G Caumon

J L Mallet

3D Stratigraphic models: representation and stochastic modelling

Keywords: Stratigraphic modeling, uncertainty, time-stratigraphy, inverse modeling

Several representations have been proposed to model the 3D architecture and petrophysical properties of stratigraphic formations from subsurface data. All methods aim at finding a consistent way of describing heterogeneities by defining a curvilinear coordinate system conform to the sedimentary layers. Classically, this coordinate system is implicitly defined by a grid conforming to the strata and to the faults; alternatively, the Geochronological model (Geochron) explicitly defines this coordinate system, without using such a grid. This paper compares the grid-based stratigraphic modeling methods with the Geochron method. Their ability to represent complex fault networks is discussed, proving the higher representative power of the Geochron model. Because the visualization of uncertainty calls for generating several possible images of the subsurface, we also discuss grid perturbation methods, and introduce a new method to stochastically perturb a Geochron model.

Stratigraphic representations

The goal of a 3D geological model is to describe the geometry and petrophysical properties of the subsurface in a consistent manner with observation data. Stratigraphic formations, which host most of the world's hydrocarbon reserves, call for specific 3D modeling techniques. Such methods are not equivalent, since they may introduce model simplifications, especially in domains affected by faults.

Vertical scaling

Pioneering stratigraphic representations are based on vertical scaling. For instance, [START_REF] Johnson | Putting geology into reservoir simulations: a three-dimensional modeling approach[END_REF] use a 3D Cartesian Grid to represent formations, but rescale the z coordinate to examine correlations between wells. This model is then upscaled to a coarse corner-point geometry flow simulation grid. [START_REF] Swanson | A new geological volume computer modeling system for reservoir desciption[END_REF] directly creates a curvilinear simulation grid with constant cell area but with variable thickness (Fig. 1-A). The grid is aligned on sequence boundaries, and the layering inside sequences can mimic typical stratigraphic styles (Fig. 1-A). Vertical scaling can be applied only to non-faulted formations, and cannot handle overturned structures.

Stratigraphic gridding

Corner-point stratigraphic grids, in which grid blocks have an irregular geometry, can conform to strata, faults, or meandering channels [START_REF] Shtuka | Petrophysical simulation within an object-based reservoir model[END_REF][START_REF] Chambers | Geologic Modeling, Upscaling and Simulation of Faulted Reservoirs Using Faulted Stratigraphic Grids[END_REF]. These grids are suited to the representation of complex geological structures, provided that there is no constraint on the shape of their cells. Unfortunately, flow simulation using finite differences techniques calls for homogeneous cell sizes and sub-orthogonal cell corners. Therefore, a compromise must be found between geological accuracy and grid quality. For this, the choice of a grid construction method is paramount. Bennis et al (1996) proceed fault block by fault block, which is not practical for a large number of faults, and assumes that fault blocks have the same topology as a cube. Instead, the grid can be warped to honor horizons, and then cut by the fault network and interpolated (DeBaun et al, 1996;Fig. 1-B). This approach generates ill-shaped cells around faults, so an alternative is to compute a quadrangulated mesh on a horizon (2D parameterization), then extrude this mesh to the whole volume (Bombarde et al, 1996; Fig. 1-C) along extrusion lines (also called fibers or pillars). Curvilinear fibers [START_REF] Cognot | Fiber based sgrid construction[END_REF][START_REF] Fremming | 3d geological model construction using a 3d grid[END_REF] can be aligned on the fault network for more accuracy. None of these approaches easily accounts for pseudo-horizontal fault contacts between a synthetic an antithetic fault (Y, X or λ contacts). Attempts have been made to address this difficult problem, for instance by combining two 3D parameterizations [START_REF] Cognot | Fiber based sgrid construction[END_REF]). Geological accuracy is then possible, but the resulting grid cell geometry is not acceptable for flow simulation. Therefore, most current approaches first create a grid conform to the synthetic faults, and then use cut-based methods to generate stair-stepped antithetic faults.

Geo-chronological space

In structurally complex areas, the construction of a stratigraphic grid is extremely difficult and tedious to achieve, if only possible. The limited 3D grid resolution and the pseudoorthogonality between cell faces required by flow simulators necessarily introduce simplifications and violations of common geological sense (faults networks are not made of orthogonal and evenly spaced faults). Therefore, while faults have a tremendous impact on subsurface heterogeneities, their geometry and slip are often coarsely approximated in stratigraphic grids, independently of the construction method.

The problem of modeling faulted stratigraphic formations and that of generating reservoir grids need not be considered simultaneously: generating first a model of stratigraphic architecture and petrophysical properties can actually be done before thinking about the generation of a volumetric mesh. This idea is central to the concept of Geochronological Space (Geochron) proposed by Mallet (2004): for any point x inside a 3D geological model, the goal is to define three spatial functions u(x), v(x), t(x). u(x) and v(x) correspond to the paleo-geographic coordinates of point x at deposition time, before any tectonic deformation. t(x) corresponds to the geological time at which the point x was deposited. The parameter t is constant along isochronous sedimentary surfaces such as maximum flooding surfaces. The Geochron model is a 3D mathematical formulation of 2D Wheeler diagrams used in sedimentology [START_REF] Wheeler | Time-stratigraphy[END_REF]. Geochron is also a generalization over previous stratigraphic modeling approaches, since the areal gridding of a stratigraphic grid implicitly defines a (u, v) coordinate system on horizons (Fig. 2). However, in Geochron, the gradient of the t coordinate is not assumed to be vertical or aligned on the fault network (Fig. 2), as is the case for the layer stacking direction in stratigraphic grids. In the Geochron model, the stacking of layers can be orthogonal to stratigraphic horizons or conform to a flexural slip deformation mechanism. This means that distances can be conserved between chronostratigraphic and physical spaces, event if faults are not vertical or have a complex layout. The Geochron model can be created by interpolating the (u, v, t) coordinates on a tetrahedral mesh [START_REF] Moyen | Paramétrisation 3D de l'espace en Géologie sédimentaire : Le modèle Geochron[END_REF]. Geostatistical algorithms can be applied in (u, v, t) coordinates to populate the stratigraphic volume with properties. If needed, the physical space can be meshed, e.g. using techniques described by Prevost et al (2005).

Stratigraphic uncertainty modeling

Stratigraphic uncertainty modeling

Geostatistical simulation generates stochastically hundreds of petrophysical property models both on stratigraphic grids and in the Geochron framework. However, quantitative assessment of structural uncertainty is essential. Methods have been developed to perturb the geometry of stratigraphic grids, using autocorrelated random perturbations of the grid geometry [START_REF] Thore | Structural uncertainties: determination, management and applications[END_REF]. Depth and thickness variations typically affect the entire grid by moving cell corners in the vertical (or the w grid axis) direction to account for horizon uncertainty [START_REF] Corre | Integrated uncertainty assessment for project evaluation and risk analysis[END_REF]. Modeling of the uncertainty about fault positions is more difficult and may require a full grid reconstruction [START_REF] Thore | Structural uncertainties: determination, management and applications[END_REF]Lecour et al., 2001). Instead, free-form deformation techniques such as Simple Constrained Deformation [START_REF] Borrel | Simple Constrained Deformations for Geometric Modeling and DesignACM[END_REF]) can be applied to account for fault perturbations in an existing grid [START_REF] Zhang | Perturbation of fault network geometry on a stratigraphic grid[END_REF]. This approach is appropriate in reasonably complex models for small uncertainties, but cannot replace a full grid generation when structural uncertainty is very large. Methods to assess geometric uncertainty have not yet been described for the Geochron model. The implicit representation of horizons used by Moyen (2005)'s implementation is suited to an easy randomization of the position and thickness of strata: the t coordinate, available at every point in the reference model, can be incremented by some stochastic, spatially correlated scalar field computed either in GeoChron space or on the tetrahedral mesh. Uncertainty about the faults is, in the case of Geochron also, more difficult to model. When only small fault perturbations are needed to represent the geometrical uncertainty about faults, then the model updating methods described by [START_REF] Tertois | Preserving Geological Information during Real-Time Editing of Faults in Tetrahedral Models[END_REF] are adapted. When the structural uncertainty is larger, reliable methods must be defined to update locally a Geochron model while allowing for topological changes. This calls for adapting and improving existing methods for volume model updating [START_REF] Caumon | Building and Editing a Sealed Geological Model[END_REF] to Geochron models.

Future work

Stratigraphic modeling still raises a number of challenges. The Geochron model does not presently address gridding, which is required for simulation of physical processes. The work of Prevost et al ( 2005) must be pursued. Also, in all stratigraphic models, the scale and volume support would require more consideration, especially since the application of geostatistical methods assume that volumes are conserved between the physical and the chronostratigraphic spaces, which is not necessarily true.

Fig. 1 .

 1 Fig. 1. Stratigraphic gridding approaches. A: Vertical scaling according to sedimentary styles (from Swanson, 1988); B: Global grid warping followed by an interpolation; C: extrusion of a parameterized triangulated surface.

Fig. 2 .

 2 Fig. 2. Curvilinear stratigraphic coordinates in a stratigraphic grid (first line) and in a Geochron model (second line). Whereas (u, v, w) coordinates are aligned on the grid mesh, hence on some faults, (u, v, t) can be discontinuous across faults in the Geochron model, for a better accuracy.

Université de Liège -Belgium

Acknowledgments: The authors would like to thank all members of the Gocad Consortium for their support, and especially EarthDecision for providing the Gocad software. We also want to thank Richard Cognot, Laurent Deny Bruno Lévy and Rémi Moyen for their involvement in this research.


Joao Guilherme

Caldas Steinstraesser

Rodrigo Cienfuegos

José Daniel

Galaz Mora

Antoine Rousseau
email: antoine.rousseau@inria.fr

Daniel Galaz Mora

A Schwarz-based domain decomposition method for the dispersion equation

Keywords: domain decomposition method, Schwarz method, transparent boundary conditions, KdV equation

We propose a Schwarz-based domain decomposition method for solving a dispersion equation consisting on the linearized KdV equation without the advective term, using simple interface operators based on the exact transparent boundary conditions for this equation. An optimization process is performed for obtaining the approximation that provides the method with the fastest convergence to the solution of the monodomain problem.

Introduction

The Korteweg -de Vries (KdV) equation, derived by [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves[END_REF] in 1895, models the propagation of waves with small amplitude and large wavelength, taking in account nonlinear and dispersive effects. In terms of dimensionless but unscaled variables, it can be written as [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF] u t + u x + uu x + u xxx = 0 As done in [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF] (and in [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF] as a special case of their work), we will focus in this paper on the linearized KdV equation without the advective term :

u t + u xxx = 0 (1)
to which we will refer as dispersion equation.

The work developed here is inspired from [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF] and [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF]. Nevertheless, our objectives are different from theirs. In this paper we propose an additive Schwarz method (ASM) for solving the dispersion equation (1) in a bounded domain, i.e., we decompose the computational domain in subdomains and solve the timedependent problem in each one of them. Our work focuses on the formulation of appropriate and optimized conditions on the interface between the subdomains, in order to minimize the error due to the domain decomposition method (DDM) and to accelerate the convergence of the method.

The interface boundary conditions (IBCs) proposed here are based on the exact transparent boundary conditions (TBCs) for the equation [START_REF] Antoine | A review of Transparent Boundary Conditions or linear and nonlinear Schrödinger equations[END_REF], derived by [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF] and [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF]. The TBCs make the approximate solution in the computational domain coincide with the solution of the whole domain, but its exact computation is not doable in general [START_REF] Antoine | A review of Transparent Boundary Conditions or linear and nonlinear Schrödinger equations[END_REF]. [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF] and [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF] proposed numerical approximations for these conditions, seeking to reduce the error created by the introduction of artificial boundaries.

In the work presented here, we do not propose approximate TBCs for reducing the error related to the finitude of the computational domain. In fact, we intend to reduce the error created by the decomposition of the domain and the introduction of an artificial interface boundary condition, in the context of a DDM. In other words, we study the effectiveness of the boundary conditions as IBCs, not as TBCs. As a consequence, our work shall not use the same reference solution as the one used by [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF] and [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF]: for validating their approaches, they compare their approximate solution with the exact solution in the whole domain. On the other hand, our reference solution is the approximate solution computed on the computational monodomain. Moreover, in order to isolate the error due to the DDM from that originated by time discretization, we study our method locally in time, i.e., along one time step. This paper is organized as follows : In Section 2, we describe the DDM used here and we recall the exact TBCs derived by [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF] for equation [START_REF] Antoine | A review of Transparent Boundary Conditions or linear and nonlinear Schrödinger equations[END_REF]. Then, we propose approximations for them, leading to very simple mixed-type conditions (avoiding, for example, integrations in time) to be used as IBCs in the DDM. Small modifications are proposed for these IBCs such that the solution of the DDM problem converges exactly to the reference solution (the solution of the monodomain problem). In Section 3, we perform a large set of numerical tests in order to optimize the IBCs, in the sense that we search the coefficients that provide the fastest convergence for the DDM iterative process.

Resolution of the dispersion equation using a domain decomposition method

We propose in this section a DDM for solving the problem

               u t + u xxx = 0, x ∈ Ω, t ≥ t 0 u(t 0 , x) = u exact (t 0 , x), x ∈ Ω Υ 1 (u, -L) = 0, t ≥ t 0 Υ 2 (u, L) = 0, t ≥ t 0 Υ 3 (u, L) = 0, t ≥ t 0 (2) in the domain Ω = [a, b].
We firstly present a brief review of the DDM considered here, the parallel or additive Schwarz method (ASM) and then we propose IBCs for applying it to the problem solved in this paper.

The Schwarz Method

Domain decomposition methods allow to decompose a domain Ω in multiple subdomains Ω i (that can possibly overlap) and solve the problem in each one of them. Therefore, one must find functions that satisfy the PDE in each subdomain and that match on the interfaces.

The first DDM developed was the alternating or multiplicative Schwarz method, in which the IBCs for computing the solution in a subdomain are function of the most updated solution in the neighbor subdomains. We consider here a modified version of this algorithm, introduced more recently by [START_REF] Lions | On the Schwarz alternating method[END_REF] and known as parallel or additive Schwarz method. In this algorithm, the IBCs for computing the solution u k i , in the subdomain Ω i and iteration k, are always constructed using the solution u k-1 j , j = i, of the previous iteration in the neighbor subdomains.

This modification originates an inherently parallel algorithm, which one naturally implements with parallel computing. The advantages obtained with the parallelism become more evident when the number of subdomains increases [START_REF] Lions | On the Schwarz alternating method[END_REF].

In the ASM, the boundary condition for the problem in Ω i , in each interface between the subdomains Ω i and Ω j , can be written as

B i (u k+1 i ) = B i (u k j ) (3)
where (3), B i denotes the operator of the IBC. This operator allows the construction of more general Schwarz methods: in the original one, the IBCs are Dirichlet conditions (i.e., B i (u) = u ) [START_REF] Japhet | The best interface conditions for Domain Decompostion methods : Absorbing Boundary Conditions[END_REF][START_REF] Lions | On the Schwarz alternating method III: a variant for nonoverlapping sub-domains[END_REF]. Without loss of generality, in the following we consider a domain Ω decomposed in two non-overlapping subdomains, Ω 1 and Ω 2 , with Γ = Ω 1 Ω 2 .

When implementing a Schwarz method, one must define appropriate operators B i such that:

• There is a unique solution u i in each subdomain Ω i ;

• The solution u i in each subdomain Ω i converges to u| Ωi , i.e., the solution u, restricted to Ω i , of the problem in the monodomain Ω;

Moreover, one wants the method to show a fast convergence.

In fact, accordingly to [START_REF] Japhet | The best interface conditions for Domain Decompostion methods : Absorbing Boundary Conditions[END_REF], the optimal additive Schwarz method for solving the problem

A(u) = f in Ω u = 0 on ∂Ω
where A is a partial differential operator, is the one which uses as IBCs the exact TBCs for the problem, which are given by

B i (u) = ∂ ∂n i u + D2N (u)
where ∂n i is the outward normal to Ω i on Γ , and the D2N (Dirichlet to Neumann) operator is defined by

D2N : α(x) → ∂ ∂n c i v Γ
with α defined on Γ. v is solution of the following problem, solved in the complementary set of Ω i , denoted by

Ω c i      A(v) = f in Ω c i v = 0 on ∂Ω i \Γ v = α on Γ
The ASM using such exact TBCs is optimal in the sense that it converges in two iterations, and no other ASM can converge faster [START_REF] Japhet | The best interface conditions for Domain Decompostion methods : Absorbing Boundary Conditions[END_REF]. Nevertheless, these TBCs, in general, are not simple to compute both analytically and numerically. More specifically, they are nonlocal in time, so they must be approximated for an efficient numerical implementation [START_REF] Antoine | A review of Transparent Boundary Conditions or linear and nonlinear Schrödinger equations[END_REF]. These facts motivate us to look for simpler operators to use as IBCs in the ASM. We propose them based on the exact TBCs operators for the equation (1), as derived by [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF].

Interface boundary condition operators based on the exact TBCs for the dispersion equation

In [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF], TBCs are derived for the one-dimensional continuous linearized KdV equation (or Airy equation):

u t + U 1 u x + U 2 u xxx = h(t, x), t ∈ R + , x ∈ R (4)
where U 1 ∈ R, U 2 ∈ R + * and h is a source term, assumed to be compactly supported in a finite computational domain [a, b], a < b.

For the homogeneous initial boundary value problem

     u t + U 1 u x + U 2 u xxx = 0, t ∈ R + , x ∈ [a, b] u(0, x) = u 0 (x), x ∈ [a, b] +boundary conditions
the TBCs are given by [3, equations (2.17) -(2.18)]

u(t, a) -U 2 L -1 λ 1 (s) 2 s * u x (t, a) -U 2 L -1 λ 1 (s) s * u xx (t, a) = 0 u(t, b) -L -1 1 λ 1 (s) 2 * u xx (t, b) = 0 u x (t, b) -L -1 1 λ 1 (s) * u xx (t, b) = 0 (5)
where L -1 denotes the inverse Laplace transform, * the convolution operator, s ∈ C, Re(s) > 0, is the Laplace frequency and λ 1 is, among the three roots of the cubic characteristic equation obtained when solving (4) in the Laplace space and in the complementary set of [a, b], the only one with negative real part.

In this paper, we focus on the special case U 1 = 0, U 2 = 1, which results on the dispersion equation [START_REF] Antoine | A review of Transparent Boundary Conditions or linear and nonlinear Schrödinger equations[END_REF]. In this case, accordingly to [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF], the only root with negative real part is

λ(s) = λ 1 (s) = -3 √ s (6)
The computation of the TBCs ( 5) is not simple due to the inverse Laplace transform, which makes these conditions nonlocal in time. Therefore, we propose approximations of the root (6) that avoid integrations in time, making the operators considerably simpler.

Obviously, we do not expect these operators to be as accurate as the approximate TBCs proposed by [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF] (who derives TBCs for the discrete linearized KdV equation). Nevertheless, the objectives of our work and the work of [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF] are very different: while they seek to minimize the error of the computed solution (compared to the analytical one) due to the boundary conditions, we want here to apply our operators as IBCs in a DDM. Therefore, our objective lays on the convergence of the DDM to the solution of the same problem in the monodomain, independently of the errors on the external boundaries.

We use the constant polynomial P 0 (s) = c for approximating λ 2 /s (which can be seen as a (0,0) order Padé approximation). Moreover, as a consequence of (6), we can approximate the other operands of the inverse Laplace transforms in [START_REF] Blayo | Towards optimized Scwharz methods for the Navier-Stokes equations[END_REF] only in function of c :

λ 2 s = c, λ s = -c 2 , 1 λ(s) 2 = c 2 , 1 λ(s) = -c (7)
Replacing ( 7) in ( 5), using some well-know properties of the Laplace Transform (linearity and convolution) , we get the approximate transparent boundary conditions

Θ c 1 (u, x) = u(t, x) -cu x (t, x) + c 2 u xx (t, x) = 0 Θ c 2 (u, x) = u(t, x) -c 2 u xx (t, x) = 0 Θ c 3 (u, x) = u x (t, x) + cu xx (t, x) = 0 (8)
We notice that the approximation (8) has the same form as the exact TBCs for the equation (1) presented in [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF] and [START_REF] Besse | Discrete Artificial Boundary Conditions for the linearized Korteweg-de Vries Equation[END_REF], being the constant c an approximation for fractional integral operators.

We also remark that ( 8) are mixed-type boundary conditions (up to the second derivative of the solution), which, in the next section, we apply as IBCs in a DDM and we seek to optimize in order to accelerate the convergence of this method. The idea of using optimized boundary conditions in DDMs was already explored in [START_REF] Halpern | Optimized and Quasi-optimal Schwarz Waveform Relaxation for the one dimensional Schrödinger equation[END_REF] and [START_REF] Besse | Schwarz waveform relaxation method for the one-dimensional Schröedinger equation with general potential[END_REF], in the context of the Schrödinger equation.

Considering a discrete domain with mesh size ∆x and points x 0 , ..., x N and using some finite difference approximations, the operators (8) are discretized as

u 0 -c u 1 -u 0 ∆x + c 2 u 0 -2u 1 + u 2 ∆x 2 = 0 u N -c 2 u N -2u N -1 + u N -2 ∆x 2 = 0 u N -u N -1 ∆x + c u N -2u N -1 + u N -2 ∆x 2 = 0 (9)

ASM with the proposed IBCs

With the operators Θ c i defined, we now apply them in a DDM, with two nonoverlapping subdomains Ω 1 = [a, 0] and

Ω 2 = [0, b], a < 0 < b and interface Γ = Ω 1 ∩ Ω 2
Considering that we want to analyze and minimize the error due to the application of a DDM (isolating it from the error acumulated along the time steps, due to the temporal discretization), the reference solution u ref in our study is the solution of the monodomain problem (2) solved along onte time step (equation 10). Therefore, we implement a DDM to an evolution problem discretized in time (thus consisting in an ODE in space), an idea already explored by [START_REF] Blayo | Towards optimized Scwharz methods for the Navier-Stokes equations[END_REF].

               u(t0+∆t)-u(t0) ∆t + u xxx = 0, x ∈ Ω u(t 0 , x) = u exact (t 0 , x), x ∈ Ω Υ 1 (u(t 0 + ∆t), a) = 0 Υ 2 (u(t 0 + ∆t), b) = 0 Υ 3 (u(t 0 + ∆t), b) = 0 (10)
The external BCs Υ i , i = 1, 2, 3 (i.e., defined on ∂Ω i \Γ) are independent of the interface BCs. Here, we consider

Υ 1 = Θ c=1.0 1 , Υ 2 = Θ c=0.0 2 and Υ 3 = Θ c=0.0 3 , which gives Υ 1 (u, x) = u -u x + u xx = 0 Υ 2 (u, x) = u = 0 Υ 3 (u, x) = u x = 0
This choice was based on the simple form and implementation of these boundary conditions. Nevertheless, it does not have much importance in the study done here, as we want to analyze exclusively the behavior of the DDM. The only restriction for an appropriate study is that the external BCs for computing u ref must be the same Υ i , i = 1, 2, 3, used for each subdomain in the DDM, as we did in ( 11)-( 12) and [START_REF] Japhet | The best interface conditions for Domain Decompostion methods : Absorbing Boundary Conditions[END_REF].

The resolution of the problem (10) by the Additive Schwarz method and using the IBCs ( 9) is written as

               u k+1 1 (t0+∆t)-u k+1 1 (t0) ∆t + (u k+1 1 ) xxx (t 0 + ∆t) = 0, x ∈ Ω 1 u 0 1 (t 0 ) = u exact (t 0 , x), x ∈ Ω 1 Υ c 1 (u k+1 1 (t 0 + ∆t), a) = 0, Θ c 2 (u k+1 1 (t 0 + ∆t), 0) = Θ c 2 (u k 2 (t 0 + ∆t), 0), Θ c 3 (u k+1 1 (t 0 + ∆t), 0) = Θ c 3 (u k 2 (t 0 + ∆t), 0) (11)
               u k+1 2 (t0+∆t)-u k+1 2 (t0) ∆t + (u k+1 2 ) xxx (t 0 + ∆t) = 0, x ∈ Ω 2 u 0 2 (t 0 ) = u exact (t 0 , x), x ∈ Ω 2 Θ c 1 (u k+1 2 (t 0 + ∆t), 0) = Θ c 1 (u k 1 (t 0 + ∆t), 0) Υ c 2 (u k+1 2 (t 0 + ∆t), b) = 0 Υ c 3 (u k+1 2 (t 0 + ∆t), b) = 0 (12)
A simple analysis (for example in the Laplace domain) shows that the monodomain and DDM problems [START_REF] Japhet | The best interface conditions for Domain Decompostion methods : Absorbing Boundary Conditions[END_REF] and ( 11)-( 12) have an unique solution.

We also remark that our proposed DDM can be used for solving the problem (2), i.e., in a time window containing multiple time steps, by solving (11)- [START_REF] Lions | On the Schwarz alternating method[END_REF] in each time step, with the converged solution of the previous time step as initial data.

Remarks on the notation. In the following study of our proposed DDM, where we perform a spatial discretization, we introduce an extra subindex, so the solution is denoted as u k i,j , where i indicates the subdomain Ω i (or, in the case of the reference solution, i = ref , and in the convergence of the method, i = * ), j indicates the spatial discrete position and k indicates the iteration.

Spatial discretization of the problem

Concerning the spatial discretization, the monodomain Ω is divided in 2N +1 homogeneously distributed points, numbered from 0 to 2N . In all the analytical description, we consider that the two subdomains Ω 1 and Ω 2 have the same number of points, respectively x 0 , ..., x N and x N , ..., x 2N . The interface point x N is common to the two domains, having different computed solutions u k 1,N and u k 2,N in each one of them. Evidently, we expect, at the convergence of the method, that u ∞

1,N = u ∞ 2,N = u * N
As done in the initial numerical tests in the section 2.2, an implicit Finite Difference scheme is used here. For the interior points of each one of the domains, we consider a second order discretization for the third spatial derivative in equation (1):

u k+1 i,j -α i,j ∆t + -1 2 u k+1 i,j-2 + u k+1 i,j-1 -u k+1 i,j+1 + 1 2 u k+1 i,j+2 ∆x 3 = 0 ( 13
)
which is valid for j = 2, ..., N -2 in the case i = 1; for j = N + 2, ..., 2N -2 in the case i = 2; and for j = 2, ..., 2N -2 in the case i = ref .

In the above expression, α i,j is a given data (for example, the exact or the converged solution in the previous time step).

For the points near the boundaries, we use second order uncentered discretizations or the appropriate boundary condition. Considering that one boundary condition is written for the left boundary and two for the right one, we have to impose an uncentered discretization only for the second leftmost point of the domain. For example, for the point x 1 :

u k+1 1,1 -α 1,1 ∆t + -5 2 u k+1 1,1 + 9u k+1 1,2 -12u k+1 1,3 + 7u k+1 1,4 -3 2 u k+1 1,5 ∆x 3 = 0
and similarly to the other points near the boundaries.

In the resolution of the problem in Ω 1 , two IBCs are imposed (corresponding to Θ 2 and Θ 3 ) to the discrete equations for the points x N -1 and x N . On the other hand, in the resolution of the problem in Ω 2 , only one interface boundary condition is used (corresponding to Θ 1 ), being imposed to the point x N .

Remark : modification of the reference solution. Even if the DDM with the proposed IBCs is compatible with the monodomain problem (which we will see that is not the case), the solution of the DDM does not converge exactly to u ref , for a reason that does not depend on the expression of the IBCs, but on the fact that for each domain we write two boundary conditions in the right boundary and only one on the left boundary. We are using a second order centered discretization for the third spatial derivative (which uses a stencil of two points in each side of the central point), implying that we must write an uncentered discretization for the point x N +1 when solving the problem in Ω 2 . Therefore, this point does not satisfy the same discrete equation as in the reference problem. In order to avoid this incompatibility and allow us to study the behavior of the DDM, we will modify the discretization for the point u N +1 in the monodomain problem, using the same second-order uncentered expression :

u k+1 2,N +1 -α 2,N +1 ∆t + -5 2 u k+1 2,N +1 + 9u k+1 2,N +2 -12u k+1 2,N +3 + 7u k+1 2,N +4 -3 2 u k+1 2,N +5 ∆x 3 = 0
Figure 1 resumes the discretizations imposed to each point in the monodomain and the DDM problems, as described above:

Ω j = 0 1 2 N -2 N -1 N N + 1 N + 2 2N -2 2N -1 2N • • • • • • • • Ω 1 • • • ⊕ ⊕ Ω 2 ⊕ • • • • Centered 2nd order FD • Uncentered 2nd order FD External BC ⊕IBC

Corrections for the approximate IBCs

When using approximate IBCs in a Schwarz method, one should guarantee that the converged solutions u * satisfy the same equation as the solution u ref of the monodomain problem. Nevertheless, one can easily see that, in the convergence, the solution u * does not satisfy the discrete equation ( 13) on the points where the IBCs are imposed (the poins

x N -1 , x N ∈ Ω 1 and x N ∈ Ω 2 ).
As pointed out by [START_REF] Gander | Internal report no 469 -Optimal Schwarz waveform relaxation for the one dimensional wave equation[END_REF], a finite difference discretization of the IBCs requires a special treatment to be consistent with the monodomain discretization. Therefore, we will formulate modified IBCs in order to avoid this problem:

Θ c 1 (u k+1 2 ) + θ 1 = Θ c 1 (u k 1 ) + θ 1 Θ c 2 (u k+1 1 ) + θ 2 = Θ c 2 (u k 2 ) + θ 2 Θ c 3 (u k+1 1 ) + θ 3 = Θ c 3 (u k 2 ) + θ 3 ( 14
)
with θ i , θ i given by

θ 1 = ∆xc u k+1 2,N +1 -2u k+1 2,N + u k 1,N -1 ∆x 2 + c 2 ∆x ∆t u k+1 2,N -α 2,N θ 1 = -c 2 ∆x ∆t u k 1,N -α 1,N θ 2 = ∆x ∆t c 2 u k+1 1,N -α 1,N θ 2 = - ∆x ∆t c 2 u k 2,N -α 2,N θ 3 = 2 ∆x ∆t -∆x u k+1 1,N -1 -α 1,N -1 -c u k+1 1,N -α 1,N +∆x u k+1 1,N -3 -2u k+1 1,N -2 + u k+1 1,N -1 ∆x 2 θ 3 = 0
It is straightforward to verify that the DDM problem with these modifications in the IBCs insure that the converged solution u * satisfies, in every point, the same discrete equations as the solution u ref of the monodomain problem [START_REF] Japhet | The best interface conditions for Domain Decompostion methods : Absorbing Boundary Conditions[END_REF].

In addition, we notice that all the modification terms θ i , θ i , i = 1, 2, 3, are of order O(∆x) (they are composed of discrete versions of time derivatives and second spatial derivatives multiplied by ∆x). It is essential to insure that these terms are small, for the consistency with the approximate IBCs Θ i to be fulfilled.

Numerical tests for optimizing the IBCs (speed of convergence)

Our objective now is to optimize the IBCs in the sense of minimizing the number of iterations of our method until the convergence. We perform a very large set of tests in order to find the coefficient c that provide the fastest convergence. To start with, we make this study with fixed time step and space step, in order to analyze exclusively the influence of the coefficient.

As we are interested in the speed with which the solution of the DDM method converges to the reference solution, the criteria of convergence used is

e Ω,k ≤ ε with ε = 10 -9 and e Ω,k = ||u ref,N -u k N || 2 = ∆x   N j=0 u ref,j -u k 1,j 2 + 2N j=N u ref,j -u k 2,j 2  
The range of tested coefficients is [-10.0, 20.0] (chosen after initial tests to identify a proper interval), with a step equal to 0.1 between them (or even smaller, up to 0.005, in the regions near the optimal coefficients), and the maximal number of iterations is set to 100.

Test varying the initial time step and the interface position

As said above, in the first set of tests we consider a fixed time step ∆t = 20/2560 = 0.0078125 and a fixed mesh size ∆x = 12/500 = 0.024. Moreover, we consider two subsets of tests, in order to study the speed of convergence with different initial conditions and different sizes of the subdomains:

1. Tests varying the initial time step t 0 , with the interface in the center of the monodomain Ω = [-6, 6]; 2. Tests varying the position of the interface (x interf ace = -a + α(b -a), where b = -a = 6 and 0 < α < 1), for a fixed initial time t 0 = 0.78125.

In all the cases, the reference solution u ref is the solution of the monodomain problem [START_REF] Japhet | The best interface conditions for Domain Decompostion methods : Absorbing Boundary Conditions[END_REF].

The results are summarized in Figure 2, with the number of iterations plotted as function of the coefficient c (for the positive coefficients). We can see a very similar behavior of all the curves, with two minima whose position do not depend on t 0 and α (approximately, c = 0.20 and c = 4.5). For c < 0, the curves are very similar, with two minima located at c = -0.10 and c = -1.35, approximately. Moreover, the minima closest to zero (c = -0.10 and c = 0.20) are both associated with very discontinuous peaks, while the other two minima are associated with smoother curves. A detail of the curves around each positive minima are shown in Figures 2c -2d and 2e -2f. Finally, we remark that, for some curves, the minimal number of iterations is associated with the coefficients closest to zero, and, for other ones, to the other minimum, but the minimal number of iterations are very similar (between 5 and 7).

Tests varying ∆t and ∆x

After verifying that the method behaves similarly for several initial conditions (i.e., for several values of t 0 ) and various positions of the interface, now we keep these parameters fixed (t 0 = 0 and α = 0.5) and make new tests with different values of ∆t (with fixed ∆x = 12/250) and different values of ∆x (with fixed ∆t = 0.02).

The number of iterations as functions of the coefficient, for some of the tests, are shown in Figure 4, in the case of positive coefficients. The results for negative coefficients are similar.

Figure 5 presents the optimal positive coefficient for each ∆t or ∆x (for one fixed value for the other coefficient). Considering the observation we did before about the similar results (i.e. the number of iterations until the convergence) for the four optimal coefficients, we only took into account, for the construction of this curve, the positive minimum farther from zero: it was done because, as shown in Figure 4, these minima have a strong dependency on ∆t or ∆x, and we will seek to study this relation. 5 suggests a dependence of the optimal coefficient on (∆t) ν and (∆x) η , with 0 ≤ ν ≤ 1 and η < 0. In fact, performing some regressions with ∆t or ∆x fixed, we conclude that ν = 2 3 and η = -1 provide really well-fitted regression curves (with the coefficients of determination R 2 bigger than 0.99), both for the negative and the positive coefficients (although each one of these cases correspond to different curves). Therefore, we seek to model a function respectively for the positive and the negative optimal coefficients. The coefficients of determination of each regression are R 2,+ = 0.9999894 are R 2,-= 0.9998993, showing an excellent representation.

c opt (∆t, ∆x) = κ + α(∆t) 2 3 + β 1 ∆x + γ (∆t)
In order to validate the expressions (15) and ( 16), we used them to compute the optimal coefficients for several points (∆t, ∆x), with ∆t ∈ [0.0005, 0.3] and ∆x ∈ [12/5000, 12/50]. For almost all the points in the considered domain, the computed optimal coefficient provides a fast convergence to the monodomain solution, with less than 20 iterations, what is also observed in the case of the negative coefficients. The numbers of iterations observed are not always the smallest ones that we could find (cf. Figures 2 to 4), because the expressions (15) and ( 16) are regressions constructed from optimal coefficients obtained among a discrete set of possible values. Nevertheless, they give a very good approximation for the optimal c for each (∆t, ∆x), and one could search around a small region around the computed c opt to obtain an even faster convergence.

The results presented in this section show that the DDM proposed here is able to provide a fast convergence toward the solution of the monodomain problem. Furthermore, using the corrected IBCs [START_REF] Zheng | Numerical solution to a linearized KdV equation on unbounded domain[END_REF], this convergence is exact. Therefore, we reached our goals of solving the dispersion equation in a finite domain divided in two subdomains.

Moreover, the results of the optimization tests are very satisfying regarding a more general application of our method. Firstly, for fixed spatial and temporal discretizations, we obtained optimal coefficients for the method independently of the initial solution and the size of the subdomains (i.e., independently of the initial instant and the position of the interface). Secondly, we obtained good regression expressions for the optimal coefficient as function of ∆t and ∆x, which could allow the application of the model, with fast convergence, in other computational frameworks.

Conclusion and outlook

We presented and implemented in this paper an additive Schwarz method for the resolution of an one dimensional dispersive evolution equation, using as interface conditions between the subdomains some operators constructed based on the exact transparent boundary conditions for this equation. Although not as accurate (in the role of TBCs) as the ones proposed in the works we are based on (providing better TBCs was not our objective here), these approximate conditions stand out for its simple form and implementation and the fast convergence that they provide for the Schwarz method. Moreover, we also proposed small corrections to them, which insure that the solution of the DDM problem converges exactly to the solution of the monodomain problem. Finally, we verified that the speed of convergence depends on the time step, the mesh size and the (only) coefficient for constructing the approximate interface conditions; thus, via an optimization process, we obtained and validated regression expressions that provide the optimal coefficient (i.e., the one that provides the fastest convergence) in function of ∆t and ∆x.

Natural continuations of the work presented here would be the study of the method using more complex operators as IBCs, using for example higher-orders Padé approximations for λ 2 /s and considering different approximations for left and right boundary conditions. Moreover, we can extend this study for other problems, for instance the linearized KdV equation, which adds an advective term on the equation solved here, as well as other models of wave propagation. Finally, we can seek the development of global in time Schwarz methods, using optimized Scwharz waveform relaxation methods.

Figure 1 :

 1 Figure 1: Scheme indicating the discretization imposed to each point in the monodomain and the DDM problems

Figure 2 :

 2 Figure 2: Number of iterations until the convergence as function of the coefficient of the TBC, in the case of positive coefficients

Figure 3

 3 Figure3shows the evolution of the error, as function of the iterations, for the five positive coefficients c that gave the fastest convergences, for a fixed initial instant and a fixed position of the interface. For other values of t 0 and α this graph is similar, concerning the number of iterations and the fact that the convergence is more regular for the coefficients closest to zero, compared to the other optimal coefficients.

Figure 3 :

 3 Figure 3: Error evolution with the iterations for the fastest results

  (a) Fixed ∆x = 12 250 (b) Fixed ∆t = 0.02

Figure 4 :

 4 Figure 4: Number of iterations until the convergence as function of the coefficient of the TBC (for positive coefficients)

Figure 5 :

 5 Figure 5: Optimal coefficients as function of the time step and the space step

Figure

  Figure5suggests a dependence of the optimal coefficient on (∆t) ν and (∆x) η , with 0 ≤ ν ≤ 1 and η < 0. In fact, performing some regressions with ∆t or ∆x fixed, we conclude that ν = 2 3 and η = -1 provide really well-fitted regression curves (with the coefficients of determination R 2 bigger than 0.99), both for the negative and the positive coefficients (although each one of these cases correspond to different curves). Therefore, we seek to model a function

Acknowledgement

This study was conducted under the Marine Energy Research & Innovation Center (MERIC) project CORFO 14CEI2-28228, and thanks to the support of international partnerships department of Inria, through fundación Inria Chile.

The authors also want to thank Philippe Bonneton and Véronique Martin for fruitful discussions related to this work.


Laure Berti-Equille
email: laure.berti-equille@univ-amu.fr

Angela Bonifati
email: angela.bonifati@univ-lyon1.fr

Tova Milo
email: milo@post.tau.ac.il

Machine Learning to Data Management: A Round Trip

With the emergence of machine learning (ML) techniques in database research, ML has already proved a tremendous potential to dramatically impact the foundations, algorithms, and models of several data management tasks, such as error detection, data cleaning, data integration, and query inference. Part of the data preparation, standardization, and cleaning processes, such as data matching and deduplication for instance, could be automated by making a ML model "learn" and predict the matches routinely. Data integration can also benefit from ML as the data to be integrated can be sampled and used to design the data integration algorithms. After the initial manual work to setup the labels, ML models can start learning from the new incoming data that are being submitted for standardization, integration, and cleaning. The more data supplied to the model, the better the ML algorithm can perform and deliver accurate results. Therefore, ML is more scalable compared to traditional and time-consuming approaches. Nevertheless, many ML algorithms require an out-of-the-box tuning and their parameters and scope are often not adapted to the problem at hand. To make an example, in cleaning and integration processes, the window sizes of values used for the ML models cannot be arbitrarily chosen and require an adaptation of the learning parameters. This tutorial will survey the recent trend of applying machine learning solutions to improve data management tasks and establish new paradigms to sharpen data error detection, cleaning, and integration at the data instance level, as well as at schema, system, and user levels.

I. DETAILED OUTLINE OF THE TUTORIAL

The tutorial is organized into an introductory SWOT analysis (Section I-A), four main parts (described in detail from Section I-B to Section I-E), and final conclusions (Section I-F) as follows.

A. ML for Data Management: SWOT Analysis

The tutorial start with an introduction to the relevant concepts in Machine Learning from the data management perspective. We explore the use of ML techniques as a tool to express and quantify data patterns and knowledge transfer for representing and analyzing data, using examples from literature in database cleaning/repair, data integration, and query inference. We first provide an overview of the opportunities and limitations, alongside with the computational challenges associated with ML techniques applied to data management. We articulate the tutorial into four main parts related to the presented levels of data management: (1) instance-based; (2) schemabased; (3) system-based, and (4) user interaction-based data management.

B. ML in Instance-Based Data Transformation Tasks

In this part, we illustrate the role played by ML techniques in data processing at the data instance level, through ML applications to data cleaning, database repairing, and data fusion. This list of topics is by no means exhaustive albeit representative of the diversity of problems where ML tools have proved useful. We will review recent DB/ML research leveraging:

• Clustering applied to anomaly detection and data cleaning [START_REF] Chung | A Data Quality Metric (DQM): How to Estimate the Number of Undetected Errors in Data Sets[END_REF], [START_REF] Song | Turn Waste into Wealth: On Simultaneous Clustering and Cleaning over Dirty Data[END_REF], [START_REF] Zhang | Time Series Data Cleaning: From Anomaly Detection to Anomaly Repairing[END_REF], detection of patterns of glitches [START_REF] Berti-Equille | Discovery of complex glitch patterns: A novel approach to Quantitative Data Cleaning[END_REF], [START_REF] Berti-Equille | A masking index for quantifying hidden glitches[END_REF], replacement of erroneous or missing values, and deduplication [START_REF] Bilenko | Learnable Similarity Functions and their Applications to Clustering and Record Linkage[END_REF]; • Classification applied to database repairing [START_REF] Volkovs | Continuous Data Cleaning[END_REF], [START_REF] Yakout | Don't be SCAREd: Use SCalable Automatic REpairing with Maximal Likelihood and Bounded Changes[END_REF], regression classification used in record linkage [START_REF] Hu | Improving Temporal Record Linkage Using Regression Classification[END_REF], and kNN for data fusion; • Semi-and supervised learning models for similarity and blocking functions used in deduplication and record linkage [START_REF] Bhattacharya | Entity Resolution. Encyclopedia of Machine Learning and Data Mining[END_REF], [START_REF] Bilenko | Adaptive Blocking: Learning to Scale Up Record Linkage[END_REF], and active learning in entity resolution [START_REF] Whang | Pay-as-you-go Entity Resolution[END_REF];

• Bayesian analysis applied to data cleaning [START_REF] De | BayesWipe: A Scalable Probabilistic Framework for Improving Data Quality[END_REF], probabilistic inference in data repairing [START_REF] Rekatsinas | Holistic Data Repairs with Probabilistic Inference[END_REF], disambiguation (conflict resolution), and data fusion [START_REF] Dong | Data fusion: resolving conflicts from multiple sources[END_REF], [START_REF] Dong | Integrating Conflicting Data: The Role of Source Dependence[END_REF], [START_REF] Pradhan | Staging User Feedback toward Rapid Conflict Resolution in Data Fusion[END_REF]; and • Model optimization and statistical model training with guarantees [START_REF] Krishnan | ActiveClean: Interactive Data Cleaning For Statistical Modeling[END_REF] used in learning from samples for progressive data cleaning. Finally, we will discuss the pros and cons of ML applications to instance-based data management tasks and we will also review some work in ML to address noisy data labels and model robustness and discuss its applicability to data quality and integration.

C. ML in Schema-Based Data Transformation Tasks

In this part, we focus on the application of ML techniques to schema-driven tasks in data management, such as schema and constraint inference, schema mapping and query specification. Again, our list is not meant to be exhaustive but aims at fostering the discussion on the usage of learning in all its facets in the above tasks. The presentation will include the following topics:

• Schema and schema mapping discovery techniques [START_REF] Andritsos | Information-Theoretic Tools for Mining Database Structure from Large Data Sets[END_REF], [START_REF] Bex | Learning Deterministic Regular Expressions for the Inference of Schemas from XML Data[END_REF], [START_REF] Kimmig | A Collective Probabilistic Approach to Schema Mapping Discovery[END_REF], [START_REF] Miller | Schema Discovery[END_REF] to fight database decay and facilitate data integration; • Usage of input data examples to help the user specify complex data transformation tasks [START_REF] Bonifati | Interactive Schema Mapping Specification with Exemplar Tuples[END_REF], [START_REF] Jin | Foofah: Transforming Data By Example[END_REF]; • Usage of machine learning in data source reconciliation [START_REF] Doan | Reconciling Schemas of Disparate Data Sources: A Machine-Learning Approach[END_REF]; • Learning in rule discovery and information extraction [START_REF] Flach | Database Dependency Discovery: A Machine Learning Approach[END_REF], [START_REF] Hellerstein | The MADlib Analytics Library or MAD Skills, the SQL[END_REF], [START_REF] Li | Regular Expression Learning for Information Extraction[END_REF], [START_REF] Rostin | A Machine Learning Approach to Foreign Key Discovery[END_REF]; and • Query specification paradigms leveraging grammar induction techniques [START_REF] Antonopoulos | Definability Problems for Graph Query Languages[END_REF], [START_REF] Bonifati | Learning Join Queries from User Examples[END_REF], [START_REF] Bonifati | Learning Path Queries on Graph Databases[END_REF]. We will review recent approaches for data transformation, schema, and constraint discovery that can benefit from learning based on input data examples. We will discuss the advantages and limitations of these techniques and pinpoint a few extensions.

D. ML in System-Oriented Data Management Tasks

In this part, we will discuss about a recent trend in our community to use ML techniques for obtaining "trained database systems", i.e., databases that can learn from past query workloads (or past query executions and optimized plans) the behavior to be adopted in upcoming querying and tuning tasks. We will put under lenses the many ML approaches for database query optimization and bulk data processing systems by highlighting their advantages and their possible impact on full-fledged database systems. We will (not exhaustively) discuss about the following trends in the DB community:

• Predicting query answering based on the past history of queries [START_REF] Park | Database Learning: Toward a Database that Becomes Smarter Every Time[END_REF]; • Predicting the decisions of a query optimizer [START_REF] Schleich | Learning Linear Regression Models over Factorized Joins[END_REF] and performing database tuning [START_REF] Van Aken | Automatic Database Management System Tuning Through Large-scale Machine Learning[END_REF] by leveraging ML techniques; • Feature engineering and labeling are bottlenecks in ML techniques that hinder their adoption in database-oriented tasks; we will review works to facilitate those tasks [START_REF] Anderson | Input Selection for Fast Feature Engineering[END_REF], [START_REF] Bach | Learning the Structure of Generative Models without Labeled Data[END_REF], [START_REF] Ratner | Snorkel: A System for Lightweight Extraction[END_REF]; • Finally, we will focus on the connections between ML and Databases and the unsolved challenges in this area [START_REF]Frontiers in Massive Data Analysis[END_REF], [START_REF] Ré | Machine Learning and Databases: The Sound of Things to Come or a Cacophony of Hype? SIGMOD[END_REF].

E. ML in User-Guided Data Management Tasks

In this part, we will discuss the limitations of pure ML approaches and how users can help to complement the efforts. As illustrative examples, we will examine common data management tasks such as entity resolution and data cleaning [START_REF] Assadi | DANCE: Data Cleaning with Constraints and Experts[END_REF], [START_REF] Bergman | Query-Oriented Data Cleaning with Oracles[END_REF], [START_REF] Doan | Human-in-the-Loop Challenges for Entity Matching: A Midterm Report[END_REF], [START_REF] Geerts | The LLUNATIC Data-Cleaning Framework[END_REF], [START_REF] Wang | A Sample-and-Clean Framework for Fast and Accurate Query Processing on Dirty Data[END_REF], [START_REF] Yakout | Guided Data Repair[END_REF], [START_REF] Zhang | Crowdmatcher: Crowd-Assisted Schema Matching[END_REF]. We will consider the interplay between ML-based algorithms and crowd-sourcing, and highlight where users' input is essential. Specifically, we will discuss three dimensions of the problem:

• How users can help in improving the data itself, e.g., by detecting errors [START_REF] Thirumuruganathan | UGuide: User-Guided Discovery of FD-Detectable Errors[END_REF], gathering missing data and choosing among possible data repairs [START_REF] Assadi | DANCE: Data Cleaning with Constraints and Experts[END_REF], [START_REF] Yakout | Guided Data Repair[END_REF]; • How they can assist in gathering meta-data that facilitates improved data processing [START_REF] Bergman | Query-Oriented Data Cleaning with Oracles[END_REF], [START_REF] Wang | A Sample-and-Clean Framework for Fast and Accurate Query Processing on Dirty Data[END_REF]; • How can we find and identify the most relevant crowd to complement the ML efforts in a given data management task [START_REF] Doan | Human-in-the-Loop Challenges for Entity Matching: A Midterm Report[END_REF], [START_REF] Zhang | Crowdmatcher: Crowd-Assisted Schema Matching[END_REF].

F. Lessons Learned and Perspectives

We foresee two major outcomes from this tutorial. In the short term, we expect that this tutorial will lead to a more effective use of ML techniques in data management applications. In the long term, we hope that understanding the benefits and limits of the application of ML to the modeling, representation, and analysis of data will lead to a better interaction between data management and ML when designing the next-generation database management systems.

II. PRESENTERS' BIOGRAPHY

  Laure Berti-Equille received her Ph.D. degree in Computer Science from University of Toulon in France in 1999. From 2000-2010, she was a tenured Associate Professor at University of Rennes 1, and a 2-years visiting researcher at AT&T Labs Research in New Jersey, USA, as a recipient of the prestigious European Marie Curie Outgoing Fellowship (2007-2009). From 2011-2017, she joined IRD, the French Institute of Research for Development, as a Research Director. From 2014-2017, she was a Senior Scientist at Qatar Computing Research Institute (Hamad Bin Khalifa University). She is now is a full Professor at Aix-Marseille University (AMU) in France. Her interests are at the intersection of large-scale data science, data analytics, and machine learning with a focus on data quality and truth discovery research. She initiated the very first workshop editions on information and data quality in information systems (IQIS 2005) and in databases (QDB 2009 and 2016) in conjunction with SIGMOD and VLDB respectively, and co-organized the first French workshops on Data and Knowledge Quality in conjunction with EGC (Extraction et Gestion de Connaissances) in 2005, 2006, 2010, and 2011. Laure is serving as an associated editor of the ACM Journal on Data and Information Quality and served as a Program Chair of the International Conferences on Information Quality (ICIQ) in 2012 and 2016. She has received various grants from the French Agency for National Research (ANR), the French National Research Council (CNRS), and the European Union. Angela Bonifati received her Ph.D. degree in Computer Science from Politecnico di Milano in 2002. After graduating she worked as a postdoctoral researcher at the INRIA research institute in Paris. She then obtained a permanent position as a researcher at the Italian National Research Council in 2003. She is now a full Professor in France (since 2011), currently at University of Lyon 1. Her research focuses on advanced database applications such as data integration and exchange, web and graph databases, query inference by considering both structured and semi-structured data models. She has been visiting professor in several foreign universities, such as Stanford University, UBC and Saarland University. Angela served as the Program Chair of several international conferences, including ICDE 2011 (Semi-structured data Track) and ICDE 2018 (Information Extraction and Data Cleaning and Curation Track), WebDB 2013, and XSym 2009. She is currently associate editor of the VLDB Journal, ACM Transactions on Database Systems (TODS) and Distributed and Parallel Databases. She has been the recipient of the prestigious Palse Impulsion Starting

Grant at the University of Lyon (IDEX) in 2016. She has received grants from the French and Italian Ministry of Science and the French National Research Council (CNRS). Tova Milo received her Ph.D. degree in Computer Science from the Hebrew University, Jerusalem, in 1992. After graduating she worked at the INRIA research institute in Paris and at University of Toronto and returned to Israel in 1995, joining the School of Computer Science at Tel Aviv university, where she is now a full Professor. She is the head of the Database research group and holds the Chair of Information Management. She served as the Head of the Computer Science Department from 2011-2014. Her research focuses on large-scale data management applications such as data integration, semi-structured information, Data-centered Business Processes and Crowd-sourcing, studying both theoretical and practical aspects. Tova served as the Program Chair of several international conferences, including PODS, VLDB, ICDT, XSym, and WebDB, and as the chair of the PODS Executive Committee. She served as a member of the VLDB Endowment and the PODS and ICDT executive boards and as an editor of TODS and the Logical Methods in Computer Science Journal. Tova has received grants from the Israel Science Foundation, the US-Israel Binational Science Foundation, the Israeli and French Ministry of Science and the European Union. She is an ACM Fellow, a member of Academia Europaea, a recipient of the 2010 ACM PODS Alberto O. Mendelzon Test-of-Time Award, the 2017 VLDB Women in Database Research award, the 2017 Weizmann award for Exact Sciences Research, and of the prestigious EU ERC Advanced Investigators grant.