L Cordier

B Abou

El Majd

J Favier

Calibration of POD Reduced-Order Models using Tikhonov regularization

Keywords: POD Reduced Order Model, Calibration, Tikhonov regularization, Optimization

. Finally, we show that for this flow configuration this procedure is the most effective in terms of reduction of errors.

INTRODUCTION

Within a relatively short period of a few years, reduced-order modeling by Proper Orthogonal Decomposition (POD) has gained a significant importance in fluid mechanics and turbulence. This sudden increase of interest is mainly linked to recent developments of technological facilities (computers, data acquisition systems) which now allow easier and faster data recording, ready-to-use for the POD (space-time correlations). However, in the majority of the applications, the studies are not carried out up to a complete and accurate representation of the flow dynamics by POD, even if the procedure is now well-known. The latter requires a Galerkin projection of the governing equations (in general the Navier-Stokes equations) onto the spatial POD functions, to obtain after truncation in the POD subspace, a set of ordinary differential equations representing the dynamics of the original system. Then, how can we explain that the most promising approaches in terms of applications: optimization and optimal control problems, real-time flow control, parametric studies, continuation methods,. . . are hardly used in the literature? The major barrier to the expansion of POD Galerkin approach is essentially the lack of accuracy of the reduced-order models obtained. Indeed, it is often difficult to represent with a sufficient accuracy, the dynamics at short times of the original system with the parameters of control used to determine the POD basis. In these conditions, what can be expected from the ability of POD-based reduced-order models to represent the dynamics of the system at long times (bifurcation studies, for instance) or to predict the dynamics of the system for parameters of control different from those used to determine the POD basis (flow control applications)? Recently, various studies [START_REF] Galletti | Low-order modelling of laminar flow regimes past a confined square cylinder[END_REF][START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF][START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF][START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF][START_REF] Galletti | Accurate model reduction of transient and forced wakes[END_REF][START_REF] Kalb | An intrinsic stabilization scheme for proper orthogonal decomposition based lowdimensional models[END_REF], presenting numerical methods termed as calibration, appeared in the literature in order to improve the accuracy of POD-based reduced-order models thanks to solutions of optimization problems. The idea is simple, since the temporal dynamics of the POD model is known in advance, it is possible to use this information to correct whole or part of the coefficients issued from POD Galerkin. However, according to our knowledge of the literature, there is no trace of any comparative study of these methods. It is then difficult to claim which is the most effective method in terms of reduction of errors. The initial objective of this paper is to use a bench mark configuration, simple from the point of view of dynamics, to analyze thoroughly these various approaches, and, if possible, to release a more effective strategy. Incidentally, this approach led us to extend the method of calibration based on Tikhonov regularization [START_REF] Tikhonov | Solutions of ill posed problems[END_REF] recently used in [START_REF] Weller | Robust model identification of actuated vortex wakes[END_REF] to the first and second-order [START_REF] Alekseev | The Analysis of an Ill-Posed Problem Using Multiscale Resolution and Second Order Adjoint Techniques[END_REF].

This manuscript is organized as follows. Section 2.1 introduces the Proper Orthogonal Decomposition in the general context of approximation functions. Then, the POD Galerkin approach is described for a generic controlled flow configuration (section 2.2.1). In the next subsection (section 2.2.2), the equations of the reduced-order model based on POD are simplified in order to give an overall picture of the methods of calibration. In section 2.3, the bench mark flow configuration used in this study is first described, then the necessity of introducing methods of calibration to improve the accuracy of the POD model is motivated. Section 3 is dedicated to the presentation of the existing methods of calibration, within an unified framework. The various choices of definitions of errors between the calibrated dynamics and that of the original system are first presented in section 3.1. Then, we show in section 3.2 that in the case of affine functions of errors, the minimization of the normalized errors leads to the resolution of a linear system. To conclude section 3, the method of calibration suggested in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] is presented in section 3.3. The various methods of calibration used so far in the literature are then compared on the bench mark configuration in section 4. Finally, we introduce in section 5 the method of calibration based on the Tikhonov regularization, which we have developed in this paper, and present the results of a comparative study to determine the most effective method of calibration.

REDUCED-ORDER MODELING BASED ON A POD GALERKIN APPROACH

Proper Orthogonal Decomposition

Given a set of data, elements of a high dimensional space (potentially infinite), the main idea of the Proper Orthogonal Decomposition (POD) is to determine a subspace of reduced dimension which is optimal in the sense that the error of projection on this subspace is minimal. To formulate this statement more precisely, we introduce H a Hilbert space with inner product (•, •) H and induced norm • H . In the majority of the applications, and it will be our case besides, H corresponds to L 2 (Ω), with Ω the spatial domain. Let U = {u(x, t m ) = u m } m=1,...,Nt be a set of N t snapshots taken over a time interval [0, T ], with x ∈ Ω. Then, the aim of the POD is to find a subspace S of dimension N POD ≪ N t , such that the error E ( u -P S u H ) is minimized † . Here, E (•) denotes an average operator over m, for instance an ensemble average (E (u) = 1 Nt Nt m=1 u m ), and P S the orthogonal projection onto S. The procedure is then equivalent to minimize the expression

1 N t Nt m=1 u m -P S u m 2 H = 1 N t Nt m=1 u m - N POD j=1 a P j (t m )Φ j 2 H
, where {Φ j } j=1,...,N POD is a basis for the subspace S and {a P j } j=1,...,N POD refer to temporal coefficients corresponding to the POD expansion (as indicated by the superscript P ). It can be shown ([10] or [START_REF] Cordier | Proper Orthogonal Decomposition: an overview[END_REF] for instance) that this minimization problem leads to the eigenvalue problem:

RΦ j = λ j Φ j j = 1, • • • , N POD , where R = E (u ⊗ u * ).
Here, ⊗ denotes the dyadic product between two vectors u and u * where the * superscript indicates complex conjugate. Since R is linear, self-adjoint and positive semi-definite on H, the spectral theory applies and guarantees that:

1. we may choose Φ j to be orthonormal, i.e. a P j (t m ) = (u m , Φ j ), 2. we have

λ 1 ≥ λ 2 ≥ • • • ≥ λ N POD ≥ 0 where λ j (j = 1, • • • , N POD ) are the largest N POD eigenvalues of R.
This approach is called direct method and corresponds to the formulation introduced originally in [START_REF] Lumley | Atmospheric Turbulence and Wave Propagation. The structure of inhomogeneous turbulence[END_REF]. However, when the input data come from numerical simulations, it is much more efficient to use an alternate way of computing the POD eigenfunctions. This method, known as method of snapshots [START_REF] Sirovich | Turbulence and the dynamics of coherent structures[END_REF], consists of writing the POD modes as linear combinations of the snapshots:

Φ j (x) = Nt m=1 b m j u m (x).

The vectors b

j = b 1 j , • • • , b Nt j T
are then determined as the solutions of a new eigenvalue problem given by:

Cb j = λ j b j ,
where C is a N t × N t correlation matrix with C ij = 1 Nt u i , u j . This matrix is self-adjoint, as for the direct method. It follows that the vectors b j are orthogonal with respect to the inner product defined by:

(b j , b k ) T = 1 N t Nt m=1 b m j b m k . † Remind that for any orthogonal projection P S we have u 2 H = u -P S u 2 H + P S u 2 H . Then, minimizing E u -P S u 2 H is equivalent to maximizing E P S u 2 H .
By extension, the inner product associated with the direct method will henceforth be denoted (Φ j , Φ k ) Ω where by definition:

(Φ j , Φ k ) Ω = Ω Φ j (x) • Φ k (x) dx,
with • the notation of the standard Euclidean inner product.

POD Reduced-Order

Model based on Galerkin projection 2.2.1. General description For incompressible flows, the motion of the fluid is described by the incompressibility condition and the Navier-Stokes equations,

∇ • u = 0, ∂ t u = N (u) -∇p with N (u) = -(u • ∇) u + 1 Re ∆u.
In these equations, all variables (u velocity vector and p pressure) are assumed to be nondimensional and Re is the Reynolds number.

The POD Reduced-Order Model (POD ROM) is then constructed by applying the Galerkin projection to the governing equations. Since by construction the eigenfunctions Φ i are divergence-free for an incompressible flow, they can be used as test functions to derive the variational formulation of the Navier-Stokes equations:

(∂ t u, Φ i ) Ω = (N (u), Φ i ) Ω -(∇p, Φ i ) Ω . (1)
Using the relation ∇ • Φ i = 0, the pressure term can be rewritten as a boundary term:

(∇p, Φ i ) Ω = Ω Φ i • ∇p dx = Ω ∇ • (pΦ i ) dx = ∂Ω pΦ i • n dx = P i ,
where n is the outward unit normal at the boundary surface ∂Ω. If the snapshots u m employed for the POD are zero on the boundary, then Φ i = 0 on ∂Ω and the pressure term vanishes. In most of the applications [START_REF] Noack | A hierarchy of low-dimensional models for the transient and post-transient cylinder wake[END_REF][START_REF] Galletti | Low-order modelling of laminar flow regimes past a confined square cylinder[END_REF][START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF], the contribution of the pressure term is simply neglected as a first approximation.

To carry on our developments, we have to specify the variable on which the POD will be applied. Beyond the idea of a correct representation of the flow physics, we would like to simplify as much as possible the analytical expressions of the coefficients associated to the POD ROM. In the previous paragraph, we point out that prescribing Φ i = 0 on the boundary is relevant for the pressure term. We will thus use a lifting procedure discussed thoroughly in [START_REF] Cordier | Two typical applications of POD: coherent structures eduction and reduced order modelling[END_REF]: if the boundary conditions are time-independent (uncontrolled flow for instance), the original set of snapshots U is replaced by U ′ = {u mu m } m=1,...,Nt where u m = E(u); if the boundary conditions are time-dependent (flow controlled at the boundary for instance), the POD is applied to U ′′ = {u mγ m u cu m } m=1,...,Nt where γ m = γ(t m ) characterizes the time evolution of the control and u c is the actuation mode. In practice, this mode is determined as a particular solution of the Navier-Stokes equations, where the boundary conditions on the controlled part of ∂Ω are equal to the spatial evolution of the control, and where the boundary conditions on the uncontrolled part are set to zero (see [START_REF] Bergmann | Optimal control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models[END_REF] for an application to the controlled cylinder wake).

In the general formulation, the velocity expansion of u over the POD modes Φ j writes:

u(x, t) = u m (x) + γ(t)u c (x) + N POD j=1 a j (t)Φ j (x).
(

) 2
By substituting (2) into the Galerkin projection (1), we obtain after some algebraic manipulations [START_REF] Bergmann | Control of the circular cylinder wake by Trust-Region methods and POD Reduced-Order Models[END_REF] the following expression for the reduced-order model in the presence of control:

ȧR i (t) = A GP i + B GP ij a R j (t) + C GP ijk a R j (t)a R k (t) -P i (t) (3a)
+D GP i γ(t) + E GP i + F GP ij a R j (t) γ(t) + G GP i γ 2 (t) a R i (0) = a P i (0) = (u(x, 0) -u m (x) -γ(0)u c (x), Φ i ) Ω , (3b)
where the Einstein summation is used and all subscripts i, j, k run from 1 to N gal . Here, N gal < N POD corresponds to the number of Galerkin modes retained in the reduced-order model. This number of modes is assumed to be sufficient to reproduce accurately the flow (see section 2.3 to evaluate N gal ). The coefficients

A GP i , B GP ij , C GP ijk , D GP i , E GP i , F GP ij
and G GP i depend explicitly on Φ, u m and u c (see Appendix I). The superscripts R are introduced for the temporal coefficients a i are to indicate that these coefficients are obtained by integrating in time the system of ODEs [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF]. Similarly, we introduce the superscripts GP for the coefficients of the POD ROM to stress that these coefficients are obtained directly by Galerkin projection.

Simplification of the POD ROM

In practice, the POD ROM (3) will not necessarily be integrated in time with the coefficients determined by the method of POD Galerkin. Indeed, whole or part of the coefficients may either be unknown (in the case of experimental data for instance) or known with an insufficient level of accuracy to reproduce correctly the original dynamics (see section 2.3 for an example). It is thus necessary to identify, or in other words calibrate, whole or part of the coefficients. Since the structure of (3a) is not modified by the pressure term [START_REF] Noack | Reduced Order Models for Analysis and Control of Fluid Flows[END_REF], we can consider without restricting the generality P i (t) = 0.

In section 3, the various methods of calibration encountered in the literature will be presented within a unified framework. Therefore we introduce a

R (t) = a R 1 (t), • • • , a R N gal (t)
T , vectorial solution of (3), to rewrite compactly (3a) as:

ȧR i (t) = f i (y i , a R (t)) + g i (z i , a R (t), γ) i = 1, • • • , N gal , (4a)
where

f i (y i , a R (t)) = A i + B ij a R j (t) + Q ijk a R j (t)a R k (t), (4b)
and

g i (z i , a R (t), γ) = D i γ(t) + E i + F ij a R j (t) γ(t) + G i γ 2 (t) (4c) with j = 1, • • • , N gal and k = 1, • • • , j.
In (4a), y i and z i denote the unknown coefficients corresponding to respectively the uncontrolled and controlled part of (3a), i.e.

y i =             A i B i1 . . . B iN gal Q i11 . . . Q iN gal N gal             ∈ R Ny i and z i =          D i E i F i1 . . . F iN gal G i          ∈ R Nz i ,
where

N yi = 1+N gal + N gal (N gal +1) 2
and N zi = 3+N gal . In (4b), the coefficients Q ijk correspond to the symmetric part of C ijk only, i.e.

Q ijk = 1/2 (C ijk + C ikj ) for i, j = 1, • • • , N gal and k = 1, • • • , j.
This modification of the expression of the quadratic term is justified by the fact that it is impossible to differentiate C ijk from C ikj by any technique of calibration.

In vectorial formulation, the controlled POD ROM given by (4a) is finally:

ȧR (t) = f (y, a R (t)) + g(z, a R (t), γ), (5)
where for the uncontrolled contribution:

f =      f 1 f 2 . . . f N gal      ∈ R N gal ; y =      y 1 y 2 . . . y N gal      ∈ R Ny with N y = N gal N yi ,
and, for the controlled contribution:

g =      g 1 g 2 . . . g N gal      ∈ R N gal ; z =      z 1 z 2 . . . z N gal      ∈ R Nz with N z = N gal N zi .
Here, f and g can be evaluated easily using the coefficients of the model y and z (see Appendix II).

For the sake of clarity, the rest of the paper is limited to uncontrolled flows, i.e. γ = 0 or g = 0, in [START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF]. One of the interests of ( 5) is to clearly demonstrate that for a given value of γ the extension to controlled flows is straightforward (see [START_REF] Weller | Robust model identification of actuated vortex wakes[END_REF] for a recent application of the calibration techniques to a controlled wake flow).

Two-dimensional cylinder wake flow at Re = 200

The reduced-order modeling approach based on POD is now applied to a two-dimensional incompressible cylinder wake flow at Re = 200. The database was computed using a finiteelement code (DNS code Icare, IMFT/University of Toulouse, see [START_REF] Favier | Contrôle d'écoulements : approche expérimentale et modélisation de dimension réduite[END_REF] for details) and contains N t = 200 two-dimensional snapshots of the flow velocity, taken over a period T = 12 i.e. over more than two periods of vortex shedding (T vs = 5). Typical iso-values of the longitudinal Following the discussion of section 2.2.1, the method of snapshots introduced in section 2.1 is applied to the velocity fluctuation. The first six spatial eigenfunctions Φ i are represented in Fig. 2 and the relative kinetic energy is plotted in logarithmic scale for the first 40 POD modes in Fig. 3. The energy is clearly concentrated in a very small number of modes: the first six POD modes are sufficient to represent 99.9% of the flow energy and we thus consider N gal = 6 to derive the POD ROM.

The POD ROM ( 5) is then integrated in time with a classical fourth-order Runge-Kutta scheme and a time step of 10 -3 T . A set of predicted time histories for the mode amplitudes a R i (t) is obtained, and compared to the set of POD temporal eigenfunctions a P i (t). As shown in Fig. 4, the original dynamics is globally well reproduced but the accuracy is not perfect. Indeed, the most energetic mode is well reconstructed, while for the higher modes the maxima remain over-estimated. Besides that, a constant phase shift can quickly occur between the dynamics of the model and that of the original one (see mode 3 for instance). If the final objective is to use the POD ROM in a phase control strategy, then these errors can lead to the failure of this control strategy. This relatively bad performances of the model may be attributed to i) the structural instability of the Galerkin projection [START_REF] Iollo | Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations[END_REF][START_REF] Rempfer | On low-dimensional Galerkin models for fluid flow[END_REF][START_REF] Noack | A hierarchy of low-dimensional models for the transient and post-transient cylinder wake[END_REF], ii) the truncation of the POD basis (dissipative scales associated to higher POD modes are not sufficiently present in the model), iii) the fact that the set of data does not perfectly respect the weak formulation (5): inaccurate treatment of the boundary terms and the pressure contribution, incompressibility not verified (experimental data), iv) an insufficient numerical precision to compute the POD ROM coefficients.

It is thus necessary to introduce calibration techniques in order to reproduce accurately the dynamics of reference using the POD ROM.

EXISTING CALIBRATION METHODS

Many methods of calibration have already been proposed in the literature. Let us quote for example the different techniques based on least-square minimization [START_REF] Galletti | Low-order modelling of laminar flow regimes past a confined square cylinder[END_REF][START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF][START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF], those consisting in solving a constrained optimization problem, iteratively [START_REF] Bergmann | Optimisation aérodynamique par réduction de modèle POD et contrôle optimal[END_REF] or simultaneously [START_REF] Galletti | Accurate model reduction of transient and forced wakes[END_REF], the recent method termed intrinsic stabilization introduced in [START_REF] Kalb | An intrinsic stabilization scheme for proper orthogonal decomposition based lowdimensional models[END_REF], and finally the calibration procedure suggested by [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF]. To compare fairly the previous procedures on the case of the cylinder wake flow (see section 4), we first introduce various criteria of error (section 3.1), along the lines of [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF]. Then, we emphasize in section 3.2 that the problem of minimization reduces to a linear system when the errors are affine functions of y. Finally, the calibration procedure suggested in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] is presented, as the Tikhonov regularization method proposed in this paper (section 5) constitutes an alternative to their approach. 5) is to represent, as accurately as possible, the dynamics given by the POD temporal eigenfunctions. It is then natural to seek the coefficients y which minimize the error

e (1) (y, t) = a P (t) -a R (t),
under the constraint of the Cauchy problem defined by:

(P C ) ȧR (t) = f (y, a R (t)), a R (0) = a P (0).
Since e (1) ∈ R N gal and is time-dependent, we rather seek to minimize

I (1) (y) = e (1) (y, t) 2 Λ To , where • To is a time average operator over [0, T o ] (T o ≤ T ) and • Λ is a norm of R N gal .
• To corresponds in this paper to the arithmetic time-average on N equally spaced elements of the interval [0, T o ]:

g(t) To = 1 N N k=1 g(t k ) with t k = (k -1) ∆t and ∆t = T o N -1 .
For the norm, we introduce Λ ∈ R N gal ×N gal the symmetric definite positive matrix associated to • Λ and define for any e ∈ R N gal :

e 2
Λ = e T Λe. The matrix Λ introduced in the definition acts as a weight function: it allows to balance the importance of specific POD modes. When Λ = I N gal for example, all POD modes have the same importance in terms of error.

The minimization of I (1) under the constraints of P C corresponds to a nonlinear constrained optimization problem. Until now, this problem is solved only for Λ = I N gal i.e. minimizing:

I (1) (y) = 1 N N k=1 N gal i=1 a P i (t k ) -a R i (t k ) 2 .
In [START_REF] Bergmann | Optimisation aérodynamique par réduction de modèle POD et contrôle optimal[END_REF], the constrained optimization problem built on I (1) was solved iteratively to find optimal eddy viscosities. In [START_REF] Galletti | Low-order modelling of laminar flow regimes past a confined square cylinder[END_REF], the same iterative approach is used to determine a linear model for the pressure contribution of the Galerkin projection. More recently, a similar approach was used in [START_REF] Galletti | Accurate model reduction of transient and forced wakes[END_REF] to determine the constant and linear coefficients of the POD ROM. However, in this case, the constrained optimization problem was solved simultaneously with a pseudo-spectral discretization of the variables.

State calibration method

From a mathematical point of view, the minimization problem based on I (1) is not well posed: several solutions may coexist and convergence is not even guaranteed (see [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] for some arguments). Therefore, it was proposed in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] to suppress the dynamical constraint in the definition of e (1) .

After integration in time of the POD ROM, the error e (1) can be rewritten as:

e (1) (y, t) = a P (t) -a P (0) - t 0 f (y, a R (τ )) dτ.
To suppress the nonlinear constraint, the occurrence of a R in the last term can be replaced by a P . We then introduce a new error e (2) defined as:

e (2) (y, t) = a P (t) -a P (0) - t 0 f (y, a P (τ )) dτ.
In the literature, the minimization of I (2) (y) = e (2) (y, t) 2 Λ To was considered in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] to determine all the coefficients of the model and more recently in [START_REF] Bourguet | Reduced-order modeling for unsteady transonic flows around an airfoil[END_REF] to evaluate the constant and linear coefficients for an unsteady transonic flow. In both cases, all modes have the same contribution, i.e. Λ = I N gal , and the following error is minimized:

I (2) (y) = 1 N N k=1 N gal i=1 a P i (t k ) -a P i (0) - t k 0 f i (y, a P (τ )) dτ 2 .

Flow calibration method

The third criterion of error is obtained by taking the temporal derivative of the e (1) criterion:

d dt e (1) (y, t) = ȧP (t) -f (y, a R (t)),
and by replacing a R by a P in order to suppress the nonlinear constraint. The following error is thus obtained:

e (3) (y, t) = ȧP (t) -f (y, a P (t)).
The idea suggested by [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] to minimize

I (3) (y) = e (3) (y, t) 2 Λ
To seems natural because it is equivalent to impose that the temporal POD eigenfunctions a P are solutions of the flow model given by f . For this reason, this least-square procedure was proposed independently by [START_REF] Galletti | Low-order modelling of laminar flow regimes past a confined square cylinder[END_REF] to evaluate a linear model for the pressure term coming from the Galerkin projection and by [START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF] to identify all the coefficients of the model starting from experimental data obtained by PIV. All these applications were carried out for Λ = I N gal i.e. with the aim of minimizing:

e (3) (y, t) 2 IN gal To = 1 N N k=1 N gal i=1 ȧP i (t k ) -f i (y, a P i (t k ) 2 .

Affine functions of errors

For the state calibration method (section 3.1.2) and the flow calibration method (section 3.1.3), the corresponding errors are affine with respect to y. We can then demonstrate (see Appendix III) that minimizing

I (i) (y) = e (i) (y, t) 2 Λ To for i = 2, 3
gives rise to the linear system

A (i) y = b (i) , (6)
where A (i) ∈ R Ny×Ny and b (i) ∈ R Ny are defined in Appendix III.

Calibration procedure proposed in [3]

The general idea is to determine the coefficients y

(i)
α , which characterize uniquely the calibrated model, as the solution of the optimization problem based on the functional:

J (i) α (y) = (1 -α)E (i) (y) + αD(y) with i = 2, 3.
α ∈ [0, 1] is a weighting parameter; E (i) (y) is a measure of the normalized error between the behavior of the data i.e. a P (t), and the behavior of the polynomial model defined by y whose state is a R (t); D(y) is a measure of the difference between the coefficients of the model y and the coefficients obtained from the Galerkin projection y GP . For α = 0, the calibrated model is fully optimized (we will see in section 4 that in this case the linear system ( 6) is ill-conditioned) and for α = 1, the coefficients from the Galerkin projection are recovered. E (i) and D are defined as:

E (i) (y) = e (i) (y, t) 2 Λ To e (i) (y GP , t) 2 Λ To = I (i) (y) I (i) (y GP ) , and
D(y) = y -y GP 2 Π y GP 2 Π
, where • Π is a semi-norm on the polynomial vector space. For any y ∈ R Ny , y Π is defined by

y 2 Π = y T Πy,
where Π ∈ R Ny×Ny is a non-negative symmetric matrix. For Π = I Ny , all the coefficients are calibrated and have the same weight in the calibration. For values different from I Ny , a partial calibration is possible (see [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] for the only example present in the literature). In section 4, our numerical experiments will be carried out for Π = I Ny .

Finally, it can be shown (Appendix IV) that for i = 2 or 3, the minimization of

J (i)
α amounts to solve the linear system:

A (i) α y = b (i) α , (7)
with

A (i) α = 1 -α I (i) (y GP ) A (i) + α y GP 2 Π Π, and
b (i) α = 1 -α I (i) (y GP ) b (i) + α y GP 2 Π Πy GP .
The questions which remain open are:

1. How to fix the weighting parameter α, and what is the optimal choice? 2. How to apply this procedure of calibration in the case of experimental data, for which the coefficients coming from the Galerkin projection are usually not available?

Table I. Normalized errors E (i) and costs of the calibration D. Comparison between the results obtained by: i) minimizing I (1) under the constraint of PC , ii) minimizing I (3) with the determinations of the linear (L), constant and linear (C and L) and eddy-viscosity terms (V ) and iii) Intrinsic Stabilization [START_REF] Kalb | An intrinsic stabilization scheme for proper orthogonal decomposition based lowdimensional models[END_REF].

Method of calibration

Control terms

E (1) (y) E (2) (y) E (3) (y) D(y)
Minimization of I (1) C and L 2.60 10 -2 2.43 10 -1 2.44 10 -1 1.53 10 -1 under the constraint of PC Minimization of I (3) L 6.78 10 -2 8.85 10 -1 3.86 10 -1 3.28 10 -2 Minimization of I (3) C and L 3.19 10 -2 2.46 10 -1 2.68 10 -1 3.27 10 -2 Minimization of I (3) V 7.85 10 -1 8.72 10 -1 9.18 10 -1 1.19 10 -2 Intrinsic Stabilization [START_REF] Kalb | An intrinsic stabilization scheme for proper orthogonal decomposition based lowdimensional models[END_REF] C and L 6. 3) with determination of all the coefficients (constant, linear and quadratic). The linear system is not regularized.

APPLICATION TO THE CYLINDER WAKE FLOW

The methods of calibration presented in section 3 are applied to the cylinder wake flow of section 2.3. In section 4.1, the coefficients of the model are found by first minimizing I (1) under the constraint of P C , and then by minimizing I (3) . In the subsequent section 4.2, the results determined by minimizing J

(2) α and J

(3) α are presented. The choice of the weighting parameter α and its impact on the POD ROM are discussed. (1) under the constraint of P C and minimization of I (3) In Table I, the normalized errors and costs of the calibration are reported for the minimization of I (1) , under the constraint of P C , and for the minimization of I (3) . In that case, different control parameters are considered: determination of the linear, constant and linear, eddy viscosity terms of the model. The results are compared with those obtained by the intrinsic stabilization scheme recently suggested in [START_REF] Kalb | An intrinsic stabilization scheme for proper orthogonal decomposition based lowdimensional models[END_REF].

Minimization of I

It is found that the most effective method of calibration in terms of reductions of normalized errors corresponds to the minimization of I (1) under the constraint of the Cauchy problem P C , for any specific criterion E (i) considered. For the minimization of I (3) , the results confirm the intuitive idea that the normalized error decreases as the number of calibration variables increases. Indeed, for this method of calibration, the lowest normalized error is systematically obtained for the determination of constant and linear coefficients. For the corresponding costs of calibration, it is found that for the minimization of I (1) under the constraint of P C , √ D is approximately equal to 15 % whereas the cost is only of 3 % in the case of the minimization of I (3) . Since the normalized errors are of the same order of magnitude in both methods, we are tempted to claim that the minimization of I (3) , with the determination of the constant and linear coefficients, is more effective than the other approaches of calibration. However, the economic argument (ratio of savings over costs) commonly used in flow control to assess the efficiency of control strategies, makes here little sense, because the main objective of the calibration is to determine POD ROMs of improved accuracy. For the methods based on identification of parameters, the cost corresponds to the numerical implementation of the calibration and not to the variation of the coefficients from their value determined by POD Galerkin, as the conclusions drawn by [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] may let it think. The cost of calibration, as it is defined by D, is interesting, but especially as an indication, as it is difficult in practice to use this criterion to determine the method of calibration to apply (see section 4.2). Lastly, concerning the intrinsic stabilization scheme, one notes that, on this flow configuration, it is always less effective in terms of normalized errors than other methods.

Since the minimization of I (1) , under the constraint of P C , and the minimization of I (3) with determination of the constant and linear coefficients seem both as effective, we have tried to use the calibration based on minimization of I (3) to determine all the coefficients. Figure 5 shows that the POD ROM calibrated in this way diverges quickly during the numerical integration. This behavior, which can seem surprising for this simple kind of dynamics, will be explained in section 4.2 by the ill-conditioning of A

(3) 0 (see Fig. 8). α

E (1) y (2) α E (2) y (2) α E (3) y (2) α E (1) y (3) α E (2) y (3) α E (3) y (3)

Minimizations of J

(2) α and J

(3) α

In this section, the calibration is carried out by minimizing the functionals J

(2) α and J

(3) α

suggested in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF]. Their idea was to determine the coefficients of calibration as solutions of an optimization problem aiming at minimizing a weighted average of the normalized error, and a term measuring the variation of the coefficients from their value obtained by POD Galerkin. The main difficulty is to evaluate efficiently, and possibly using an optimal procedure, the value of the weighting parameter α. Figure 6 (top) represents the evolutions of the normalized errors and the calibrations costs obtained by minimizing J

(2) α and J

(3) α as α is varied (α ∈ {0.05, 0.1, • • • , 1}). Note that by definition of the functional

J (i) α , only E (2) (y (2) α ), E (3) (y (3) α ), D(y (2)
α ) and D(y

(3)
α ) are monotone functions of α. In addition, since the curves of errors and costs vary extremely rapidly when α tends to 1, we have made the same choice as in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF], and introduced also a variation of these variables according to a parameter δ defined in increasing bijection with α on [0, 1]. The weighting parameter α is defined by:

α = δ ζ (i) (1 -δ) + δ with ζ (i) = I (i) (y GP ) I (i) (0) ,
and is displayed as a function of δ in Fig. 7, with δ ∈ {0.05, 0.1, • • • , 1}. The results computed with a linear variation of δ on [0, 1] are represented in Fig. 6 (bottom). Numerically, when α tends to 0, the trend tends towards a decrease of the normalized error (better calibrated model), and towards an increase of the cost of calibration (model more modified). In practice, it is thus difficult to exploit these curves to choose the value of the appropriate weighting parameter α. Of course, it is always possible to arbitrarily choose a criterion, ensuring a specific balance between the reduction of the normalized error and the cost of calibration. However, as it has already been discussed in the previous section, the cost of the calibration measured by the criterion D is not fully relevant. Then, we may wonder if it would not be more convenient to simply minimize the normalized error by setting α = 0. The answer is given in Fig. 8, by showing that matrices A

(2)

α and A

α become very ill-conditioned as α goes to 0. The suggestion of [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF] to add the term of calibration cost D to the functional can be interpreted as a particular method of regularization of the matrix A (i) , which is, by nature, ill-conditioned. Moreover, the drawback of the procedure is the choice of the weighting parameter α which is not straightforward. In addition, minimizing J

(2) α and J

(3) α is impossible when coefficients resulting from Galerkin projection are not available (case of the majority of the experimental data). In the next section, we thus propose another method of regularization directly based on the minimization of I (3) .

CALIBRATION BY TIKHONOV REGULARIZATION

Filter factors and discrete Picard condition

The minimization of the functional I (3) amounts to solving the linear system A (3) y = b (3) where A (3) and b (3) are given in Appendix III. For the sake of clarity, the superscript is omitted and we thus write simply the linear system as Ay = b. In practice, the right-hand side is contaminated by approximation errors related to the numerical evaluation of the timederivatives of the POD eigenfunctions (remind that e (3) (0, t) = ȧP (t)). To understand the influence of these errors on the solution of the linear system, the concept of filter factors is introduced. To do so, the Singular Value Decomposition [START_REF] Golub | Matrix Computations. 3 edn[END_REF] is applied to the matrix A:

A = U ΣV T = Ny j=1 u j σ j v T j ,
where

U = u 1 , • • • , u Ny and V = (v 1 , • • • , v N y
) are orthogonal matrices containing the left u j and right v j singular vectors, and where Σ = diag σ 1 , • • • , σ Ny is a diagonal matrix with the singular values σ j arranged in non-increasing order such that

σ 1 ≥ • • • ≥ σ Ny ≥ 0.
Since U and V are orthogonal matrices (U U T = V V T = I Ny ), the solution y of the linear system is given by ‡ :

y = Ny j=1 1 σ j u T j bv j = Ny j=1 h j 1 σ j u T j bv j with h j = 1 for j = 1, • • • , N y ,
where h j are the filter factors. This relation clearly illustrates the numerical difficulties encountered when the linear system is solved without precautions. Indeed, if the Fourier coefficients |u T j b|, corresponding to the smaller singular values σ j , do not decrease sufficiently fast compared to the singular values, the solution is dominated by the terms in the sum corresponding to the smallest σ j . This behavior can be assessed by inspecting the discrete Picard condition plotted in Fig. 9: for j ≃ 80, the singular values decay faster than the Fourier coefficients |u T j b|. As a result, the solution obtained presents many oscillations around zero, and thus appears to be completely random (see Fig. 10 for the solution without regularization). To fix this, the first idea is to modify the filter factor h j so that it behaves like an ideal low-pass ‡ It can be shown easily (see Appendix III) that y is also the solution of the linear least-squares problem miny Ayb 2 2 . 3) , with the determination of all the coefficients (constant, linear and quadratic).

The linear system is regularized by Tikhonov with L = I and y0 = y GP . Note that for N gal = 6, Ny = 168.

filter:

h j = 1 if j ≤ 80, 0 if j > 80.
For the cylinder wake flow, this procedure may be appropriate since the temporal dynamics is relatively simple (see the abrupt falling-off in the singular values for j ≃ 80). However, this method is not suitable for more complex dynamics (3-D turbulent flow for instance) for which the singular values decrease continuously. It is thus necessary to modify the filter factors in a more sophisticated way.

Tikhonov regularization

Undoubtedly, the most common and well-known method of regularization is the Tikhonov regularization [START_REF] Hansen | Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF]. The idea is to seek the regularized solution y ρ as the minimizer of the following weighted functional

Φ ρ (y) = Ay -b 2 2 + ρ L (y -y 0 ) 2 2
, where the first term corresponds to the residual norm, and the second to a side constraint imposed on the solution. ρ is called the regularization parameter and L represents the discrete approximation matrix of a differential operator. This matrix is typically either the identity matrix of order N y (derivative of order zero), or a banded matrix of dimension (N yd) × N y approximation of the derivative operator of order d. In particular, for d = 0, d = 1 and d = 2, the method is termed zeroth, first and second-order Tikhonov regularization respectively. Thereafter, these operators will be denoted

L = I (d = 0), L = F OD (d = 1) and L = SOD (d = 2).
Intuitively, the regularization can be seen as a balance between two requirements:

1. y ρ should give a small residual Ay ρb, 2. L (y ρy 0 ) should be small with respect to the 2-norm. 3) , with the determination of all the coefficients (constant, linear and quadratic) and application of the Tikhonov regularization (L = I ; y0 = y GP ). The "corner" of the L-curve is at ρ = 6.78 10 -8 .

By using the same type of argument that that of section 5.1 to justify the introduction of filter factors h j , it becomes possible to prove that for y 0 = 0, the regularized solution y ρ can be written as follows:

y ρ = Ny j=1 h j 1 σ j u T j bv j with h j = σ 2 j σ 2 j + ρ if L = I Ny , and
y ρ = Ny-d j=1 h j 1 σ j u T j bx j + Ny j=Ny-d+1 u T j bx j with h j = γ 2 j γ 2 j + ρ if L = I Ny .
Here γ j (j = 1, • • • , N yd) are the generalized singular values of (A, L) and x j the jth column of X ∈ R Ny×Ny (see Appendix V for the definition of the Generalized Singular Value Decomposition).

The regularization parameter ρ needs now to be computed. To do so, the L-curve method implemented in the package Regularization Tools [START_REF] Hansen | Regularization Tools: A Matlab package for analysis and solution of discrete ill-posed problems[END_REF] is used throughout the paper. The L-curve method is based on the analysis of the curve representing the semi-norm of the regularized solution Ly ρ 2 , versus the corresponding residual norm Ay ρb 2 . In most of the cases, this curve exhibits a typical L shape (see Fig. 11). The corner of the L-curve represents a fair compromise between the minimization of the norm of the residual (horizontal branch) and the semi-norm of the solution (vertical branch). In [START_REF] Hansen | Regularization Tools: A Matlab package for analysis and solution of discrete ill-posed problems[END_REF], the detection of the corner is based on the maximization of the curvature of the L-curve. Comparison between the temporal evolutions of the projected ✷ (POD) and predicted (POD ROM) mode amplitudes. The POD ROM is calibrated by minimizing I (3) , with the determination of all the coefficients (constant, linear and quadratic). The linear system is regularized by Tikhonov with L = I and y0 = y GP .

Comparison of the different types of Tikhonov regularization

Table II. Normalized errors E (i) and costs of the calibration D. Comparison between the results obtained by minimizing I (3) , with the determination of all the coefficients (constant, linear and quadratic), for different types of Tikhonov regularization. The case L = F OD is not reported because the numerical integration of the calibrated model is diverging.

Type of Tikhonov regularization In this section, the various types of Tikhonov regularization introduced in section 5.2 are compared on the configuration of the cylinder wake flow. POD index number i

√ E (1)(y) E (2) (y) E (

Energetic content

No calibration Calibration POD eigenvalues Figure 13. Comparison between the modal energetic contents obtained before and after calibration. The POD eigenvalues are reported for reference. The POD ROM is calibrated by minimizing I (3) , with the determination of all the coefficients (constant, linear and quadratic). The linear system is regularized by Tikhonov with L = I and y0 = y GP .

and costs of the calibration obtained by zeroth and second-order Tikhonov regularization. The case of first-order Tikhonov regularization (L = F OD) is not mentioned because it is diverging, despite the regularization carried out. On average over all the criteria of normalized errors, the case L = I and y 0 = y GP appears to be the most effective. Consequently, this type of Tikhonov regularization is retained for the comparison of section 5.4. In Fig. 12, the temporal evolutions of the POD modes are compared with those predicted by the calibrated reducedorder model. Contrary to the results presented in Fig. 4 there is no clear difference in the dynamics. The immediate consequence is that the modal energy distribution associated to the calibrated model now corresponds perfectly to the POD energy (see figure 13). However, it is worth mentioning that the finite-time thermodynamics (FTT) formalism recently suggested in [START_REF] Noack | A finite-time thermodynamics formalism for unsteady flows[END_REF] is more satisfactory because the calibration techniques are rather a posteriori methods whereas the FTT is a flow modeling approach in itself.

Comparison of the most effective calibration methods

To conclude, we compare in this section the three most effective methods of calibration presented in this paper on the configuration of the wake flow. Table III gives the normalized errors and the costs of the calibration obtained by minimization of I (1) under the constraint of P C , by minimization of J

(3) α for α = 0.001 and by minimization of I (3) with determination of all the coefficients of the model and application of the most effective Tikhonov regularization i.e. L = I and y 0 = y GP . Except for the E (3) criterion where E (3) (y) = 6.29 10 -2 for the minimization of J

(3) α and E (3) (y) = 6.31 10 -2 for the minimization of I (3) with Tikhonov regularization, the numerical experiments prove that the normalized errors are minimized by the calibration based on Tikhonov regularization. The difference between these methods of calibration can be analyzed in a finer way by introducing the modal errors I (j) i defined § for i = 1, • • • , N gal and j = 1, • • • , 3 as:

I (j) (y) = N gal i=1 I (j) i (y) .

The modal errors I

(1) i are represented in Fig. 14 for the various calibration techniques presented in Table III. For all POD modes, the minimization of I (3) using the Tikhonov regularization is more effective than the minimization of J

(3) α for α = 0.001. Additionally, these methods clearly outperform the minimization of I (1) under the constraint of P C for the higher POD modes. The main interest of the calibration technique based on the Tikhonov regularization is that the choice of the parameter of regularization ρ is determined by the L-curve without any intervention of the user.

Table III. Normalized errors E (i) and costs of the calibration D. Comparison between the results obtained by: i) minimizing I (1) under the constraint of PC, ii) minimizing J

(3) α with α = 0.001, and iii) minimizing I (3) with the determination of all the coefficients (constant, linear and quadratic) and application of the Tikhonov regularization (L = I ; y0 = y GP ).

Method of calibration Control terms

√ E (1)(y) E (2) (y) E (3) (y) D(y)
Minimization of I (1) C and L 2.60 10 -2 2.43 10 -1 2.44 10 -1 1.53 10 -1 under the constraint of PC Minimization of J

CONCLUSIONS AND OUTLOOK

In the first part of this paper, we have presented within a unified framework the various methods of calibration used so far in the literature to identify the coefficients of the POD ROM. Afterwards, we have applied these methods to a 2-D cylinder wake flow, to understand these techniques in details, and, if possible, to release a more effective strategy. We have thus showed that the minimization of I (1) , under the constraint of the Cauchy problem P C , is much more effective than the methods based on the minimization of I (3) , or than the Intrinsic Stabilization scheme. We have then continued by applying the procedure suggested in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF], § Rigorously, these modal errors can be introduced only when Λ is the identity matrix of size N gal . i . Comparison between the results obtained by: i) minimization of I (1) under the constraint of PC, ii) minimization of J

(3) α with α = 0.001 and iii) minimization of I (3) with determination of all the coefficients (constant, linear and quadratic) and application of the Tikhonov regularization (L = I ; y0 = y GP ). We remind that

I (1) (y) = N gal i=1 I (1) i (y).
which is based on the minimization of a weighted sum of the normalized error, and of a term measuring the variation of the coefficients of the model to their values obtained by POD Galerkin. In substance, the idea is identical to what is usually made in optimal control ( [START_REF] Gunzburger | Introduction into mathematical aspects of flow control and optimization[END_REF] for instance). Indeed, the cost functional is built as the sum of two terms, the first being a measure of the objective of the optimization, and the second corresponding to the cost of the control. However, as far as the identification is concerned, the variation of the coefficients of the calibrated model from their values obtained by POD Galerkin, is not relevant because the main objective is to improve the accuracy of the model. Also, in the case of experimental data, the coefficients resulting from POD Galerkin do not even exist in the majority of the cases. Finally, it can be shown numerically that adding the cost of the calibration to the functional to be minimized, comes down to improve the conditioning of the linear systems associated to the minimization of I (2) and I (3) . As the choice of the parameter α is relatively tricky, even arbitrary, the linear system associated to the minimization of I (3) is regularized by the method of Tikhonov. For our test-case, the zeroth-order regularization (L = I) with y 0 = y GP is the most effective among all the types of Tikhonov regularization considered. Lastly, our numerical experiments demonstrate that the Tikhonov regularization outperforms, in terms of normalized errors, the minimization of I (1) under the constraint of P C and the minimization of J

α with α = 0.001. Compared to the approach suggested in [START_REF] Couplet | Calibrated Reduced-Order POD-Galerkin system for fluid flow modelling[END_REF], the interest of the Tikhonov regularization is that the value of the parameter of regularization ρ is determined automatically, via the L-curve method.

The calibration by Tikhonov regularization presented here still needs to be tested on flow configurations corresponding to more complex dynamics: 3-D turbulent flow obtained by numerical simulations, and challenging experimental data. The extension to actuated configurations with known control parameters should be straightforward. On the other hand, calibrating a reduced-order model, so that it can represent the evolution of dynamics when the control law varies, is a more complex task. A basic idea consists in determining a POD basis starting from snapshots coming from several control laws, and then to calibrate the model for all the control laws considered. Recently, this approach was applied for the first time in [START_REF] Weller | Robust model identification of actuated vortex wakes[END_REF] to identify a robust model of actuated wakes.

APPENDIX I. POD ROM COEFFICIENTS

The coefficients of the POD ROM (3) obtained by Galerkin projection (GP) are:

A GP i = -(Φ i , (u m • ∇)u m ) Ω - 1 Re (∇ ⊗ Φ i ) T , ∇ ⊗ u m Ω + 1 Re [(∇ ⊗ u m )Φ i ] ∂Ω , B GP ij = -(Φ i , (u m • ∇)Φ j ) Ω -(Φ i , (Φ j • ∇)u m ) Ω - 1 Re (∇ ⊗ Φ i ) T , ∇ ⊗ Φ j Ω + 1 Re [(∇ ⊗ Φ j )Φ i ] ∂Ω , C GP ijk = -(Φ i , (Φ j • ∇)Φ k ) Ω , D GP i = -(Φ i , u c ) Ω , E GP i = -(Φ i , (u c • ∇)u m ) Ω -(Φ i , (u m • ∇)u c ) Ω - 1 Re (∇ ⊗ Φ i ) T , ∇ ⊗ u c Ω + 1 Re [(∇ ⊗ u c )Φ i ] ∂Ω , F GP ij = -(Φ i , (Φ j • ∇)u c ) Ω -(Φ i , (u c • ∇)Φ j ) Ω , G GP i = -(Φ i , (u c • ∇)u c ) Ω ,

II. EVALUATION OF f AND g

The values of f i and g i (see section 2.2.2 for their definitions) can be easily determined if we know the coefficients y i and z i . Indeed, f i belongs to the space of polynomials of degree 2 in N gal variables: a R 1 (t), • • • , a R N gal (t). Therefore, let

m(t) =             1 a R 1 (t) . . . a R N gal (t) a R 1 (t)a R 1 (t) . . . a R N gal (t)a R N gal (t)            
∈ R Ny i be the vector containing the natural monomial basis of this space, we can write:

f i (y i , a R (t)) = m(t) • y i .
Similarly, if we define

q(t) =          γ(t) γ(t) γ(t)a R 1 (t) . . . γ(t)a R N gal (t) γ 2 (t)          ∈ R Nz i ,
we have: g i (z i , a R (t), γ) = q(t) • z i .

We can deduce from these expressions that f can be computed at any time instant t, as the product of a block diagonal matrix M where each block is equal to m T by the vector y. Similarly, g can be obtained as the product of a block diagonal matrix Q where each block is given by q T by the vector z. (2) AND I (3) For i = 2 and 3, e (i) is an affine function with respect to y ∈ R Ny (see sections 3.1.2 and 3.1.3). Therefore, we introduce the application e (i) (•, t) : R Ny → R N gal y → E (i) (t)y + e (i) (0, t) with E (i) (t) ∈ R N gal ×Ny .

III. MINIMIZATION OF I

By identification with the expressions of the errors e (i) (i = 1, 2), one finds immediately that:

• for i = 2 E (2) (t)y = -t 0 f (y, a P (τ )) dτ and e (2) (0, t) = a P (t)a P (0)

• for i = 3 E (3) (t)y = -f (y, a P (t)) and e (3) (0, t) = ȧP (t),

where f can be evaluated using the method described in Appendix II.

Assuming that Λ is symmetric, we can prove that for i = 2 and 3:

I (i) (y) = e (i) (y, t) 2 Λ To = y T E (i) (t) T ΛE (i) (t)
To y + 2 e (i) (0, t) T ΛE (i) (t) To y + e (i) (0, t) T Λe (i) (0, t) To , = y T A (i) y -2 b (i) T y + c (i) ,

where

A (i) = E (i)T (t)ΛE (i) (t) To ∈ R Ny×Ny , b (i) = -E (i)T (t)Λe (i) (0, t) To ∈ R Ny , c (i) = e (i) (0, t) T Λe (i) (0, t) To ∈ R.
If Λ is a symmetric matrix, then A (i) is also symmetric by construction. In that case, minimizing the quadratic function I (i) is equivalent to solve the linear system defined by:

A (i) y = b (i) .

IV. MINIMIZATION OF J (i) α

The functional J (i) α can also be written as:

J (i) α (y) = χ α A I (i) (y) f1(y) +χ α Π y -y GP 2 Π f2 (y)
, by denoting

χ α A = 1 -α I (i) (y GP ) and χ α Π = α y GP 2 Π .
In Appendix III, we have demonstrated that when Λ is a symmetric matrix we have f 1 (y) = y T A (i) y-2 b (i) T y+c (i) . Similarly, it can be shown that f 2 (y) = y T Πy-2 y GP T Πy+ y GP T Πy GP if we assume that Π is symmetric.

Since f 1 and f 2 are two quadratic functions, it is simple to evaluate their gradients at y. One obtains:

∇f 1 (y) = 2 A (i) y -b (i)
and ∇f 2 (y) = 2Π yy GP .

By definition, the functional J α (y) for i = 2 or 3 is then equivalent to solve the linear system:

A (i) α y (i) α = b (i) α ,
with A (i) α = χ α A A (i) + χ α Π Π, and b (i) α = χ α A b (i) + χ α Π Πy GP . Note that for α = 0, χ α A = 1 I (i) (y GP ) and χ α Π = 0. Inserting these relations into (7), the linear system (6) is recovered.

V. GENERALIZED SINGULAR VALUE DECOMPOSITION

Let A ∈ R m×n and L ∈ R p×n be given with m ≥ n ≥ p. There exist orthogonal matrices U ∈ R m×n and V ∈ R p×p and a nonsingular matrix X ∈ R n×n such that

A = U Σ 0 0 I n-p X -1 , L = V (M, 0) X -1
where

Σ = diag(σ 1 , • • • , σ p ) and M = diag(µ 1 , • • • , µ p ) with 0 ≤ σ 1 ≤ • • • ≤ σ p ≤ 1 and 1 ≥ µ 1 ≥ • • • ≥ µ p ≥ 0.
Furthermore, it holds that σ 2 j + µ 2 j = 1 for j = 1, • • • , p. The values γ j = σ j /µ j (j = 1, • • • , p) are called the generalized singular values of (A, L). The jth column x j of X is the right singular vector associated with σ j .

Figure 1 .

 1 Figure 1. Iso-values of the longitudinal velocity fluctuation.

Figure 2 .

 2 Figure 2. The first 6 spatial POD eigenfunctions are visualized by iso-values of their norm ( Φi Ω).

Figure 3 .

 3 Figure 3. POD eigenvalues in logarithmic scale. EN t corresponds to twice the energy contained in the database (EN t = Nt j=1 λj).

Figure 4 .

 4 Figure 4. Comparison between the temporal evolutions of the projected ✷ (POD) and predicted (POD ROM) mode amplitudes. No calibration was used for the POD ROM.

3. 1 .

 1 Definitions of errors 3.1.1. State calibration method with dynamical constraints The objective of the POD-based model (

Figure 5 .

 5 Figure 5. Comparison between the temporal evolutions of the projected ✷ (POD) and predicted (POD ROM) mode amplitudes. The POD ROM is calibrated by minimization of I(3) with determination of all the coefficients (constant, linear and quadratic). The linear system is not regularized.

Figure 6 .α

 6 Figure 6. Normalized errors E (i) and costs of the calibration D obtained using the minimizations of J (2) α and J (3) α , for α varying in {0.05, 0.1, • • • , 1} (top) and δ varying in {0.05, 0.1, • • • , 1} (bottom).

Figure 7 .Figure 8 .

 78 Figure 7. Evolution of α with respect to δ.

Figure 9 .

 9 Figure 9. Visual check of the discrete Picard condition corresponding to the minimization of I (3) with determination of all the coefficients (constant, linear and quadratic) and application of the Tikhonov regularization (L = I ; y0 = y GP ).

Figure 10 .

 10 Figure 10. Comparison between solutions obtained with and without regularization, for the minimization of I(3) , with the determination of all the coefficients (constant, linear and quadratic). The linear system is regularized by Tikhonov with L = I and y0 = y GP . Note that for N gal = 6, Ny = 168.

2 Figure 11

 211 Figure 11. L-curve corresponding to the minimization of I(3) , with the determination of all the coefficients (constant, linear and quadratic) and application of the Tikhonov regularization (L = I ; y0 = y GP ). The "corner" of the L-curve is at ρ = 6.78 10 -8 .

6 Figure 12 .

 612 Figure 12. Comparison between the temporal evolutions of the projected ✷ (POD) and predicted (POD ROM) mode amplitudes. The POD ROM is calibrated by minimizing I(3) , with the determination of all the coefficients (constant, linear and quadratic). The linear system is regularized by Tikhonov with L = I and y0 = y GP .

3 )

 3 (y) D(y) L = I ; y0 = 0 4.14 10 -3 2.33 10 -1 6.27 10 -2 8.90 10 -1 L = I ; y0 = y GP 2.33 10 -3 2.33 10 -1 6.31 10 -2 5.67 10 -1 L = SOD ; y0 = 0 2.27 10 -1 2.28 10 -1 1.14 10 -1 1.22 L = SOD ; y0 = y GP 5.21 10 -3 2.32 10 -1 7.07 10 -2 3.44 10 -2

- 1

 1 Minimization of I(3) with Tikhonov C, L and Q 2.33 10 -3 2.33 10 -1 6.31 10 -2 5.67 10 -1 regularization (L = I ; y0 = y GP )

Figure 14 .

 14 Figure 14. Modal errors I (1)

P= 0 .

 0 ij Q ji dx. Here, n c is the number of components of u. When the flow is not controlled, u c = 0 and When U = {u i } i=1,...,Nt , we have u m = 0 and A GP i = 0.

  Table II reports the normalized errors

	10
	1
	0.1
	0.01
	1	2	3	4	5	6

ACKNOWLEDGEMENTS

A preliminary version of this work was presented at the workshop on "Industrial applications of low order models based on POD" organized by Angelo Iollo (IMB -Bordeaux University and MC2 -INRIA Bordeaux Sud-Ouest). This workshop held at Bordeaux on March 31 -April 2, 2008 was very stimulating and nevertheless friendly. LC is very indebted to Angelo for inviting him to give a talk. The extension of the Tikhonov regularization method to the first and second order presented in this paper was suggested by I. M. Navon (School of Computational Science and Department of Mathematics, Florida State University) during the workshop. LC is very grateful to him for this suggestion. Last but not least, LC would like to acknowledge Bernd R. Noack and the low-dimensional modeling and control team at the Technische Universität Berlin for stimulating and fruitful discussions.


Metropolitics, 5 March 2014.

The sustainable metropolis: when night enlightens day

Luc Gwiazdzinski (*)

The night is a wonderful space for renewal and replenishment, for invention and new projects for our changing cities, and for their inhabitants in search points of reference. It is time to put the night firmly on the agenda and anticipate the inevitable development of night-time activities, with the aim of thinking seriously about the overall development of the 24-hour city and a new form of sensory, temporal and temporary urbanism.

The night: a temporal space under pressure Urban nights are not deserts. In the unique temporal space that is the night, a part of social and economic life is now forever awake, creating other centralities and sociabilities. From illuminations to all-night cultural events, power cuts, unsafe roads, noise pollution, light pollution and urban violence, the night makes its way into the daytime news, for better or for worse. Over the last 15 years or so, the colonisation of the urban night-time has been accelerating and we have seen a gradual "diurnisation" of the night with the arrival of mundane daytime activities. The frontier between day and night is moving and pressure is growing under the effects of the non-stop nature of the economy and various networks. The night, like other traditional periods of rest, such as Sundays or lunch breaks, is gradually being eroded by human activity. Businesses run continuously in order to optimise their production facilities, while financial markets and the internet never stop. Light has gradually taken possession of the urban space, partly erasing the threatening darkness of our nights, allowing the prolongation of regular activities and the transformation of the public space. The nighttime economy continues to develop and the key players of this economy are becoming more organised. In Europe, night work now concerns almost 18% of employees, mainly men. In many countries, legislation on stores' opening hours has been relaxed, although in France the subject remains controversial, as illustrated by recent rulings (and sanctions) on store chains that stay open after 9 p.m. One consequence of these developments has been that the urban night has become nothing more than a three-hour lull in the activity of large cities. Although the night is not as dangerous as we are led to believe, neither is it the space of freedom and encounters dreamed of by poets. As we advance into the night, the options on offer decrease, and the city shrinks and seems to condense itself into a few clusters of streets where we find concentrations of illumination and animation. The night is expensive, and any sense of social and generational diversity is illusory. Reduced public transport limits night-time use of the metropolis and movement between centre and suburbs.

Colonised by light and daytime activities, the night -outpost of the day -is the scene of temporal conflicts between different neighbourhoods, activities, groups and individuals within the city. The city that sleeps, the city that parties and the city that works do not always make good bedfellows. Tensions arise in the archipelago of urban nights, and borders are built up that challenge and compromise harmonious living.

Night-time, sustainable development and innovation

The night reveals humankind in all its paradoxes. This presents a major challenge for the public authorities, who have to engage debate and discussions in order to promote sustainable nocturnal development, which seeks to balance economic development, environmental concerns, social cohesion and culture. It is in the urban night, that curious temporal space, that the tensions and contradictions between the economic and the social, environment and culture, are probably most visible. It is in this space that part of our ability to live together comes into play. Improving our quality of urban life cannot come at the expense of other individuals in the city and in the world, or indeed of future generations. By masking this debate, or by dismissing it as belonging exclusively to the private sphere of our lives, the economy will dictate its law to the weakest among us, who do not have the option of choosing between a taxing night job and unemployment. Far from the constraints of the day, the night is a living laboratory, enabling us to reinvent the daytime and imagine cities that are generally more accessible and hospitable. The concepts of centrality, diversity, urbanity, identity, accessibility, hospitality and habitability must be reconsidered in terms of the nocturnal and the temporary. The night, which falls within the remit of everyone and no one, can transcend institutional boundaries, and forces us to think laterally. It calls upon essential sensory dimensions, and allows us to incorporate users and artistic know-how into the production of the metropolis, and to imagine a public policy for the night that strikes a balance between the "right to the city" and the "right to the night".

An opportunity to think differently

Addressing night-time sustainable development calls for a salutary change of outlook. We can move from the vision of a dangerous territory to be controlled to one of a project space to be developed; from the commodification and spectacularisation of the night based on territorial marketing to a more humane approach; from hurried, unplanned management of usage conflicts to the long-term, strategic consideration of harmonious urban living according to different time frames; from the fun-filled night of revellers to the humdrum night of shift workers; from a sector-based approach to an integrated, transversal approach that takes account of the economic, social, environmental and cultural dimensions of the urban system; from conventional urban planning to timeoriented urban planning; and from a -supposedly rational -diurnal way of thinking to a more sensitive and uncertain nocturnal way of thinking. Beginning the process of re-examining the urban night means learning to manage the paradoxes of a hyper-modern metropolis by accommodating the needs of all: illuminating the night without causing it to disappear completely under a continuous sheet of light; preserving its original identity and its mystery; developing night-time activity without creating new usage conflicts; ensuring public safety without imposing curfews; opening up the night to economic activity while safeguarding the health of workers; ensuring a continuity between centre and suburbs without homogenising the way the city is used at night; regulating nightlife while allowing room for transgression; leaving some things unregulated without abandoning the night to market forces. It is up to us to decide, together, "how far not to go". Finally, making the most of the urban night means meeting the comfort-and informationrelated needs of all of its users, paying particular attention to those who are most excluded from it: women who have little access to the night, homeless people, young people and older people, isolated individuals with health problems, night workers, and residents of far-flung suburbs.

Initial experiments

Faced with these rapid changes, some cities adapt and experiment. Around the world, the trend is towards an increase in the frequency and service span of public transport, accompanied by the development of shared transport. From Paris to Montreal, the success of Nuit Blanche [1] events continues to grow. In Helsinki, night-time crèches have existed for many years now. In Spain, social, cultural and sports facilities and leisure opportunities for young people continue into the night, with positive effects on juvenile delinquency. In the UK, nightlife forms part of cities' marketing strategies and urban revitalisation programmes. Oslo and Rotterdam are experimenting with flexible and interactive lighting. Macau boasts of its airport open night and day, Hong Kong's public services are available online 24 hours a day, Rome proudly offers a citizen enquiry line that is open at all times, Paris still believes it is the "City of Light" and, for 2014, London is promising round-the-clock public transport. From Las Vegas to Ibiza, whole territories are specialised in 24/7 partying and nightlife. Around the world, cities wishing to attract businesses, tourists, students and the creative classes have incorporated the night into their territorial marketing strategies. Elsewhere, mediation devices, "correspondants de nuit" ("night-time mediators"), "chuchoteurs" ("whisperers") and "night-time charters" are being invented and help to pacify relations between residents and revellers and imagine other ways of organising and configuring the city at night. Major night-time events are an opportunity to experiment with new policies relating to culture, transport, lighting and public order, together with new partnerships between the public and private sectors. In Paris, Geneva and Lausanne, "General Assemblies of the Night" (États généraux de la nuit) have helped to initiate debate and outline a future night-time public space that corresponds to what Habermas described as "a symbolic place where public opinion is formed, as the result of political debate and the public use of reason". Everywhere, outside formal frameworks, users are creating and inventing. It is often at night that the world is set to rights and that people are able to break free from the constraints that the technical, rational daytime seeks to impose. It is at night that they distort and reuse objects and codes, reclaiming space and usages, in order to create a functional and metaphorical city that resists the dominant city. During the night-time -a lived-in, ephemeral and cyclic low-density territory -solutions are invented and developed that welcome a do-it-yourself attitude, the vernacular, little arrangements and exchanges here and there, "tricks", swaps, sharing and other frugal innovations that may also be of interest to the daytime city.

An integrated approach and new urbanism

Beyond these scattered initiatives, it is essential to develop an overall public policy regarding the night in order to "manage public needs and problems, and create policy". The night is of concern to everyone and is intimately linked to everyday life; it calls for partnerships and is an alternative means of approaching the creation of the city. It is an opportunity to break down the barriers between research and experimentation, between the city centre and "beyond the city", through the creation of "open urban innovation platforms" focused on creativity that bring together researchers, public authorities, professionals, artists and citizens, and which make it possible to envisage a comprehensive approach in terms of the economy, social affairs, the environment and culture. As true living observatories, these platforms could allow us to imagine a specific form of night-time governance. Potential areas for development could be: facilitating appropriation of the night by producing accurate maps of the night-time facilities and activities on offer; giving night workers, going about their jobs unseen in the shadow of diurnal corporate governance, the recognition they deserve; maintaining calm and tranquillity through natural social control linked to human presence; expanding and diversifying urban facilities by, for example, opening parks, gardens and other public places at night; improving the comfort and user-friendliness of spaces through versatile street furniture that is adapted to its environment; introducing night-time public services and defining a "time ecology" whereby each and every one us can make the most of our everyday life at night. As William Grossin points out, we are entitled to the respect of natural temporal environments, to which our biological organisms have adapted over millennia [START_REF] Grossin | Pour une science des temps. Introduction à l'écologie temporelle[END_REF].

When night enlightens day!

Through observation and awareness, projects and experiments, the night can -somewhat paradoxically -shed a new light on the city. Beyond statistical and organisational questions, between percepts and concepts, we must first of all experience the night, explore it in order to make the most of it, and agree to let go and let ourselves be enveloped by it and come round to a nocturnal way of thinking, a "trembling thought"in the words of Martinican writer Édouard Glissant -that contrasts with the certainties of day. In this sense, politicians, researchers, technicians and motivated citizens can turn to the night to refresh their outlook and perhaps discover the beginnings of possible futures for cities and society. Whether it is in terms of the economy, social affairs, the environment or culture, a part of the future of our metropolises already lies, waiting to be revealed, in the urban night.