ω s will depend on p(•,T), and that the value of p(x, 0) for x oe Ω s will depend on the boundary condition p on ‡(µ).

Remark 41 By construction, p(•, 0) will be discontinuous at ˆΩs .

We sum up the conditions on p:

• p constant over ‡(µ)

• 't, p("(t; µ),t)=ū "(T ;µ) ≠ u d

• p(•,T)=u(x, T ; µ) ≠ u d (x) away from "(T ; µ)

At that point, a few questions remain unanswered:

• if u d and u(•,T; µ) have aligned shocks, then ˆµJ is not properly defined.

• equation (5.10) suggests taking ˆµu p 0 = p(x, 0) which is discontinuous. This is very likely to create solutions with more than one shock These issues are partially solved in [START_REF] Castro | An alternating descent method for the optimal control of the inviscid burgers equation in the presence of shocks[END_REF]. By inspecting the resulting form of the derivative of J, equation (5.10), they choose to decompose the variation of the initial condition into two independent infinitesimal variations:

• the variation of the smooth part of the initial condition; this corresponds to the first term in (5.10)

• the variation of the position and intensity of the shock at t =0 ; this corresponds to the second term in (5.10) They alternate between these two optimizations and manage to mitigate the previous issues. Nevertheless, the method they propose relies on the exact form of the objective function beeing used. Also, their method does not fulfill one of our objectives here: fit this control problem in the presence of shock into a ROM framework. This is the topic of the rest of this chapter. We will use the calibration ideas developped in chapters 3 and 4

Calibration

The work done until now has highlighted the difficulties due to the non smoothness of µ ae u(x, t; µ). This makes both the development of standard optimal control strategy, and of standard ROM methods tedious. The central aspect of the method proposed in this chapter is calibration. As in chapter 3, denote v the calibrated solution:

v := ; Ω ◊ [0,T] ◊U ad ae R x, t; µ ' ae u(x ≠ "(t; µ),t; µ).
The point of view adopted until the end of this chapter is to consider that u and " are known. One can for instance think that we have solved for u using the method of chapter 3. v is then just a translated version of the known u. The focus is now put on computing the derivatives of v and " with respect to the parameter. Note that this idea of using calibrated solutions for conservation laws is not new. For instance, it was used in [START_REF] Godlewski | The linearized stability of solutions of nonlinear hyperbolic systems of conservation laws: A general numerical approach[END_REF] to study the linearized stability of solutions. The main difference with what is presented here, is that the authors eventually go back to the initial equation, as it is its stability that is beeing studied. Calibration was in that context used a detour to get properties on the solutions to the original problem. In this section, we use calibration to modify the objective function, and work solely with calibrated solutions.

Remark 42

Recall that the method proposed in chapter 3 does not necessarily capture the exact position of the shock. Unlike shock fitting methods, calibration allows for a discrepancy between the calibration parameter and the true shock position. The consequences of this discrepancy are unclear, and will not be discussed in this chapter. In consequence, we will make the strong assumption that the calibration method captures the exact position of the shock.

We will re use the notation of section 5.2.3. This time Ω ≠ ,Ω + ,Q + and Q ≠ are independent of µ.

Smoothness away from the shock

This will be a very short section. We simply want to insist on the fact that the calibration has not destroyed the smoothness property away from the shock. We can follow the steps of section 5.2.2. Let (x, t) oe Ω. We need to make sure that v(x, t; µ + "µ) and v(x, t; µ) are close. Thanks to calibration, we now know that they both stand on the same side of the shock. Then, the associated characteristics are emitted from close points on the t =0line. We use the smoothness of v(x, t = 0; µ) as a function of µ to conclude. A more rigorous proof would use the precise form of the characteristics for v and the smoothness hypothesis on µ ae "(t; µ), see next section.

Smoothness at the shock

The surface in Ω ◊ [0,T] ◊U ad where the calibrated solutions v are discontinuous is now a plane: (" 0 ,t; µ). An illustration is presented in Figure 5.6. For fixed µ, the normals to the calibrated µ x t

:= (‹ x ,‹ t )=(1, 0).
The key property of the calibrated solution is that the measure component in the derivative is set to zero. Its derivative is a smooth by part function. To put it in the notation of the uncalibrated case, we have: ˆµv(x, t; µ)=ˆµv p (x, t; µ).

The proof is trivial, we just have to follow the derivation of subsection 5.2.3.

⁄ Q v(x, t; µ)ˆµw = ≠ ⁄ Q + ˆµv(x, t; µ)w ≠ ⁄ Q ≠ ˆµv(x, t; µ)w.
We insist a little bit. Usually, the derivative of a discontinuous function makes a dirac mass pop.

Here, even if (x, t) ae v(x, t; µ) is discontinuous, we have :

U ad ae L 2 (Ω ◊ (0,T)) µ ' ae v(•, •; µ) in C 1 (U ad )
. This is it, we have found a better candidate for optimal control, than the original state variable.

The calibrated solution

Y _ _ ] _ _ [ ˆtˆµv + ˆx(ˆµvv)+ˆµˆt"xv + ˆt"xˆµv =0 in Q ≠ fi Q + ˆµv(t = 0) = ˆµv 0 on Ω ◊U ad ˆµˆt" = ˆµv (5.
12)

The first two equations are a consequence of the smoothness of µ ae v(•, •; µ) away from the shock. The last equation can be rigorously derived by following the lines of [START_REF] Bardos | Derivatives and control in the presence of shocks[END_REF]. This is presented in the appendix.

Once again, we can use characteristics to better understand the structure of this system. The first equation of (5.12) is equivalent to:

ˆtˆµv +(v + ˆt") ˆxˆµv = ≠ˆµˆt"xv ≠ ˆxvˆµv.
Note that the characteristics are the same as the ones of the equation involving v: X cal;µ .B u t unlike v, ˆµv is not constant on the characteristics, but is solution to a first order ODE. More precisely, denote h the value of ˆµv over some characteristic:

h : t ae ˆµv(X cal;µ (t),t; µ), it is a solution of: ˆth = ≠ˆxvh, ≠ˆµˆt"xv.
This equation can be explicitly solved.

Remark 44 Again, because of calibration, ˆxv is a well defined term outside of the fixed shock. The integration of the ODE does not cause any particular problem.

This interpretation in terms of characteristics will be used in the numerical section.

Objective function

With no calibration, the derivative of the objective function was given by something like:

ˆµJ(µ) ¥ ⁄ Ω ! u(x, T ; µ) ≠ u d (x)
" ˆµu(x, T ; µ)+ some pointwise value at the shock We propose a few possible objective functions adapted to the calibrated framework:

• the calibrated solution should look like some prescribed solution u d

• the shock should be located at some prescribed location " obj at t = T

• some combination of the two previous objectives Let K oe R. We choose the last option:

J K (µ)= ⁄ Ω |v (x, T ; µ) ≠ u d (x)| 2 dx + K("(T ; µ) ≠ " obj ) 2
where K is the weight given to the shock positioning objective. The computation of the derivative is straightforward:

ˆµJ K (µ)= s Ω ! v(x, T ; µ) ≠ u d (x) " ˆµv(x, T ; µ) +Kμ"(T ; µ) ! "(T ; µ) ≠ " obj " .
Unlike in the uncalibrated case, there is no problem in interpreting the first term as both quantities are functions, smooth by parts. The measure component has been set to zero thanks to calibration.

We have solved the first objective: we have modified the state variable as well as the objective function to get smooth dependencies µ ae v(•, •; µ) and µ ae J K (µ). The next topic on the line is finding an efficient way of computing ˆµJ K (µ). This involves the numerical estimation of ˆµˆt".

Remark 45

The computation of ˆµ"(T ; µ) will not be an issue, as it has a simple explicit form:

ˆµ"(T ; µ)=ˆµ"(µ, 0) + ⁄ T 0 ˆtˆµ"(t; µ).
because of the smoothness in time discussed in section 5.2.

The estimation of ∂ µ ∂ t φ

We could use the derivative of the R-H condition with respect to the parameter at the shock, see third equation of (5.12). Instead, we propose a method that can also be used for classical solutions. More precisely, we will show how the problem of finding ˆµˆt" can be treated as was the problem of finding (" n+1 ≠ " n ) in chapter 3 and F n+1 ¶ F ≠1 n in chapter 4. Let {Φ i } be some reduced basis of the calibrated solution manifold, computed offline. The ROM premise is that:

'µ oeU ad , 't oe [0,T], ÷{-i }, s.t u(•,t; µ)= ÿ i -i (t; µ)Φ i (•≠"(t; µ)). ( 5.13)
The smoothness of the calibrated solution with respect to time and to parameter can be expressed as: 'i, (t, µ) aei (t; µ) oe C 1 (U ad ◊ [0,T], R).

With this assumption, we have an explicit form for ˆµv: ˆµv(•,t; µ):(t; µ) ae ÿ i ˆµi (t; µ)Φ i (•).

We want to construct a procedure to estimate ˆµˆt" mimicking the methods of chapter 3 and 4. Towards this objective, for each smooth functional

 : ; [0,T] ◊ U ad ae R (t, µ) ' ae Â(t; µ), (5.14) define v  : I U ad ◊ [0,T] ae X (µ, t) ' ae u(•≠Â(t; µ),t; µ).
This corresponds to a badly calibrated solution. With this notation, the true calibrated solution v is denoted v " . A wrong estimation for ˆµˆt" while solving (5.12) is equivalent to working with v  where  is such that

Y _ ] _ [ Â(µ, t)= "(µ, t) ˆtÂ(µ, t)=ˆt"(µ, t) ˆµÂ(µ, t) " = ˆµ"(µ, t).
'well behaved' scheme in this context, but note that other choices are possible. This discussion resembles the one we had in chapter 1 about the fact that reduced scheme and truth scheme could be different. Similarly here, tailored scheme for the computation of ˆµv could be constructed. Nevertheless, adapting the numerical tools used for the computation of v is the simplest way to guarantee stability.

Upwinding

To solve numerically a convection dominated problem with a finite volume method, it is advised to use upwinded flux instead of centered flux, to enforce stability. It appears obvious that whatever the parameter dependency, the upwinding direction used to compute ˆµv should be the same as the one used for v. This appears clearly when the parameter is on the initial condition, see section 5.3.2. An example with varying boundary conditions is mentioned in section 5.4.3. This particular fact, that pieces of information about the computations of the state variables could be used to compute the derivative with respect to some parameter, was already used in [START_REF] Godlewski | The linearized stability of solutions of nonlinear hyperbolic systems of conservation laws: A general numerical approach[END_REF].

For a non calibrated problem, the upwinding procedure is trivially not smooth with respect to the parameter. We have no guarantee that the resulting scheme on ˆµv will be stable and we can expect that terms with irreductible n-widths would appear, as described in section 1.5.2. Calibration is helping in that respect. For instance, in the simple viscous Burgers numerical examples of chapter 3, the upwinded flux are linear functions of the calibrated solution considered. Using the premise that µ ae v(•, •; µ) is a smooth function, we conclude that the upwinding procedure has a smooth parameter dependence, and can thus be handle by ROM. In the next section, we explicit this idea for another standard stabilization mechanism: slope limitation.

Slope limiters

The development in this section will be closely related to the discussion about flux limiters in the introductive chapter, section 1.5.2. We denote T h some mesh in Ω and P 0 (T h ) some constant per element discretization space. Γ denotes a generic boundary in T h .

High order finite volume method require the estimation of the solution at the boundaries of control volumes, using reconstructed quantities. The most common procedure is the reconstruction of the gradient over a control volume, using the average of the solution in neighboring cells. Let v i be the average of the state variable v in control volume i. The values at some interface Γ are then estimated using the first order expansion :

v| Γ = v i + Ò i • (x| Γ ≠ x centroid ). ( 5.19)
Ò i is a function of the average of v in the neighboring cells, {v j }. The cells actually used are often referred to as stencil. Ò i needs to be chosen so that:

• the scheme is stable

• the scheme is accurate

• the added computational cost is reasonable

The rest of this section takes place in a one dimension context, where T h is an uniform mesh. This favorable setting offers rigorous theoretical results. Indeed, it is known [START_REF] Berger | Analysis of slope limiters on irregular grids[END_REF] that a scheme using a reconstructed gradient such that:

• Ò i =0at local maximas At local extremums, we have ˆµ" i =0.

We go back to the first order expansion, see equation (5.19). All the quantities involved are smooth with respect to µ. We can thus use the product rule to get an explicit form for the derivative of v Γ with respect to µ:

ˆµv Γ := ˆµv i (µ)+ " Õ (R i ) ˆµR i Ò i • (x| Γ ≠ x centroid ) +"(R i ) Ò Õ i • (x| Γ ≠ x centroid )
For VanLeer limitor " Õ : R ae 2 (R+1) 2 . We have shown in the two previous sections how to extend the properties of numerical schemes for convection dominated problems to the computation of the derivatives of the state variables. Once again, this is not the only viable strategy. Nevertheless, this perfectly fits within the calibration framework.

Euler 2d

We will apply the ideas of the previous sections to the problem studied in chapter 4. Suppose that we have some NACA airfoil inside a domain. It is standard in that context to take as objective functional J, the integral over the body of some function of the pressure and of the normal at the boundary:

J(µ):= ⁄ ˆΩ j(P (µ),n w )ds.
For simplicity, one can think of a parameter dependency on physical parameters such as Mach number, angle of attack etc. But using what has been done in chapter 4, recall that we can also handle parametrized wing shapes.

Initial remarks

From chapter 4, we know that the shock's existence as well as its position and shape depend on the angle of attack, the mach number and the shape of the wing. As for inviscid Burgers, we need hypothesis on the type of solutions considered:

• 'µ oeU ad , there is one shock. Denote "(µ) oe R its position on the wing • the solutions are smooth with respect to parameter variation away from the shock. The main arguments, just as for one dimensional Burgers, are the finite speed of propagation of the information, as well as the smoothness of initial and boundary conditions

• µ ae "(µ) is smooth
Without any calibration, we expect the derivative of J to contains a dirac mass at the shock's position: ˆµJ = some smooth by part function of ˆµj + "(x ≠ "(µ))[j(P (µ, n w )]| "(µ) ˆµ". This exact setting has been studied in [START_REF] Baeza | 2-d euler shape design on nonregular flows using adjoint rankine-hugoniot relations[END_REF].

Equation for the derivative

Far from the shock, the µ derivatives of the state variables are solution of the linearized equation, see [START_REF] Baeza | 2-d euler shape design on nonregular flows using adjoint rankine-hugoniot relations[END_REF]. We will use the notation of chapter 4 directly. Mimicking what has been done for Burgers, we do not try to solve for the original state variables ˆµu, but rather for the calibrated ones: ˆµ û := ˆµ (u ¶ F µ ). A rigorous derivation of the equation solved by ˆµ û is not in the scope of this paper. Nevertheless, we expect that the work done in the previous sections can be transposed exactly. We will highlight a few aspects.

Computational details

Just as for Burgers, one has to be careful on the numerical method used to solve for ˆµu (or ˆµ û). For instance, suppose that one of the parameters of the problem is the angle of attack. That is, on Γ ext , we have a Dirichlet boundary condition on the velocity of the form (u, v)=(cos(µ),sin(µ)).

The derivative of the solution will satisfy the linearized equation, with the following boundary condition on Γ ext :

(ˆµu, ˆµv)=(≠sin(µ),cos(µ)).

Intuitively, we know that giving this boundary conditions to a black box solver will give non satisfactory results. Indeed, some of the inflow and outflow boundaries are exchanged. This is not reasonable physically. As for Burgers, we need to take as upwinding direction for the derivatives the same as the upwinding direction for the original variables. The calibration helps with the smoothness of the upwinding directions.

Estimation of ∂ µ N

This is the equivalent to the estimation of ˆµ" in the previous section. Once again, we could use RH conditions. We rather propose a ROM oriented method. Let F be a family of mapping and F µ oeF the correct calibration. An error on ˆµF µ while solving for ˆµ(u ¶ F µ ) corresponds to a badly calibrated solution ũ(µ): =u ¶ G with G " = F µ . For Burgers, we were looking for an orthogonal basis satisfying Φj " oe span {ˆxΦ i }. The counterpart is here:

Ψj " oe span {Ψ i ¶ " F }
where " F are variations around the identity, among the family F of mappings considered. The optimization procedure can be conducted as follows: first compute the solution of the equation on the derivative for several values of ˆµF µ . Then, interpolate to get the ˆµF µ that maximizes the projection of ˆµ(u ¶ F µ ) onto the basis { Ψj }.

Optimal control on the shape of the wing

We can combine the work that has been done in chapter 4 and in this one to propose a procedure to solve the optimal control problem of the design of airfoils. We give here a roadmap. ˆxˆµw + ⁄ Ω◊U ad v 0 ˆµw =0.

Let w oeD (Q). We will first compute the new terms coming from calibration. We start by integrating by parts with respect to µ.

s Q ˆt"vμˆxw = ≠ s Q ˆµ (ˆt"v) ˆxw + s Σ ˆµ"ˆt"[v]ˆxw = ≠ s Q ˆµˆt"vxw ≠ s Q ˆt"μvˆxw + s Σ ˆµ"ˆt"[v]ˆxw.
Now, we integrate by parts with respect to x:

s Q ˆt"vμˆxw = s Q ˆµˆt"xvw + s Q ˆt"xˆµvw + s Σ ˆµ"ˆt"[v]ˆxw ≠ s Σ ‹ x (ˆµˆt"[v]w + ˆt"[ˆµv]w) .
The others three terms (already dealt with in [START_REF] Bardos | Derivatives and control in the presence of shocks[END_REF]) are: For ˆµv to be weak solution, the previous equation needs to be true for all w oeD(Q). As usual, we start with w such that w(•,t = 0; µ)=0, this gives us the linear PDE solved by ˆµv: ˆtˆµv + ˆx(ˆµvv)+ˆµˆt"xv + ˆt"xˆµv =0.

s Q vˆtˆµw = ≠ s ˆµvˆtw + s Σ ˆµ"[v]ˆtw = s Q ˆtˆµvw + s Σ ‹ t [ˆµv]w + s Σ ˆµ"[v]ˆtw ≠ s Ω◊U ad ˆµv(t = 0)w s Q 1 v 2 2 2 ˆxˆµw = ≠ s Q (vμv) ˆxw + s Σ ˆµ"[v 2 ]ˆxw = s Q ˆx (vμv) w + s Σ ‹ x [vˆµv]w + s Σ ˆµ"[v 2 ]
Taking w(•,t = 0; µ) non zero, we have the expected initial condition: ˆµv(•,t = 0; •)=ˆµv 0 (•; •).

The surface contributions now. Because of ˆµ" =0and ‹ t =0 , most of the terms cancel. We are left with: Chapter 6

Analysis of the two level POD method

The problem we propose to deal with in this section is the computation of POD basis of very large sets. One reviewer has brought to my attention that this issue has already been recently solved in a more general setting in [START_REF] Himpe | Hierarchical approximate proper orthogonal decomposition[END_REF]. Suppose that we have a set of N snap snapshots, all living in some N dimensional truth discretized space (say finite element space). We know that the computational cost of the POD method is roughly given by: min(N 2 N snap , N N 2 snap ). This is pretty clear when looking at the method of snapshots, see [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. i-coherent structures. ii-symmetries and transformations. iii-dynamics and scaling[END_REF].

There are several ways to circumvent this issue. The first one is to use a POD greedy type method such as the one developed in [START_REF] Martin | A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations[END_REF]. The associated convergence rate has recently been studied in [START_REF] Haasdonk | Convergence rates of the pod-greedy method[END_REF]. These methods, as all RB type methods, need a cheap, tight upper bound on the error. It is well known that this task can be difficult when dealing with non linear PDEs. Another option available in the literature is to perform the POD in parallel in the N direction, as in [START_REF] Wang | Approximate partitioned method of snapshots for pod[END_REF]. Here, we follow a third route, see [START_REF] Ballarin | Reduced-order models for patient-specific haemodynamics of coronary artery bypass grafts[END_REF], and propose a two level POD method.

Let Ξ be some fine discretization of a solution manifold denoted M embedded in some Hilbert space X. Start by creating subsets of Ξ, that we will call batches from now on and denote them {Ξ p } p . The idea is to parallelize the computation of the POD basis of Ξ using a divide and conquer approach. More precisely, we start by computing the POD basis of each batch. We then re combine the 'sub POD' basis, as best as we can. The rest of this section is devoted to the analysis of this recombination.

One particular application we have in mind is time-parameter solution manifolds. We will use this specific notations, as it will make our calculations a little more intuitive: each batch corresponds to a time simulation for one parameter µ p : Ξ p := ) u(•,t k ; µ p ),t k oe [0,T] * .

From now on, we will denote:

• P the number of batches

• N p snap the number of time-snapshots for the simulation for parameter µ = µ p • N p red the size of the POD basis corresponding to the simulation µ = µ p

• N tot the size of the final basis

• " µp i the ith basis of the batch Ξ p • " tot j the jth basis, after re combination of the " µp i .

• " stand j the jtth basis returned by the standard POD algorithm on Ξ.

What do we want to control ? let N tot be some prescribed size of the final basis. We want to minimize the following quantity: * . This is the most natural way to go in this particular setting. We model this easier problem by defining the following functional J: J :

J Ntot ({
Y _ _ _ _ ] _ _ _ _ [ R q p N p red ◊ X Ntot ae R ) - µp i * i,p , {" j } j ' ae J(-, " j )= q p q N p red i=1 . . .  -p i " µp i ≠ q Ntot j=1 e  -p i " µp i ," j f X " j . . . 2 X = q p q N p red i=1 -p i . . ." µp i ≠ q Ntot j=1 + " µp i ," j , X " j . . . 2 X
We will denote by {" tot j (-)} the optimal basis for a specific choice of weights and by {⁄ tot j (-)} the corresponding eigenvalue set. Recall that they are the eigenfunctions and eigenvalues of the Unfortunately, this is not true for the original basis {" orig }. We need to use something else:

(Π Â Π origin ≠ Π origin Π Â )= ! Π ‹ Â Π ‹ origin ≠ Π ‹ origin Π ‹ Â "
This concludes this digression and we go back to our original problem. As {" tot j } is optimal to represent the set Ξ proj defined in (6.3), we have:

q j>Ntot ⁄ tot j (⁄) AE q p q N p

ROM and big data: a common methodology

In this short bonus chapter we will discuss the new trend in the reduced order modeling community that consists in adapting 'big data' methods. The underlying idea is to replace part/all of a PDE resolution by some learning algorithm inspired by big data. We do not intend to present a thorough list of applications but have chosen a small set of examples, that illustrate the broad spectrum of possibilities. In a recent paper [START_REF] Gallinari | Reduced basis' acquisition by a learning process for rapid on-line approximation of solution to pdes': Laminar flow past a backstep[END_REF], they propose an original way to combine ROM and big data. The physical domain is decomposed, using some a priori knowledge of the solution, into disjoint subdomains. The reduced basis in each subdomain is then chosen among a set of reduced basis using some learning process. In [START_REF] Ibañez | Data-driven non-linear elasticity: constitutive manifold construction and problem discretization[END_REF], they adapt the Local Linear Embedding (LLE), that was originally developed for image processing, to the numerical resolution of PDEs. More precisely, they build a 'model free' method for elasticity problems. The modeling of the relation between displacement field and stress tensor is replaced by a learning algorithm. Finally, in [START_REF] Wang | Model identification of reduced order fluid dynamics systems using deep learning[END_REF] they have chosen to evolve the POD coefficients of a Navier-Stokes simulation, using a neural network.

The focus in this short chapter is put on a method with growing importance in a part of the reduced order modeling community. This method is referred to as Dynamic Mode Decomposition (DMD) [START_REF] Peter | Dynamic mode decomposition of numerical and experimental data[END_REF][START_REF] Tu | On dynamic mode decomposition: theory and applications[END_REF][START_REF] Rowley | Spectral analysis of nonlinear flows[END_REF]. It uses a learning algorithm in order to deal with the computational complexity incurred by the presence of non linear functions of the solution, in a ROM context. We end this chapter, by presenting a fully 'model free' solver that has been developed in the computer graphics community [START_REF] Jeong | Datadriven fluid simulations using regression forests[END_REF].

DMD

DMD aims at accelerating the online computation of non linear functions of the solution, while solving some PDE. Figure 7.1 shows a typical situation. We have computed snapshots over some solution manifold M, {u k } k and have computed their image through some non linear application F , {F (u k )} k . The objective is then to find a 'cheap' way of computing the image of any member of M through F . We have presented in the introductive chapter, section 1.5.2.1 the most common way of dealing with this in a ROM context, the Empirical Interpolation Method (EIM). DMD method, as the LLE method mentioned earlier, is of a different type. It handles the non linear dependency by building an equivalent linear model. The associated computational gain is obvious. The novelty compared to other linearization methods is its theoretical foundation. The method supposes the existence of a space, somehow related to M, on which the possibly highly non linear application F , is linear. A graphical representation can be found The method relies on the Koopman theory, and more precisely on the construction of a so-called Koopman Operator. In this section, we denote:

• X either the manifold of continuous solutions, or equivalently a 'truth' discretized space • a generic non linear application X ae X, denoted by F

• O some observable space over X, that is, a set of functionals over X. Note that these are not necessarily linear

In that framework, the dots of Figure 7.2 are the images of the elements of X through the observables O. The latter can for instance be thought of as the scalar product with some reduced basis {" i } i : Whatever the space of observables, the idea is to replace the non linear transformation on X by a linear transformation on O. For that, the key component is the Koopman operator K,a linear operator acting on the vector space spanned by the elements of O as: 'g oe span O, Kg = g ¶ F.

O := ; X ae R u ' ae <u," i > X ,
(7.1)

Note that the previous equality is an equality between functionals over X. We can now reformulate the premise of the DMD method: it supposes that the complicated non linear behavior, restricted to the reduced basis (or some other observables space) is well represented by a linear transformation. This is precisely the assumption that we discuss in this section. To get a little more context, I have chosen to reproduce a paragraph found in [START_REF] Tu | On dynamic mode decomposition: theory and applications[END_REF], which is among the most cited paper in the DMD literature:

Without these connections, the use of DMD to analyze nonlinear dynamics appears dubious, since there seems to be an underlying assumption of (approximately) linear dynamics (see Section 3.1), as in [START_REF] Abgrall | Residual distribution schemes: current status and future trends[END_REF]. One might well question whether such an approximation would characterize a nonlinear system in a meaningful way. However, so long as DMD can be interpreted as an approximation to Koopman spectral analysis, there is a firm theoretical foundation for applying DMD in analyzing nonlinear dynamics.

It is the firm theoretical foundation that we will try to understand.

The Koopman operator

We show in this section how the DMD community intends to use the knowledge of the eigenfunctions of K to solve the target objective. Suppose that there exists a complete set of eigenfunction/eigenvalue pairs {(Â j ,⁄ j ),j =1...N red } for the operator K. In other words, we suppose that:

'g oe span O, ÷{c j (g)}oeR N red , s.t ,g= N red ÿ j=1 c j (g)Â j .

Let g oeO. The objective is to be able to compute efficiently g ¶ F over X. With the previous assumption, we have:

÷{c j (g)}oeR N red ,g ¶ F = N red ÿ j=1 c j (g)(Â j ¶ F ) . (7.2)
Using the Koopman definition, and the fact that {Â j } is an eigenfunction set, we have:

g ¶ F = N red ÿ j=1 c j (g)(KÂ j )= N red ÿ j=1 ⁄ j c j (g)Â j . (7.
3)

The rest of the derivation will be done in the case where the observables are taken as the scalar products with some reduced basis {" i } i , but note that there would be no difference for other choices of observables spaces. To avoid confusion, we denote by "i the observable in O that corresponds to the scalar product with " i .

Offline, start by computing the couples {(Â j ,⁄ j )} j . We show in the next section how this is performed. Then, compute the coordinates of the observables onto the space spanned by the eigenfunctions of K:t h es e t{c j ( "i )} i,j .

Online, evaluate the observables applied to the non linear term as: 'u oeM, 'i, "i (F (u)) =<F(u)," i > X =

N red ÿ j=1 ⁄ j c j ( "i )Â j (u) (7.4) From this discussion, we need a priori N red eigenfunctions of K. We will come back on this in section 7.1.4.

In order to fit DMD into a standard offline/online paradigm, we need more assumptions on the set of eigenfunctions {Â j }. By inspecting the form of the quantities needed online, see (7.4), the functionals in the eigenfunction set either need to have a computationally cheap action on X (linear or quadratic functionals for instance) or to involve a reduced number of spatially localized estimates. This of course restricts the observable spaces O that can be considered. The next section is devoted to the search for approximate eigenfunctions.

Details on the offline stage

The objective of this section is to propose a method to compute some1 eigenfunctions of the Koopman operator. We are still in the situation where the observables { "i } are the projection onto some reduced basis {" i }. Let {u j } j be a representative/well chosen snapshot set of M in X, of cardinality M .D e fi n eU as the following N red ◊ M matrix: U ij := "i (u j )=<u j ," i > X (7.5) Compute the image through the non linear application F , and store the result into Y :

Y ij := "i (F (u j )) =< (F (u j )) ," i > X . (7.6) Denote A oe R N red ◊N red any linear model fitting X to Y . The most common choice is to use the least squares approximations:

A = argmin BoeR N red ◊N red ÎBU ≠ Y Î 2 2
given by: A = YU + (7.7)

where • + denotes the Moore Penrose pseudo inverse.

There is only one result in the DMD literature on the computation of eigenfunctions of K. It links eigenvectors of A with eigenfunctions of K under very stringent conditions. Let  be an eigenfunction of K, and ⁄ its eigenvalue. As it is in span{ "i }, the set of observables, we have:

÷{-i }oeR N red ,Â= N red ÿ i=1 -i "i .

Link with EIM

Using as observable a small set of pointwise evaluations, say {u(x i ),uoeM} i can lead to think that this method is some kind of generalization of EIM, allowing for a larger choice of observable space O. This is not the case, as DMD linearly fits: {u(x i ),i oe [1,...,N red ]}ae{F (u)(x i ),ioe [1,...,N red ]}.

The power of EIM is that it does not reduce a non linearity to a linear (or locally linear) problem, as do DMD and LLE methods.

Conclusions on DMD

I believe that the DMD's theoretical analysis forgets that the eigenfunction of the Koopman operator, if any, are out of reach. My intuition is that the eigenvectors coming from the least squares approximation performed offline have nothing to do with eigenfunctions of the Koopman operator.

I have chosen to present this method in the big data section because it is often implied in the DMD literature that if you feed enough data, you will eventually find these Koopman eigenfunctions. My opinion on this method is that it is a 'best linear fit' approximation of a non linear model. In the literature, they advise using the DMD linear proxy only locally in X. DMD can then be assimilated to a local linearization, with all the limitations that this implies. We end this bonus chapter by peaking into the reduction of complexity as imagined by another community.

Machine learning

We have chosen to end this chapter by presenting a model free method developed in the machine learning community. Indeed, as stated many times in this manuscript, ROM and machine learning have a lot in common, in terms of methodology. Tabular 7.1 recalls the equivalence between notions. The strength of reduced order modeling is the strong theoretical background In this section, we follow the lines of [START_REF] Jeong | Datadriven fluid simulations using regression forests[END_REF]. Instead of working at a macroscopic scale on averaged quantities, we use a microscopic description and study the behavior of individual particles. From now on, we fix a set of N particles characterized by their positions and velocities.

't, {X k (t)=(x k (t),v k (t)),koe [1 ...N ]} .

How does a standard machine learning method goes ?

• find, in a training step, a correct set of features characterizing the state and environment of a particle

• in a training set still, find a good regressor. It returns the quantity of interest from the feature vectors

• online, for each evaluation, compute the feature vectors then use the regressor to classify/cluster/predict A first naive idea, is to use brute force, with no modeling whatsoever. Let k be one particle. Let J(k) be the index of some 'relevant' neighbors. Use as feature vector the relative positions and velocity of particles in J(k) to update the velocity and the speed of particle k. In compact form, we want to build a regressor that does the following:

{x j (t n ),v j (t n ),joe J(k)}ae ! x k (t n+1 ),v k (t n+1 ) " .
This specific choice of feature vector uses translation invariance, through the choice of 'local neighbors'. The main problem here is that with no modeling imposed, we are not using any inertia invariance property. This problem is too complex, and as a consequence, we go from 'no model' to 'rough model' still in the lines of [START_REF] Jeong | Datadriven fluid simulations using regression forests[END_REF]. The method chosen is derived from the Smoothed Particle Hydrodynamics (SPH) method, see for instance [START_REF] Leon | A numerical approach to the testing of the fission hypothesis[END_REF].

The features are taken as averaged quantities around each particle. For instance, let A be some quantity of interest. It is evaluated at each point x of Ω as:

A(x)= ÿ joeJ(x) m j fl j A j W (x ≠ x j )
where W is a smooth 'window' function, (m j ,fl j ) are respectively the mass and the density associated to particle j and J(x) is some neighborhood of x. The reason for this specific form is clear when you evaluate the density at x:

f(x)= ÿ joeJ(x)
m j W (x ≠ x j ). Now, let some particle X(t)=(x(t),v(t)). The time derivative of the velocity of this particle is given by d dt v (x(t),t)= ˆtv + v •Òv

We know that this time derivative will depend on viscous and pressure forces. The choice of features follows naturally:

• Φ visc ¥≠∆v(x)

• Φ pressure ¥Òp(x)

More advanced models could consider more forces such as surface tension for instance. The regressor in that situation is chosen as a function that estimates the acceleration of particle j from the local feature vectors. It is constructed, using some learning algorithm, during the training stage: ! Φ visc (x j ,t n ), Φ pressure (x j ,t n ) " ae a j (t n+1 )

The last thing to do in order to close the method is to choose an approach to update the full state vector: I ) v j (t n ),a j (t n ),a j (t n+1 ) * ae v j (t n+1 ) )

x j (t n ),v j (t n ),v j (t n+1 ),a j (t n ),a j (t n+1 ) * ae x j (t n+1 )

When using this 'model free' approach, we are of course far from solving the Navier-Stokes equation. As mentioned in the introduction, in this framework we loose all error estimations, as well as key physical properties such as incompressibility, positive pressure etc.

Conclusion

This concludes this short chapter. We have presented big data ideas, sometimes applied directly in the ROM framework, or at least that follow a similar methodology and computational cost reduction objective. We have also, by dissecting the DMD method, presented evidence on the fact that, when facing a challenging problem, feeding data to an empiric model is not (always) enough. Where EIM correctly preserves the non linear nature of a transformation and provides rigorous error estimates, the DMD does linearly fits, whatever the amount of data assimilated.

Chapter 8 Conclusion

Looking back at the topics discussed in this thesis, it seems, at first glance, like we have ended up pretty far from the initial objective stated in the introductory chapter. We take advantage of this concluding chapter to try once more to give some coherence to the work that has been presented, and to conclude on the next major steps towards the objective.

In the introductive chapter, I have highlighted the fact that a raw ROM approach, directly applied to the target problem, was doomed to fail. The first problem, the geometric variability, has been discussed in chapter 2 and in section 3.6. To solve it, we have introduced the notion of local solution manifolds and local Kolmogorov n-width. The offline section was pretty straightforward, and the focus has been put on the construction of an efficient matching method. The second problem is not specific to the target objective, and the issues raised can be transposed to the resolution of a wide variety of problems. The starting point was to notice that because of the profusion of ROM related methods, and the fact that they can be combined, and used one of top of the other, some of the literature tends to forget some of the key, mandatory, requirements for ROM. For instance, if the solution manifold has a large n-width, no POD/RB/EIM combo will ever give satisfactory results. We have discussed the illustrative case of a steep convection problem in chapters 1 and 3. We have shown how advanced numerical stabilization mechanisms for CFD problems also fall into this category. Thus, this issue needs to be dealt with to solve the target problem. We have proposed in the end of the introductive chapter two classes of methods. Chapters 3, 4 and 5 are devoted to the analysis of one of them, the so-called calibration.

Calibration is sometimes being criticized because of its supposed lack of robustness. From what I have understood, these comments are often made because calibration can be seen, at first glance, as a variation around the shock fitting method. Indeed, the calibration function can be seen as a 'best known shock position'. In that setting, the physically relevant speed, given by the RH condition, is replaced by an empiric optimization procedure. Of course, seen as such, this method does not seem very appealing. It suffers the same drawbacks as shock fitting (namely the difficulty to extend to higher dimensions), and is even worse because it uses a non physical shock speed.

In section 3.8, we propose an alternative interpretation: calibration as an h or p adaptive method, tailored for ROM. We try to show here why this is a more suited analogy than shock fitting, and answer at the same time some of the robustness concerns. First of all, the calibrated function does not need to capture the exact the position of the shock. Using a larger calibrated manifold (with space derivatives for instance) would allow for some error in the calibration function. Also, calibration can be used for more general problems than hyperbolic problems with shocks, see section 3.9.

Shed in that light, we have one common property of all the methods proposed in this thesis. They are a transposition of h or p adaptivity, that end up either in tailored domain decomposition, calibration, or both. Instead of refining the computational mesh or of increasing the degree of the (polynomial or Fourier) underlying basis, as usually performed in standard adaptive methods, we directly tailor a basis. This fits perfectly with the ROM framework, as the premise is that we have an a priori idea of the (local or global) solution manifolds at hand. We are not seeing the underlying computational mesh, but look directly for an adapted basis at a continuous level.

The important ingredient of h and p adaptive methods is the construction of a refinement function. In the most simple cases, the latter can be taken as the zones where the solutions encounters large variations. This can for instance be measured by zones with large gradients, or where a reconstructed Hessian has a large component. Other, more involved, methods require the computation of error estimators. As the output of say a FE method is the best one can say for a given computational mesh, and given trial space and test space, these error estimators necessarily involve subcell computations. The learning phase of the ROM framework helps us in that respect. To avoid any additional expensive, online, procedure, we rather precompute and store offline quantities that help estimate the quality of a mesh (or more precisely a basis) to represent a given solution manifold. The error estimator we have chosen in this thesis is the most natural one and is given by the best projection error. Other choices could be numerically investigated.

With this interpretation, the robustness issues that can be (and have been) raised for the methods developed in this thesis have their exact counterpart in standard adaptive methods, and thus, although relevant, these issues are certainly not sufficient to reject calibration as a numerically viable procedure. What is true though, is that more numerical experiments should be performed to assess the overall method. The intuition is that one should not perform adaptation (and thus should not update the calibration parameter) at each time step, as this implies numerical errors. There is a compromise to be found between reducing the n-width of the calibrated solution manifold, but not having to update the calibration parameter at each time step. One natural option would be to balance the estimation of the numerical errors due to the underlying mesh interpolation, with the best projection error on the reduced basis.

I now discuss a (pretty big) missing ingredient in this thesis: the lack of numerical experiments in some of the chapters. The domain decomposition method proposed in chapter 2 needs to be numerically investigated, and this has not been done in this manuscript. The reason is that I have tried to solve the matching problem and the stability issues inherent to CFD computations simultaneously. A more realistic next step is to apply this method to simpler, smoother problems. One evidence showing that this is a reasonable method is the fact that the ORBEM method shares many components with the Arlequin method [START_REF] Ben | Approches locales-globales méthode arlequin[END_REF], and that the latter has been successfully implemented on solid mechanics examples.

The optimal control method for solutions with shock proposed in chapter 5 also requires numerical investigation. Optimal control for the one dimensional Burgers equation is a well studied problem, and the comparison of the results obtained with the reduced method we propose is an objective within hand's reach. In the end of the chapter, we extend the method to the optimal design of an airfoil. There is still a long way to go before a complete framework and numerical experiments. We refer to the conclusion of the chapter for the unresolved issues. In any case, the roadmap has been set up and the scariest part, which is the resolution of an equivalent calibrated problem has been successfully tested in the numerical section of chapter 4.

1. 1

 1 One possible starting point for engineering models . . . . . . . . . . . . . . . . . 1.2 The chosen model problem for chapter 2 . . . . . . . . . . . . . . . . . . . . . . . 1.3 One possible empirical optimization algorithm: the surface method . . . . . . . . 1.4 Solid line: Graphical representation of the solution manifold M. Dotted line: A possible trial space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 One way of constructing a reduced basis: the Frechet differentiable case . . . . . 1.6 Localized Reduced Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Solid line: truth solution; Dotted line: Best projection onto some reduced basis; Dashed line: Output of a system with no proper stabilization . . . . . . . . . . . 1.8 Trajectories for the finite dimensional model problem . . . . . . . . . . . . . . . . 1.9 Analysis of the model problem. Left: projection onto the x ≠ y plane. Right: projection onto the x ≠ z plane . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 1

 1 Snapshots of the solution to the unsteady viscous Burger equation with u 0 = ⁄ + sin(x), ⁄ =1.3, ‹ =4, ' =0.04 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Calibrated set of the above snapshots for u 0 = ⁄ + sin(x), ‹ =4, ' =0.04 . . . . 3.3 Eigenvalues of the POD decomposition of the original set of snapshots ( in red) and of the calibrated set of snapshots (in green) . . . . . . . . . . . . . . . . . . . xiii 3.4 3rd (left) and 6th (right) POD modes for the calibrated (green) and original (red) simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 intro items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 A few values of the quantities (3.57) as a function of ∆".T h ex axis is scaled to multiples of c ú ∆t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 A few values of the quantities (3.57) as a function of ∆".T h ex axis is scaled to multiples of c ú ∆t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Relative L 2 -error of the solution as a function of time for different values of the reduced basis. The three curves close to the x-axis (almost overlapping at this scale) are the associated best approximation errors. . . . . . . . . . . . . . . . . 3.9 Relative L 2 best projection errors, on the training set . . . . . . . . . . . . . . . 3.10 Relative L 2 error of the reduced solutions, on the training set . . . . . . . . . . . 3.11 This method requires an overlapping decomposition of Ω . . . . . . . . . . . . . . 3.12 Snapshot sets at different steps of the offline stage. From left to right: snapshots with no calibration; centered snapshots; truncated snapshots . . . . . . . . . . . 3.13 The indicator function, that depends strongly on the solution manifold at hand . 3.14 Snapshots in the resulting "coarse" solution manifold M 0 . . . . . . . . . . . . . 3.15 Top left : projection of one snapshot, with no calibration, on a basis of cardinality 15; Top right : projection of the truncated snapshot onto an adapted basis; Bottom : projection of the complement on a small Fourier basis . . . . . . . . . . . . . . 3.16 On possible ouput of an offline calibration procedure such as algorithm 5 . . . . . 3.17 The output of the hierarchical clustering algorithm, for the snapshot set presentedFigure 3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Snapshots for various times, and various convection parameters c(µ) . . . . . . . 3.19 Reference mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Meshes on which the physical solutions u n =û n ¶ F n are defined . . . . . . . . . 3.21 Fine mesh on which the offline stage is performed . . . . . . . . . . . . . . . . . . 3.22 Left: A snapshot near the beginning of the simulation; Right: a snapshot near the end of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23 The truncated mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.24 Truncated versions of the solutions presented in Figure 3.22 . . . . . . . . . . . . 3.25 caption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26 Green: the 'true' inflow direction; Blue: the guessed angle . . . . . . . . . . . . . 4.1 Position of the shock for various AoA and Mach numbers. Coloured lines: Barycenters of the cells in which the shock is located; Black lines: Fitted line through these barycenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The solutions of the problem for AoA={0.0 ¶ , 1.0 ¶ , 2.0 ¶ , 3.0 ¶ } and Mach={0.81, 0.82, 0.83 } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The x velocity component at the wing in the uncalibrated case : a few POD basis 4.4 1st, 3th and 5th POD basis at the wing in the uncalibrated case for the full domain Ω111 4.5 Physical domain Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The reference domain Ω, and one possible instance Ω(µ):=F ≠1 µ ( Ω) . . . . . . . 4.7 The x velocity component at the wing in the calibrated case : a few POD basis . 4.8 1st, 3th and 5th POD basis in the calibrated case for the left subdomain . . . . . 4.9 Truth solution for velocity component with Mach=0.81 and AoA=3.0 ¶ . . . . . . 4.10 The identity mapping velocity component on a flat domain . . . . . . . . . . . . 4.11 The mapped solution for velocity component on a flat domain . . . . . . . . . . 4.12 Reference domain Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.13 Physical domain Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.14 Left: fi 3 in the original formulation, with homogeneous Neumann boundary condition; Right: fi 3 for a more suitable boundary condition . . . . . . . . . . . . . . 128 4.15 Modification of weights and projection functions to get smoother transitions on Γ1 and Γ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.16 The mapped solution for velocity component on a curved domain . . . . . . . . . 130 4.17 One of the entries of the Jacobian matrix, namely (J F ≠1 n ) 11 . Left: with no additional smoothing ingredients; Right: with some smoothing ingredients . . . . . . 131 4.18 Comparison of the outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.1 We restrict ourselves to solutions with at most one shock . . . . . . . . . . . . . 139 5.2 Away from the shock, the foot of the characteristics are close . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4 Ω ◊ [0,T] ◊U ad in the non calibrated case . . . . . . . . . . . . . . . . . . . . . . 143 5.5 Solving for p(x, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.6 Ω ◊ [0,T] ◊U ad in the calibrated case . . . . . . . . . . . . . . . . . . . . . . . . 147 5.7 Characteristics in the calibrated case . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.1 Divide and conquer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.1 Non linear operator acting on the solution manifold . . . . . . . . . . . . . . . . 168 7.2 The action of F , on the subspace, is simpler . . . . . . . . . . . . . . . . . . . . . 168

Figure 1 . 1 :

 11 Figure 1.1: One possible starting point for engineering models

Figure 1 . 2 :

 12 Figure 1.2: The chosen model problem for chapter 2

Figure 1 . 3 :

 13 Figure 1.3: One possible empirical optimization algorithm: the surface method

Figure 1 . 8 :Figure 1 . 9 :

 1819 Figure 1.8: Trajectories for the finite dimensional model problem

6 ΩFigure 2 . 1 :

 621 Figure 2.1: The chosen model problem. Here, N obs =6

Figure 2 . 6 :Figure 2 . 7 :

 2627 Figure 2.6: The chosen reference mesh. In red: Ωint ; in black: Ωtampon

Figure 2 . 8 :Figure 2 . 9 :

 2829 Figure 2.8: Three snapshots taken from M

Figure 2 . 10 :Figure 2 . 11 :

 210211 Figure 2.10: Possible generic subdomains Ωj

37 )Figure 3 . 1 :

 3731 Figure 3.1: Snapshots of the solution to the unsteady viscous Burger equation with u 0 = ⁄ + sin(x), ⁄ =1.3, ‹ =4, ' =0.04

  .1.

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Calibrated set of the above snapshots for u 0 = ⁄ + sin(x), ‹ =4, ' =0.04

Figure 3 . 4 :Figure 3 . 5 :

 3435 Figure 3.4: 3rd (left) and 6th (right) POD modes for the calibrated (green) and original (red) simulations

"

Figure 3 . 6 :

 36 Figure 3.6: A few values of the quantities (3.57) as a function of ∆".T h ex axis is scaled to multiples of c ú ∆t.

Figure 3 . 7 :

 37 Figure 3.7: A few values of the quantities (3.57) as a function of ∆".T h ex axis is scaled to multiples of c ú ∆t.

Figure 3 . 8 :

 38 Figure 3.8: Relative L 2 -error of the solution as a function of time for different values of the reduced basis. The three curves close to the x-axis (almost overlapping at this scale) are the associated best approximation errors.

Figure 3 .

 3 Figure 3.10: Relative L 2 error of the reduced solutions, on the training set

Figure 3 . 14 :Figure 3 . 15 :

 314315 Figure 3.14: Snapshots in the resulting "coarse" solution manifold M 0

  4, and is plotted on the top left picture. The top right figure shows a solution for c(x, y) ' ae 1+-y. Third figure shows a solution for c(x, y) ' ae 1+-(y ≠ ȳ) 2 ,w h e r eȳ denotes the mid height of the rectangle Ω. This results in solutions presenting various front shapes, and it is exactly fitted for the F t;µ introduced in equation (3.72).

Figure 3 . 18 :

 318 Figure 3.18: Snapshots for various times, and various convection parameters c(µ)

Figure 3 . 21 : 3 . 9 . 1

 321391 Figure 3.21: Fine mesh on which the offline stage is performed

Figure 3 . 22 :

 322 Figure 3.22: Left: A snapshot near the beginning of the simulation; Right: a snapshot near the end of the simulation

Figure 3 . 23 :

 323 Figure 3.23: The truncated mesh 3.22 on Ω int are presented in Figure 3.24.We have to calibrate the snapshots offline. For each snapshot u(•,t n ), the parameters F n can be chosen, as for the other examples, using some a priori knowledge. Here, it is the known inflow direction at time t n . A more realistic possibility is to use the algorithm proposed in section 3.7. The computational cost issues that were raised are still valid. It is even worse as we are dealing with a two dimensional problem. In order to make the method of Algorithm 5 computationally tractable, we propose a small variation. It relies on the use of filtering. For instance, let ÷ be some smooth indicator function of some small neighborhood of 0. One can smooth any function defined on Ω,s a yu, by looking at ū defined as:

Figure 3 . 24 :

 324 Figure 3.24: Truncated versions of the solutions presented in Figure 3.22

Figure 3 . 25 :

 325 Figure 3.25: Top: 2nd POD mode; Middle: 5th POD mode; Bottom: 10th POD mode Left: Uncalibrated case; Right: Calibrated case

Figure 3 . 26 :

 326 Figure 3.26: Green: the 'true' inflow direction; Blue: the guessed angle

Figure 4 . 1 :

 41 Figure 4.1: Position of the shock for various AoA and Mach numbers. Coloured lines: Barycenters of the cells in which the shock is located; Black lines: Fitted line through these barycenters

(4. 3 ) 2 .

 32 'T oe [1,...,N ] distribute the functions of Φ T to each node of T . Denote by Φ T i the local nodal residual for the node M i oe T . By construction one must have ÿ MioeT Φ T i =Φ T . (4.4)

  (a) Mach=0.81,AoA=0.0 ¶ (b) Mach=0.81,AoA=1.0 ¶ (c) Mach=0.81,AoA=2.0 ¶ (d) Mach=0.81,AoA=3.0 ¶ (e) Mach=0.82,AoA=0.0 ¶ (f) Mach=0.82,AoA=1.0 ¶ (g) Mach=0.82,AoA=2.0 ¶ (h) Mach=0.82,AoA=3.0 ¶ (i) Mach=0.83,AoA=0.0 ¶ (j) Mach=0.83,AoA=1.0 ¶ (k) Mach=0.83,AoA=2.0 ¶ (l) Mach=0.83,AoA=3.0 ¶

Figure 4 . 2 :Figure 4 . 3 :Figure 4 . 4 :

 424344 Figure 4.2: The solutions of the problem for AoA={0.0 ¶ , 1.0 ¶ , 2.0 ¶ , 3.0 ¶ } and Mach={0.81, 0.82, 0.83 }

Figure 4 . 5 :

 45 Figure 4.5: Physical domain Ω

Figure 4 . 6 :

 46 Figure 4.6: The reference domain Ω, and one possible instance Ω(µ):=F ≠1 µ ( Ω)

Figure 4 . 7 :

 47 Figure 4.7: The x velocity component at the wing in the calibrated case : a few POD basis

Figure 4 . 8 :

 48 Figure 4.8: 1st, 3th and 5th POD basis in the calibrated case for the left subdomain

Figure 4 . 9 :

 49 Figure 4.9: Truth solution for velocity component with Mach=0.81 and AoA=3.0 ¶

Figure 4 . 10 :

 410 Figure 4.10: The identity mapping velocity component on a flat domain solution u(•; µ) is known, we can compute its shock position and slope: a 0 (µ) and a 1 (µ).W e use the G-H mapping, see equation (4.12) on both ΩR and ΩL , and run the modified CFD code.Note that with this preliminary approach, the N T term is not updated at each pseudo time step. Figure4.11 shows the resulting solution, that we denote û(µ). One can observe that the general behavior is correct. The shock is more or less located at the correct position and it

Figure 4 . 11 :

 411 Figure 4.11: The mapped solution for velocity component on a flat domain

4 . 7 . 2 . 1

 4721 Figure 4.12: Reference domain Ω

Figure 4 . 13 :

 413 Figure 4.13: Physical domain Ω

Figure 4 .

 4 Figure 4.14: Left: fi 3 in the original formulation, with homogeneous Neumann boundary condition; Right: fi 3 for a more suitable boundary condition Towards the same objective, we do not want any stretching of the solution around the left boundary and close to the shock. Indeed, we need smooth transitions to neighboring domains.

Figure 4 .

 4 [START_REF] Jane Barthelmie | Modelling and measurements of power losses and turbulence intensity in wind turbine wakes at middelgrunden offshore wind farm[END_REF] for an hypothetical stretching of 4/3. The dashed red lines correspond to a non stretched mapping.

Figure 4 . 15 :

 415 Figure 4.15: Modification of weights and projection functions to get smoother transitions on Γ1 and Γ3

Figure 4 . 16 :

 416 Figure 4.16: The mapped solution for velocity component on a curved domain

Figure 4 . 17 :

 417 Figure 4.17: One of the entries of the Jacobian matrix, namely (J F ≠1 n ) 11 . Left: with no additional smoothing ingredients; Right: with some smoothing ingredients

Figure 4 . 18 :

 418 Figure 4.18: Comparison of the outputs

Figure 5 . 1 :

 51 Figure 5.1: We restrict ourselves to solutions with at most one shock

Figure 5 . 2 :

 52 Figure 5.2: Away from the shock, the foot of the characteristics are close

Figure 5 . 4 :

 54 Figure 5.4: Ω ◊ [0,T] ◊U ad in the non calibrated case

Figure 5 . 6 :

 56 Figure 5.6: Ω ◊ [0,T] ◊U ad in the calibrated case ‡(µ) are defined as: ‹ := (‹ x ,‹ t )=(1, 0).

  ˆxw s Ω◊U ad v 0 ˆµw = ≠ s Ω◊U ad ˆµv 0 w + s Σ0◊U ad ˆµ"[v 0 ]wWe regroup all volume contributions:

  s Σ ˆµˆt"[v]w s Σ ˆt"[ˆµv]w s Σ [vˆµv]wWe then just have to use: [vˆµv]=[v]ˆµv +[ˆµv]v, and the classical Rankine Hugoniot: v = ˆt". We are left with:ˆµˆt"[v]=[v]ˆµvwhich is the same as result as the one obtained by formally differentiating the RH condition. Summing up, ˆµv is solution of the following PDE:ˆtˆµv + ˆx(ˆµvv)+ˆµˆt"xv + ˆt"xˆµv =0 in Q ≠ fi Q + ˆµv(t = 0) = ˆµv 0 on Ω ◊U ad ˆµˆt" = ˆµvwhich is the desired result.

FFigure 7 . 1 :

 71 Figure 7.1: Non linear operator acting on the solution manifold

Figure 7 . 2 {ÂFigure 7 . 2 :

 7272 Figure 7.2: The action of F , on the subspace, is simpler

  ioe [1,...,N red ], 7.1. DMD or, in the spirit of EIM, as the evaluation of a function X at N red discretization points in Ω: O := ; X ae R u ' ae u(x k ),x k oe Ω,k oe [1,...,N red ].

  X} ,

	and for such that	" :	; ([0,T], D) aeF (t, µ) ' ae "(t; µ)
	1. the transformed solution manifold,
		M =	)	"(t; µ)

≠1

(u(•,t; µ)) ,µoeD,toe [0,T] * has a faster decaying n-width 2. for all µ, we can find a well posed equation satisfied by "(•; µ) ≠1 (u(•, •; µ)). This equation will be called 'calibrated equation'. A necessary condition is the smoothness of: 'µ, t ae "(t; µ) ≠1 .

  µ ' ae F t;µ so that the calibrated solution manifold is better behaved. For most of the examples we have studied (and we will study) in this thesis, an a priori knowledge is enough to select the F t;µ offline. In this section, we handle the situation when they are out of reach and propose an alternative, generic algorithm.Let Ξ be a representative snapshot set of M of cardinal say N snap .W e i n t r o d u c e t h e problematic by presenting in Algorithm 4 a simple greedy algorithm.

Data: Representative snapshot set Ξ of M D Result: Calibration parameters {F v , for v oe Ξ} V 0 := span u 0 ; k = 1; while k<N snap do 'F oeF,I (F ):=Îu k

  Representative snapshot set Ξ of M D Result: Calibration parameters F k for u k oe Ξ Let {Ξ p } be some decomposition of the original Ξ ; 'p, {F p,k } k Ω Algorithm 5({Ξ p } ); Let " p 0 ({F p,k } k ) be the first POD basis of the pth batch using the calibration parameter {F k,p } k ;

¶Fj ) ˆFj

Data:

  Note that even if the problem is two dimensional, the family F is a one parameter family. The definition of the calibration manifold M int,F ,D naturally follows:

	3	x y	4	ae Ω ' ae 3	sin(◊(t; µ)) cos(◊(t; µ)) cos(◊(t; µ)) ≠sin(◊(t; µ))	43	y x	4

  koe [1,...,N red ]}

	Result: Ωhyper ,N hyper Initialize Ωhyper := t ioeIini Êi ;
	repeat
	'û oe Ξ M F ,D , {-k (û),k oe [1,...,N red ]} := arginf {-k ,k oe[1,...,N red ]} I p Ωhyper (û) !) -k ,koe [1,...,N red ] *" ;
	i := argsup poeN Ωhyper := Ωhyper sup ûoeΞ M F ,D t Êi Î	q N red k=1 -k (û)" k ≠ ûÎ L p (ŵp) ;
	until convergence;

  on Γi ,t monotone and smooth

	fi i fi i ˆfii ˆn	=1 on Γi+1 =0 on Γi+2 . =0 on Γi≠1	(4.24)

  .2.

	t
					(x, t)
	t =0	Z	µ t,0	(x)Z	µ+"µ t,0	(x)	x

  Remark 43 This is the freezing method of section 3.2.2, in the simple translation case.As for Burgers equation, we can interpret this problem in terms of characteristics. Equation (5.11) can be written in what is usually called non conservative form as:

	ˆµv is solution of the following system:
	It is easy to derive the equation satisfied by v.W ek n o w :
	ˆtu(x, t; µ)=ˆtv(x + "(t; µ),t; µ)+ˆt"(t; µ)ˆxv(x + "(t; µ),t; µ)
	ˆxu(x, t; µ)=ˆxv(x + "(t; µ),t; µ)
	The calibrated Burgers equation is thus given by:
	ˆtv + ˆt"(t; µ) ˆxv +	ˆx 3	v 2 2	4	=0.	(5.11)
	t
	t = T
	t =0	" 0			x
	Figure 5.7: Characteristics in the calibrated case

ˆtv +(ˆt"(t; µ)+v) ˆxv =0.

Let µ oeU ad . Let X cal;µ be the solution of the following simple ODE:

ˆtX cal;µ (t)=v (X cal;µ (t),t; µ)+ˆt"(t; µ).

Until X cal;µ enters the shock, we have: v(X cal;µ (t),t; µ)=v(X cal;µ (0), 0; µ).

A simple illustration is presented in Figure

5

.7. As the original solution, v is constant on characteristics. The difference is that the latter are no longer are straight lines.

  " j } j ):= ÿ As already mentioned, minimizing J over all orthogonal basis in X is the usual POD algorithm, see the introductive chapter, section 1. The resulting optimal basis is in fact in span) u(•,t k ; µ p ),poe [1 ...P],koe [1 ...N p snap ] * .Here we rather look for a global basis {" tot j } embedded in span . This is natural in the divide and conquer strategy. We refer to Figure6.1 for an illustration of this process.

	p	N p snap ÿ k=1	. . . . . . u(•,t k ; µ p ) ≠	j=1 Ntot ÿ	+	u(•,t k ; µ p )," j	,	X " j	X . . . 2 . . .	.	(6.1)
						)	"
			{" tot j }
	{" µ0 i }	{" µ1 i }				{" µ P i }
	{u(•,t,µ 0 )}	{u(•,t,µ 1 )}				{u(•,t,µ P )}
	Figure 6.1: Divide and conquer
	Several different methods could be implemented here. We choose to look for {" tot j } as a weighted POD of ) "

µp i ,poe [1 ...P],ioe [1 ...N p red ] * µp i ,poe [1 ...P],ioe [1 ...N p red ]

  " µp u(•,t k ; µ p ) ≠ Π " stand Π " µp u(•,t k ; µ p ) ; µ p ) ≠ Π " stand u(•,t k ; µ p ) " µp Π " stand u(•,t k ; µ p ) ≠ Π " stand Π " µp u(•,t k ; µ p )

	Chapter 7	snap k=1	. . Π . . 2
		q p q AE 2 +2 p q N p q N p snap k=1 snap k=1 . . . Π " µp . Π . ! u(•,t ". . 2 . 2
	q j>Ntot ⁄ tot j (⁄) AE 2 AE 2	q n>Ntot ⁄ stand n q n>Ntot ⁄ stand n	+2 +4	q p q p	q N p snap k=1 q N p snap k=1	. . . 1 3 . Π ‹ " p Π ‹ " stand ≠ Π ‹ " stand Π ‹ " p . .Π ‹ " stand u(•,t k ; µ p ) . . . 2 + . . .Π ‹ 2 u(•,t k ; µ p ) . . 2 . . . " p u(•,t k ; µ p ) .

k

Table 7 .

 7 1: Equivalence between ROM and machine learning that ensures among other things, stability results and rigorous error estimations.

	Community	Reduced Order Modeling	Machine Learning
	Stages	Offline Online	Training stage Prediction/Classification
	Offline	Reduced basis construction	Feature selection
	Online	Reduced basis Reduced Scheme	Feature vector Regressor
	Error estimation	A priori error estimation A posteriori error estimation	Validation set ??

This a translation of Préambule.

Big-data approaches to solve CFD problems will be discussed in chapter 7.

One sometimes considers a space time formulation, which results in a different solution manifold, see for instance[START_REF] Billaud | Dynamical model reduction method for solving parameter-dependent dynamical systems[END_REF].

At a b u l a re x p l i c i t i n go t h e rs i m i l a r i t i e sb e t w e e nR O Ma n dm a c h i n el e a r n i n ga l g o r i t h m si sp r e s e n t e di n chapter 7.

Recall that the reference high cost, is the cost of a fine computation, with N degrees of freedom.

The only example using a different sampling strategy we are aware of is presented in[START_REF] Kunisch | Optimal snapshot location for computing pod basis functions[END_REF].

We note that POD uses the Hilb ert space structure, where RB can b e p erformed in a any Banach space.

Other choices such as the Petrov-Galerkin method can also be used in a ROM context.

We insist that for problems where the sho ck/front's p osition is constant in time or parameter wise, the current discussion does not apply, and the n-width of the solution manifold is not an issue.

In the bonus chapter 7, we present one such method, namely Dynamic Mode Decomposition.

Note that this non linearity can originate from the continuous differential form or from the numerical scheme.

We have seen in section 1.4 that Galerkin gives a priori estimates in X norm, independent of the size of the reduced space.

High order means here more than 2nd order

These results depend of course of the way you define the energy transfer between modes.

Note that the instability of the toy example discussed in this section is not of the same nature as the one of aG a l e r k i nm e t h o df o rN a v i e r -S t o k e se q u a t i o n . T h es y s t e md e s c r i b e dh e r ei sn o tl i n e a r l ys t a b l e .

An interesting open question is whether this concept of 'underlying truth solution, independent of the stabilization mechanism', can be interpreted in terms of determining forms, see[START_REF] Foias | A determining form for the two-dimensional navier-stokes equations: The fourier modes case[END_REF]

L[START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF]

Note that, of course, this choice of an explicit scheme involves a limitation on the time step due to a CFL condition that can be severe for an accurate finite element or finite difference scheme but reveals to be moderate in the reduced basis framework.

Indeed there is no reason why using the same discretization in time for the truth solution and for the reduced basis scheme

This work is the result of a collaboration with R. Crisovan and R. Abgrall from UZH Zurich. An article version of this chapter is available, see[START_REF] Cagniart | Model order reduction for hyperbolic problems: a new framework[END_REF]. Some of the numerical experiments of this chapter have been done using Freefem++[START_REF] Hecht | New development in freefem++[END_REF].

Note that we have no results on the existence nor on the number of eigenfunctions of K.

Remerciements 3.9. Rotating obstacle IsoValue -0.304438 -0.274152 -0.243867 -0.213581 -0.183296 -0.15301 -0.122725 -0.0924396 -0.0621541 -0.0318687 -0.00158323 0.0287022 0.0589877 0.0892731 0.119559 0.149844 0.180129 0.210415 0.2407 0.270986 0.301271 0.331557 0.361842 0.392128 0.422413 0.452698 0.482984 0.513269 0.543555 0.57384 0.604126 IsoValue -0.304438 -0.274152 -0.243867 -0.213581 -0.183296 -0.15301 -0.122725 -0.0924396 -0.0621541 -0.0318687 -0.00158323 0.0287022 0.0589877 0.0892731 0.119559 0.149844 0.180129 0.210415 0.2407 0.270986 0.301271 0.331557 0.361842 0.392128 0.422413 0.452698 0.482984 0.513269 0.543555 0.57384 0.604126 IsoValue -0.304438 -0.274152 -0.243867 -0.213581 -0.183296 -0.15301 -0.122725 -0.0924396 -0.0621541 -0.0318687 -0.00158323 0.0287022 0.0589877 0.0892731 0.119559 0.149844 0.180129 0.210415 0.2407 0.270986 0.301271 0.331557 0.361842 0.392128 0.422413 0.452698 0.482984 0.513269 0.543555 0.57384 0.604126 IsoValue -0.304438 -0.274152 -0.243867 -0.213581 -0.183296 -0.15301 -0.122725 -0.0924396 -0.0621541 -0.0318687 -0.00158323 0.0287022 0.0589877 0.0892731 0.119559 0.149844 0.180129 0.210415 0.2407 0.270986 0.301271 0.331557 0.361842 0.392128 0.422413 0.452698 0.482984 0.513269 0.543555 0.57384 0.604126 IsoValue -0.304438 -0.274152 -0.243867 -0.213581 -0.183296 -0.15301 -0.122725 -0.0924396 -0.0621541 -0.0318687 -0.00158323 0.0287022 0.0589877 0.0892731 0.119559 0.149844 0.180129 0.210415 0.2407 0.270986 0.301271 0.331557 0.361842 0.392128 0.422413 0.452698 0.482984 0.513269 0.543555 0.57384 0.604126 IsoValue -0.304438 -0.274152 -0.243867 -0.213581 -0.183296 -0.15301 -0.122725 -0.0924396 -0.0621541 -0.0318687 -0.00158323 0.0287022 0.0589877 0.0892731 0.119559 0.149844 0.180129 0.210415 0.2407 0.270986 0.301271 0.331557 0.361842 0.392128 0.422413 0.452698 0.482984 0.513269 0.543555 0.57384 0.604126

Cette thèse s'est effectuée au laboratoire Jacques Louis Lions (

Appendix

Let Ξ be some proper sampling of D, of cardinality N snap . The objective is to extend the proof of the existence of an optimal POD basis to the case X = L 2 (Ω).D e fi n er as: r :

; Ω ◊ Ω ae R (x, y) ' ae q N snap k=1 u(x; µ k )u(y; µ k )

The optimal basis, if it exists, is given by eigen functions of the following operator : R :

; X ae X Â ' ae x ' ae ( s Ω Â(y)r(x, y)dy). The existence of a complete eigenfunction set of R is a consequence of the Hilbert-Schmidt theorem, see for instance [START_REF] Renardy | An introduction to partial differential equations[END_REF]. The first thing is to prove that the image of R is included in L 2 (Ω). We then need to show that R is a bounded, self-adjoint and compact operator L 2 (Ω) ae L 2 (Ω).

Plugging this into (3.70), we have:

We can then express the gradient of our functional J as

All of the previous derivatives with respect components of F oeFexist if and only if the solutions in the original solution manifold have a smooth dependence on F .

We will explicit it in the case of translations, in the periodic one dimensional case. F is the family of translations, with parameter " oe Ω.

So the previous derivation is rigorous if and only if the original solution manifold M is embedded in C 1 (Ω). For discontinuous solution, one can use a similar algorithm, but restricting the computations to Ω d a subdomain of Ω away from the discontinuity. This idea is used in chapter 4 of this manuscript.

Algorithm

We are looking for a minimum of the functional J M . A standard quasi Newton algorithm goes a follows: Data: Representative snapshot set Ξ of M D Result: Calibration parameters F k for u k oe Ξ Let {F 0 k } k oeF N snap be an initial guess ; n := 0 ; repeat Compute ⁄({F n k }) and "({F n k }); Compute ÒJ({F n k }) using equation (3.71); Approximate the Hessian B, using BFGS for instance ; "F := ≠B ≠1 ÒJ({F n k }); 'k, F n+1 k+1 Ω line search (F n k ,"F); n Ω n +1; until some computational/accuracy threshold;

Algorithm 5: Finding offline calibration parameters This method is computationally very expensive. We propose a method to mitigate this issue. It is a divide a conquer type of method. We first decompose our snapshot set Ξ into smaller batches {Ξ p }. We can then perform the previous algorithm onto each batch independently. The fusion step can be done using the same algorithm.

L 1 minimization

The objective functional is here given by:

Once again, a standard change of variable gives:

In each control volume i, we choose a set of N quad i quadrature points {x i,j ,joe [1,...,N quad i ]} and the corresponding weights {" i,j ,joe [1,...,N quad i ]}.W eh a v e :

This is handled as in [START_REF] Abgrall | Robust model reduction by L1norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems[END_REF] by recasting it as a linear programming problem. For now, the size of the problem is of order N , the number of control volumes of the mesh. We will see in section 4.6.3 how to reduce the computational cost.

Remark 38

The upper bound on t suggested by the equation (5.6) is the same as the one appearing when discussing the breakdown of classical solutions.

We go back to the initial objective: the smoothness of

for classical solutions. We use the fact that the solutions considered are constant on the characteristics:

) ≠ u 0,µ (Z µ t,0 , 0) Once again, we decompose the right hand side into two quantities:

) ≠ u 0,µ+"µ (Z µ t,0 ) the foot of the two characteristics are close u 0,µ+"µ (Z µ t,0 ) ≠ u 0,µ (Z µ t,0 ) the transported informations are close

We use the previous result on the smoothness of µ ae Z µ t,0 :

We conclude that for fixed (x, t), µ ae u(x, t; µ) is smooth. By inspecting the actual form of the derivative, we conclude that µ ' ae u(•,

The previous analysis has been done for classical solutions. We provide formal arugments to show that this result extends to solutions with shock, but away from the shock. Let (x, t) oe Ω ◊ [0,T] away from the shock for parameter µ, i.e such that x " = "(t; µ). As (t, µ) ' ae "(t; µ) is a smooth function, we know that there is a neighborhood of µ,s a yV(µ), such that:

The argument illustrated in Figure 5.3 then applies just as for classical solutions. We conclude that:

is smooth, for solutions satisfying the hypothesis described in the begining of section 5.2.

Control with no calibration

We need to introduce new notations before going further in the analysis. For all µ, denote ‡(µ) the locus of the {("(t; µ),t oe [0,T]}. It is a smooth curve. Let Ω ≠ (µ) and Ω + (µ) be a non overlapping cover of Ω such that ‡(µ) is one of their common boundaries. For instance, for Ω=(0, 1), one can take

and

We also need the following notations:

Away from the shock (i.e where v is smooth), we have:

(5.15) Thus, away from the shock:

A wrong calibration means a bigger component onto the derivative of the underlying ROM basis. We are in the proper setting to propose a method with the desired properties. Let { Φi } i be some orthogonal basis. We choose it such that:

; q i q j < Φi , Φ j > 2 not too small Φi " oe span {ˆxΦ j }, away from the shock .

(5.16)

The first property ensures that { Φi } captures enough of the calibrated solution manifold. The quality of the estimation on ˆµˆt" will depend on the second property.

A thorough investigation of the construction of the basis { Φi } is out of the scope of this chapter. We just mention the simplest method we can think of. One could use some variant of the Gram-Schmidt orthogonalization procedure on the basis {Φ j } to remove the components parallel to span {ˆxΦ j }. We have no guarantee that the resulting basis would still satisfy the first property of (5.16).

Remark 46 A rough offline calibration procedure makes the construction of { Φi } complicated, as the x derivatives are in the basis.

We now assume the existence of such a basis. We have everything we need to construct an algorithm similar to the one used in chapter 3. Start by computing (ˆµv) n+1 for several guesses for ˆµˆt" and compute the corresponding norm of the orthogonal projection onto { Φi }.T h e n interpolate to get an optimal estimate of ˆµˆt". More precisely, we are choosing " as

where Π Φ denotes the orthogonal projection onto the basis ) Φi * , and Ω sub is some domain away from the shock. This idea of restricting the physical domain before an optimization procedure has already been used in chapter 4.

Remark 47

This method, unlike R-H conditions, works for solutions where the shock has not yet appear. We just have to define (t; µ) ae "(t; µ) as the line of highest gradient that eventually forms a shock. This ensures the required smoothness in µ and t.

On the computation of ∂ µ v

The last objective is to propose a well behaved numerical method to solve equation (5.12) at time t n+1 , for a fixed value of ˆµˆt". For simplicity, one can think of an explicit Euler time discretization:

The purpose of the following sections is to try to extend the properties of a scheme used to compute v, to the computation of ˆµv. This is the most natural choice in order to have a

• the reconstruction process does not create new local maximas is a TVD scheme.

The most common gradient limitation procedure is the usage of a modified gradient:

where Ò i is a standard gradient reconstruction, ENO or LS for instance and where the limiter " is chosen as a function of the ratio of forward and backward differences:

Many different choices are possible for ". We will restrict ourselves to the smooth ones that guarantee the TVD property. One example is the Van-Leer limitor given by: " : R ae

We denote by Ò some standard gradient reconstruction process. In control volume i:

We denote by Ò Õ i the derivative of Ò i with respect to µ,w h e ni te x i s t s :

For central differences, least squares (or any process independent of the solution), we know that this application is smooth. For instance, for central difference, the reconstructed gradient is chosen as:

2

. This application is trivially smooth, and the derivative is given by:

For ENO, WENO and other scheme more involved (with a notion of direction) the smoothness of the gradient reconstruction process is not guaranteed in the general case. We need stronger hypothesis, that will be satisfied because of the calibration process. These hypothesis are:

• the local maximas do not move with the parameters

• the direction of the flow, in the calibrated domain, does not change with the parameters A direct consequence of the first point, is that the limiting process is smooth, even at local maximas. A direct consequence of the second point is that ENO type gradient reconstructions will be smooth with respect to the parameter. We now have Ò a smooth application. Let's show " is also a smooth function of the parameter:

Away from local extremums, we have:

First, the offline phase Data: Continuous collection of wing shapes Result: Calibrated basis Select a moderate number of wing shapes ; Compute the solutions using a fine solver ; Restrict the parameter range so that all considered candidates satisfy the hypothesis of section 5.4.1 ; Pick some reference wing shape, as in chapter 4 ; Map the fine solutions to the reference mesh ; Store scheme informations, such as upwinding direction or gradient limitation ;

Algorithm 9: Offline Then, the online phase Data: Continuous collection of wing shapes, calibrated reduced basis Result: Optimal wing shape Set up, as in 4, the fine solver with modified flux boundary conditions, gradient limitations etc. ; Start the optimization algorithm, using the method developed in this chapter; The output is some 'optimal mapping' ; Deduce an optimal wing shape; Algorithm 10: Online

Conclusion

In this chapter, we have constructed a method to solve optimal control problems in the context of solutions with shocks. With the proper calibration and an appropriate choice of objective function, we have shown that we recover the smoothness of µ ae J(µ) which is at the core of any optimal control problem. Using toy examples, we have shown that some specific care had to be put in the construction of numerical schemes for the computation of the derivatives of the solutions (calibrated and non calibrated). We have shown how to use calibration to extend the stability properties of standard numerical scheme to the computation of the derivatives. Another objective of this chapter was to use ROM. This task is completely solved by calibration, see section 5.3.5. We have adapted the ideas of chapter 3 and 4 to propose an alternative of the R-H conditions for the computation of ˆµ".

All the ideas developed in this chapter need to be numerically studied. The first step is to check the premise for an utilization of ROM, see equation (5.13). It could for instance be done in the context of the numerical section in chapter 3. Also, the method proposed to estimate ˆµˆt" has to be investigated. It relies on the construction of what we have denoted { Φi }. Finally we need to test using the characteristics of the fine solver used for v to compute ˆµv.

The ultimate application of what has been done in chapter 4 and this one is to perform wing shape optimization, such as described in section 5.4.5. There is still a long way to go, as all ingredients have only been skimmed over. Nevertheless, I believe that this is an adapted framework to solve this real-life problem.

Appendix

We will prove the result given in equation (5.12). The proof is just a variation of the one given in the calibrated case in [START_REF] Bardos | Derivatives and control in the presence of shocks[END_REF]. ˆµv is a weak solution if and only if:

6.1. A posteriori error bound operator R(-) defined as: R(-):

and that we have the well known POD property:

Remark 48 Of course, we do not have the standard ROM property between the decay of the {⁄ tot j } j and the global error. This is actually the purpose of this note. We will prove two things:

• we can control the global error by a combination of the local 'in batch' errors and of the error made when 'combining' the PODs

• the loss incurred by this divide an conquer strategy is controlled: there exists a constant C (data set independent) such that

For the sake of readability, we will from now on denote:

• {Π " tot (-) } the orthogonal projection onto {" tot j (-),joe [1 ...N tot ]}

• the projections onto orthogonal sets will be denoted Π ‹

A posteriori error bound

We know that the operator norm of Π " tot is smaller than 1.W et h u sh a v e :

. depends only on the in batch approximations, we have:

To handle the second term, we consider the following dataset:

Let R Ξproj be the standard POD operator, defined in the introductive chapter 1, applied to the previous dataset: R Ξproj :

For (p, k), denote {x p k,i } i the coordinates of u(•,t k ; µ p ) onto the basis {"

We use another key property of the POD method:

The proof of this well known result is recalled in the appendix. We thus have:

This is the same operator as R(⁄) defined equation ( 6.2) with the following choice of weights:

Denote

) " tot j (⁄) * and {⁄ tot (⁄)} the corresponding eigenvector and eigenvalue sets. We use standard property:

and thus reach the first desired result:

We can now safely use the two level POD, as we have a rigorous upper bound on the global error. One possible algorithm is presented below:

Data: Fine discretization Ξ of some solution manifold M, ' some threshold Result: Reduced basis such that

q j>Ntot ⁄ tot j (⁄) AE ' 2 ; Algorithm 11: Performing two level POD

Comparison with the standard POD

We now turn to the second objective. We want to guarantee that for a fixed accuracy, the basis resulting from a two level POD is not much bigger than the one resulting from a standard POD. We fix N tot . In this section, we restrict ourselves to the case N p red Ø N tot .I n d e e d ,t oh a v ea rigorous bound, we need to be able to handle the case P =1, and the case with identical batches. We will denote: Π " stand the orthogonal projection onto {" stand j ,joe [1 ...N tot ]}. We have one easy inequality:

This uses the standard POD equality and the fact that for a fixed size, the spaces chosen on each subbasis are better adapted than a global one as well as N p red Ø N tot . The natural follow up is to compare {⁄ stand } and {⁄ tot (⁄)}. Recall that ⁄ tot (⁄) is the output of a standard POD algorithm applied to the dataset Ξ proj defined in (6.3). Intuitively, Ξ proj should be 'smaller' than the original dataset, so we expect the {⁄ tot (⁄)} to be not far from the original {⁄ stand }.

Following this initial remark, we start with a simpler problem. Let Ξ be any dataset in X and let {Â n } n be some orthogonal basis in X. We want to compare the decay of the eigenvalues of the POD algorithm applied to Ξ versus the one applied to {Π Â v, v oe Ξ}. Denote

• {" orig } and {⁄ orig } the eigenvector/eigenvalues sets for the original dataset • {" proj } and {⁄ proj } the eigenvector/eigenvalues sets of the projected dataset • Π the projection on the first N basis in both original and projected cases We know that:

The optimal property of the basis {" proj } gives:

Using the fact that the orthogonal projector has unit norm, we have

As {" proj } is the POD basis of the projected set, we know that " proj oe span {Â n }, and thus that:

We have once again used N p red Ø N tot . As a conclusion, we went from

For manifolds for which the standard POD gives quickly decaying eigenvalues, the size of the final basis with a two level POD method is guaranteed not to be much bigger than the one obtained with the standard POD algorithm.

Computational cost

What have we gained ? For simplicity, let's say that each batch has size What is the tradeoff then ? The bigger P , the cheaper the sub POD basis computations. But in order to have decent accuracy, one must take enough basis at the recombination step. Typically, one must take for each batch the same number of basis as in the final basis, see section 6.2.

Hypothetical situation. We have 20 parameters. Each time simulation consists in 300 time steps, and we are expecting a final basis of size 40. Using standard POD costs (20 ú 300) 2 N . This double POD costs (20 ú 300 2 + (40 ú 20) 2 )N . We have reduced the total computational cost by 20.

Appendix

Let Ξ p be one batch and R Ξ P the standard POD operator on this set: R Ξp :

Using the fact that {" µp i } is a set of eigenfunctions of R Ξp and is orthogonal, we have:

Denote for all triplet (i, k, p), x p k,i :=<u(•,t k ; µ p )," µp i > X .W eh a v et h ed e s i r e dr e s u l t :

x p k,i x p k,j = ⁄ p i " i,j .

DMD

Using the linearity of the Koopman operator, we know:

Let

• u k oe X a solution in the training set (used to build A)

• U k the N red dimensional vector

where r k oe R N red is residual of the linear fit. We can think of it as the least square residual. Now,

The first equality comes from equation (7.8). Second equality comes from equation (7.9). The last equality comes from the fact that  is an eigenfunction of K. If the residual of the least squares approximation vanishes, we have:

Equation (7.11) is equivalent to:

-T AU = ⁄-T U.

We are ready to state the result: if  is an eigenfunction of K such that:

• Â is in the span of the observables

• the linear fit (say least squares approximation) is exact on the training set {u k } k

• U is of rank N red . This is equivalent to UU T invertible, as we expect M ∫ N red .T h e underlying idea is that {u k } needs to be a big enough sample then  can can be found by computing the eigenvectors of A T . More precisely, there existseigenvector of A T such that

Before moving on to a simple illustrative example, we discuss the hypotheses made in this section. First of all, as the residual will most likely not vanish, we have no result guaranteeing that the error made on the approximate eigenfunction is controlled by the residual error. More importantly, we have no proof of the existence of eigenfunctions of K in the space spanned by the observables, for a general function F : X ae X.

Discrete system

The theory of Koopman operator was first developed for discrete systems. One two dimensional system example is developed in [START_REF] Rowley | Connections between koopman and dynamic mode decomposition[END_REF]. The model non linearity is chosen as:

for some ⁄, µ and c. Suppose that we take z 1 , z 2 and z 2 1 as observables. Define " 1 and " 2 as:

(7.13)

Easy computations show that both these functions are eigenfunctions of the Koopman operator:

The associated eigenvalues are ⁄ and µ respectively. In this carefully tailored example, " 1 and " 2 are the desired Koopman eigenfunctions in the span of the observables. Unfortunately, even in this toy example, we do not really solve the computational cost issue, as the evaluation of

still requires the evaluation of z 1 , z 2 and z 2 1 . We now go back to a more reasonable example, using the usual reduced order modeling framework.

A more realistic example

Let M be some continuous solution manifold to some PDE, embedded in a well chosen Hilbert space X. An eigenfunction  in the span of the observables satisfies:

I

÷{i } i , s.t  = q N red i=1i "i 'u oeM, KÂ(u)=⁄Â(u)= ¶ F (u).

When O is the space of scalar products with a reduced basis, this condition becomes:

÷⁄, ÷{i }oeR N red , s.t 'u oeM,<⁄ u≠ F (u), .14) Recall that in order to have a proper offline/online decomposition, we need enough of such eigenfunctions. We can state a necessary and sufficient condition: there exists N oe N, ) Â j oe span O,j oe [1,...,N red ] * and {⁄ j oe R,j oe [1,...,N]} such that: 'u oeM, ;

'j oe [1,...,N],< F (u), j > = ⁄ j <u, j > 'j>N, <F(u), j > =0 . (7.15) This is a very stringent condition, and we cannot expect it to be satisfied for general F .Iw i l l try to make my point a little clearer: in my opinion, the DMD method does not constitute a proof of the validity of the linear model. It is merely a way of finding the components with a 'close to linear' behavior in a non linear system.

We can now conclude on what's left to be done in order to solve the initial objective. We split its resolution into two disjoint components:

• a robust local reduced model around one wind turbine. For this, we can either transpose the stability and accuracy features of a fine solver to a ROM model: this will most likely involve some form of calibration. The other option is to construct a self sufficient reduced scheme, with a tailored stabilization mechanism, see section 1.6.1.

• a robust and accurate matching method. As discussed in chapter 2, it should allow for independent rotations in each domains and translations between domains Unfortunately, this thesis has not lead to firm conclusions on either of these aspects. Nevertheless, I believe that some of the ideas and methods discussed have paved the way to a new class of ROM methods, that could be used to solve the target problem, and more generally to solve problems that were previously out of reach.