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The determination of background velocity by full-waveform inversion (FWI) is known to be hampered by the local minima of the data misfit caused by the phase shifts associated with background perturbations. Attraction basins for the underlying optimization problems can be computed around any nominal velocity model, and they guarantee that the misfit functional has only one (global) minimum. The attraction basins are further associated with tolerable error levels representing the maximal allowed distance between the (observed) data and the simulations (i.e., the acceptable noise level). The estimates are defined a priori, and they only require the computation of (possibly many) the first- and second-order directional derivatives of the (model to synthetic) forward map. The geometry of the search direction and the frequency influence the size of the attraction basins, and the complex frequency can be used to enlarge the basins. The size of the attraction basins for the perturbation of background velocities in classic FWI (global model parameterization) and the data-space reflectivity reformulation (migration-based traveltime [MBTT]) are compared: The MBTT reformulation substantially increases the size of the attraction basins (by a factor of 4–15). Practically, this reformulation compensates for the lack of low-frequency data. Our analysis provides guidelines for a successful implementation of the MBTT reformulation.

International audience; In this paper, we provide an a priori optimizability analysis of nonlinear least squares problems that are solved by local optimization algorithms. We define attraction (convergence) basins where the misfit functional is guaranteed to have only one local-and hence global-stationary point, provided the data error is below some tolerable error level. We use geometry in the data space (strictly quasiconvex sets) in order to compute the size of the attraction basin (in the parameter space) and the associated tolerable error level (in the data space). These estimates are defined a priori, i.e., they do not involve any least squares minimization problem, and only depend on the forward map. The methodology is applied to the comparison of the optimizability properties of two methods for the seismic inverse problem for a time-harmonic wave equation: the Full Waveform Inversion (FWI) and its Migration Based Travel Time (MBTT) reformulation. Computation of the size of attraction basins for the two approaches allows to quantify the benefits of the latter, which can alleviate the requirement of low-frequency data for the reconstruction of the background velocity model.

International audience; Faults and geological barriers can drastically affect the flow patterns in porous media. Such fractures can be modeled as interfaces that interact with the surrounding matrix. We propose a new technique for the estimation of the location and hydrogeological properties of a small number of large fractures in a porous medium from given distributed pressure or flow data. At each iteration, the algorithm builds a short list of candidates by comparing fracture indicators. These indicators quantify at the first order the decrease of a data misfit function; they are cheap to compute. Then, the best candidate is picked up by minimization of the objective function for each candidate. Optimally driven by the fit to the data, the approach has the great advantage of not requiring remeshing, nor shape derivation. The stability of the algorithm is shown on a series of numerical examples representative of typical situations.; Les écoulements dans les milieux poreux peuvent être radicalement modifiés par la présence de failles ou de barrières géologiques.De telles fractures peuvent être modélisées comme des interfaces qui interagissent avec la matrice environnante. Nous proposons une nouvelle technique pour l'estimation de l'emplacement et des propriétés hydrogéologiques d'un petit nombre de grandes fractures dans un milieu poreux à partir de mesures distribuées de pression ou de flux données. À chaque itération, l'algorithme construit une courte liste de candidats par comparaison d'indicateurs de fracture. Ces indicateurs quantifient au premier ordre la décroissance d'une fonctiond'écart aux données; ils sont peut coûteux à calculer. Le meilleur candidat est ensuite isolé par minimisation de la fonctionobjectif pour chaque candidat. Guidée de façon optimale par la reproduction des données, l'approche a le grand avantage de ne pas nécessiter de remaillage, ni de dérivation de forme. La stabilité de l'algorithme est montrée sur une série d'exemples numériquesreprésentatifs de situations typiques.

The paper develops a reliable numerical method to solve inverse dynamical problem of seismic waves’ propagation on the base of nonlinear least squares formulation which is widely known as Full Waveform Inversion (FWI). The key issue on this way is correct reconstruction of macrovelocity component of the model with input seismic data without time frequencies less than 5 – 7 Hz and reasonable source – recievers offsets. To provide correct macrovelocity reconstruction we modify regular nonlinear least squares formulation used in standard versions of FWI by decomposing the model space into two subspaces slowly varying in space functions (propagators p) which do not change direction of propagation of seismic energy, but governs travel times; sharply changing in space functions (space reflectivity r) which do not change travel time, but turn propagation direction towards acquisition. In turn this reflector is represented as application of migration operator to new unknown called time reflectivity s connecting with space reflectivity by migration operator M(p): This decomposition modifies the cost function in the following way: which opens the possibility to implement minimization with respect to propagator and reflectivity as two different step-by-step processes. We present results of comparative analysis of numerical Singular Value Decompositions for both cost functions which reveal much more sensititve of the modified one with respect to the perturbation of a space of propagators. We use the modified cost function to do the FWI for well known Marmoussi2 dataset. The numerical results are presented and discussed.

This article deals with the inversion of spectroscopic measurements to obtain temperature and concentration profiles of a gas. The Radiative Transfer Equation (RTE) is discretized as a one-dimensional equation to simulate a spectrum and adjoint state method is used to compute the sensitivity matrix. The Singular Value Decomposition of the sensitivity matrix enables analyzing information content of a spectrum, and defining the number of retrievable parameters for a noise level. This number and the rank of the sensitivity matrix indicate the difficulty of data inversion. When inversion seems difficult, regularization is necessary and information about profiles is included using parameterization. This method is applied on CO2 radiation, where inversion of simulated data for Gaussian distributions is studied to know what to do with real case data. Finally, real case inversion is performed for spectroscopic measurements carried out on the exhaust of a motor.

(Received 11 January 2002; Revised 17 May 2002; In final form 15 February 2003)We address in this article the choice of both the step and the curve of the parameter space to be used in the linesearch part of descent algorithms for the minimization of least squares objective functions.Our analysis is based on the curvature of the path of the data space followed during the line search.We define first a new and easy to compute ‘‘maximum curvature step’’, which gives a guaranteed valueto the residual at the next iterate, and satisfies a linear decrease condition with ! ¼ 1=2.Then we optimize the ‘‘worst possible situation’’, by moving from one iterate to the next along a geodesicof the output set.Preliminary numerical comparisons of the proposed algorithm with the Gauss–Newton algorithm arepresented.

The link between Mixed Finite Element (MFE) and Finite Volume (FV) methods applied to elliptic partial differential equations has been investigated by many authors. Recently, a FV formulation of the mixed approach has been developed. This approach was restricted to 2D problems with a scalar for the parameter used to calculate fluxes from the state variable gradient. This new approach is extended to 2D problems with a full parameter tensor and to 3D problems. The objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy. This is achieved by defining one new variable per element. For the 2D case with full parameter tensor, this new formulation exists for any kind of triangulation. It allows the reduction of the number of unknowns to the number of elements instead of the number of edges. No additional assumptions are required concerning the averaging of the parameter in hetero- geneous domains. For 3D problems, we demonstrate that the new formulation cannot exist for a general 3D tetrahedral discretization, unlike in the 2D problem. However, it does exist when the tetrahedrons are regular, or deduced from rectangular parallelepipeds, and allows reduction of the number of unknowns. Numerical experiments and comparisons between both formulations in 2D show the efficiency of the new formulation. Copyright © 2003 John Wiley & Sons, Ltd.

A new resolution of parabolic and elliptic partial differential equations (PDEs) based on the mixed finite element approximation on triangles has been recently developed [24] , [25] . This new approach reduces the number of unknowns from fluxes or Lagrange multiplier defined on edges to a single unknown per element. In this paper, we analyze this transformation mathematically, and describe in details how to handle singular elements and singular edges. For these singular elements, the standard mixed method on triangles can always be made equivalent to a finite volume formulation, where the finite volumes are obtained by aggregation of finite elements across singular edges. The positive definiteness of the system matrix obtained with the new formulation is analyzed in details. A criterion is given concerning the property of this matrix which show that its conditioning is related to the shape of the triangle and the contrast in parameters from one element to the adjacent ones. Numerical experiments are performed for elliptic and parabolic PDEs. The comparisons between an iterative solver (PCG) and a direct solver (unifrontal/multifrontal) show that the direct solver is more efficient. Moreover, its performance is not correlated with the system matrix conditioning. It appears that the new formulation requires significantly less CPU time for elliptic PDEs and is competitive for parabolic PDEs. The new formulation remains also accurate enough even in nearly singular situations.

When estimating hydraulic transmissivity the question of parametrization is of great importance. The transmissivity is assumed to be a piecewise constant space-dependent function and the unknowns are both the transmissivity values and the zonation, the partition of the domain whose parts correspond to the zones where the transmissivity is constant. Refinement and coarsening indicators, which are easy to compute from the gradient of the least squares misfit function, are introduced to construct iteratively the zonation and to prevent overparametrization.

Output least squares stability for the diusion coecient in an elliptic equation in dimen- sion two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided. Mathematics Subject Classication. 62G05, 35R30, 93E24.

Migration‐based traveltime (MBTT) formulation provides algorithms for automatically determining background velocities from full‐waveform surface seismic reflection data using local optimization methods. In particular, it addresses the difficulty of the nonconvexity of the least‐squares data misfit function. The method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity. We present an implementation of the MBTT approach for a 2-D finite‐difference (FD) full‐wave acoustic model. Numerical analysis on a 2-D synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least‐squares approach, even when starting from very poor initial guesses. This enlargement of the domain of attraction for the global minima of the least‐squares misfit has a price: each evaluation of the new objective function requires, besides the usual FD full‐wave forward modeling, an additional full‐wave prestack migration. Hence, the FD implementation of the MBTT approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size.

Resume Pour resoudre des problemes d'equations aux derivees partielles decrivant l'ecoulement en milieu poreux sature, on montre comment construire un schema a une inconnue par maille a partir de la formulation en elements finis mixtes de Raviart-Thomas de plus bas degre. La nouvelle formulation sera etudiee sur une triangulation quelconque pour des problemes elliptiques en presence de termes puits/source. Le but est d'utiliser la methode des elements finis mixtes avec moins d'inconnues et ce sans aucune autre approximation ou integration numerique. On donne, a la fin de la Note, les criteres necessaires pour que la matrice finale, obtenue avec la nouvelle formulation, soit definie positive.

In order to define an optimal true‐amplitude prestack depth migration for multishot and multitrace data, we develop a general methodology based on the least‐squares data misfit function associated with a forward model. The amplitude of the migrated events are restored at best for any given geometry and any given preliminary filtering and amplitude correction of the data. The migrated section is then the gradient of the cost function multiplied by a weight matrix. A study of the Hessian associated with this data misfit shows how efficiently to find a good weight matrix via the computation of only few elements of this Hessian. Thanks to this matrix, the resulting migration operator is optimal in the sense that it ensures the best possible restoration of the amplitudes among the large class of least‐squares migrations. Applied to a forward model based on Born, ray tracing, and diffracting points approximation, this optimal migration outperforms or at least equals the classic Kirchhoff formula, since the latter belongs to the class of least‐squares migrations and is only optimal for one shot and an infinite aperture. Numerical results illustrate this construction and confirm the above expectations.

Regularization of nonlinear ill-posed inverse problems is analyzed for a class of problems that is characterized by mappings which are the composition of a well-posed nonlinear and an ill-posed linear mapping. Regularization is carried out in the range of the nonlinear mapping. In applications this corresponds to the state-space variable of a partial differential equation or to preconditioning of data. The geometric theory of projection onto quasi-convex sets is used to analyze the stabilizing properties of this regularization technique and to describe its asymptotic behavior as the regularization parameter tends to zero.

Nous considerons la determination, au sens des moindres carres, d'une fonction u dans un convexe ferme K a partir de la mesure z d'une quantite Tu dependant lineairement de u. Nous regularisons ce probleme par la norme L2 de u (coefficient alpha) et la semi-norme BV de la variation bornee de u (coefficient beta). Nous formulons d'abord les conditions d'optimalite du probleme regularise. Puis nous montrons qu'il admet, pour des valeurs donnees de alpha et beta, des solutions qui dependent de facon stable des donnees z. Nous etudions enfin le comportement asymptotique lorsque alpha=beta –> 0 : comme on pouvait s'y attendre, les solutions regularisees convergent vers la solution de norme L2+BV minimale du probleme non regularise. Le taux de convergence est beta**1/2 lorsque la solution de norme minimale est sufisamment reguliere.

In this paper the class of weakly nonlinear inverse problems is introduced. These problems are characterized by the property that the second derivative of the parameter-to-observation mapping can be bounded by the square of the first derivative of that mapping. Using geometric techniques it is shown that weakly nonlinear inverse problems behave similarly to linear inverse problems. In particular, their Tikhonov regularization leads to a family of quadratically well-posed problems. Examples involving the determination of source terms in semilinear reaction diffusion equations are given.

We address the problem of monitoring a linear functional (c, x)Eof an unknown vectorx of a Hilbert spaceE, the available data being the observationz, in a Hilbert spaceF, of a vectorAx depending linearly onx through some known operatorAeℒ(E; F). WhenE=E 1×E 2,c=(c 1 0), andA is injective and defined through the solution of a partial differential equation, Lions ([6]–[8]) introduced sentinelsseF such that (s, Ax)Fis sensitive to x1 eE 1 but insensitive to x2 e E2. In this paper we prove the existence, in the general case, of (i) a generalized sentinel (s, σ) e ℱ ×E, where ℱ ⊃F withF dense in 80, such that for anya priori guess x0 ofx, we have 〈s, Ax〉ℱℱ + (σ, x0)E=(c, x)E, where x is the least-squares estimate ofx closest tox 0, and (ii) a family of regularized sentinels (s n , σ n ) e F×E which converge to (s, σ). Generalized sentinels unify the least-squares approach (by construction !) and the sentinel approach (whenA is injective), and provide a general framework for the construction of “sentinels with special sensitivity” in the sense of Lions [8]).

We present an approach called multiple migration fitting (MMF) designed to automatically determine 2-D background velocities from prestack seismic data. In this approach, we maximize a scalar similarity index (SI) for a collection of migrated sections obtained by various illuminations of the same earth. Numerical investigation shows that this index is a rather smooth, nonoscillatory function of velocity that tends to be a maximum for good velocity profiles, and hence is amenable to maximization by local gradient techniques. This maximization will be practically feasible, as we prove that the exact gradient of SI can be computed at an additional cost of only twice that required for the computation of the collection of migrated sections, independently of the number of velocity unknowns. Application to synthetic data shows that MMF leads to enhanced background velocities and stacked migrated sections.

Selected groundwater flow scenarios are used in a two-way comparison between the mixed hybrid finite element method and the standard finite element method (also called the conforming finite element method). The simulations presented are performed in the bidimensional case with a triangular space discretization because of its practical interest for hydrogeologists. The basic idea of the mixed procedure is to approximate both the hydraulic potential and the velocity simultaneously and to satisfy an exact water balance for each element. By contrast, the conforming finite element method calculates the potential field everywhere and then calculates the velocity by differentiation of the potential. The conventional approach results in an elementwise constant velocity which can be subject to significant problems because of the normal component discontinuity of the velocity. The mixed hybrid finite element method provides velocities everywhere in the field, as well as potentials at the center of each element and each edge. Moreover, the normal component of the velocity field is continuous between adjacent elements. The results of the simulations are presented in the form of streamlines. To avoid the problem of velocity discontinuity, the method of Cordes and Kinzelbach (1992) is used; it allows the construction of a continuous velocity field from potentials obtained by the conforming finite element method. The comparison studies show that the mixed hybrid finite element is superior to the conforming method in terms of accuracy. It is also superior to the conforming method in terms of computational effort. The potential fields obtained by the mixed hybrid and the conforming finite element methods are the same.

In this paper convergence and rate of convergence results for nonlinear constrained ill-posed inverse problems formulated as regularized least squares problems are given.

International audience; We transpose an optimal control technique to the image segmentation problem. The idea is to consider image segmentation as a parameter estimation problem. The parameter to estimate is the color of the pixels of the image. We use the adaptive parameterization technique which builds iteratively an optimal representation of the parameter into uniform regions that form a partition of the domain, hence corresponding to a segmentation of the image. We minimize an error function during the iterations, and the partition of the image into regions is optimally driven by the gradient of this error. The resulting segmentation algorithm inherits desirable properties from its optimal control origin: soundness, robustness, and flexibility.; Nous transposons une technique de contrôle optimal au problème de segmentation d'image. L'idée est de considérer la segmentation d'image comme un problème d'estimation de paramètre où le paramètre à estimer est la couleur des pixels de l'image. Nous utilisons la technique de paramétrisation adaptative qui construit itérativement une représentation optimale du paramètre en régions uniformes qui forment une partition du domaine, et correspondent ainsi à une segmentation de l'image. Nous minimisons une fonction d'erreur au cours des itérations, et le partitionnement de l'image en régions est piloté de façon optimale par le gradient de cette erreur. L'algorithme de segmentation résultant hérite de propriétés intéressantes de son origine contrôle optimal: fondement, robustesse et flexibilité.

This study provides a stability theory for the nonlinear least-squares formulation of estimating the diffusion coefficient in a two-point boundary-value problem from an error-corrupted observation of the state variable. It is based on analysing the projection of the observation on the nonconvex attainable set.

This paper is devoted to the introduction and analysis of regularization in state space for nonlinear illposed inverse problems. Applications to parameter estimation problems are given and numerical experiments are described.

International audience; Multiphase flow modelling is a major issue in the assessment of groundwater pollution. Three-phase flows are commonly governed by mathematical models that associate a pressure equation with two saturation equations. These equations involve a number of secondary variables that reflect the fluid behaviour in a porous medium. To improve the computational efficiency of multiphase flow simulators, several simplified reformulations of three-phase flow equations have been proposed. However, they require the construction of new secondary variables adapted to the reformulated flow equations. In this article, two different approaches are compared to quantify these variables. A numerical example is given for a typical fine sand.

Summary An automatic numerical method is proposed for simultaneous determination of relative permeabilities and capillary pressure from the results of a single two-phase flow experiment in a range of velocities representative of field flow conditions. The experiment is a standard imbibition or drainage displacement with an imposed difference of pressure or flow rate. The experimental data used are the outlet production, the pressure drop between the two faces, and local saturation profiles at different times along the core. Relative permeabilities and capillary pressure are estimated with a least-squares technique. Use of a high-order numerical scheme significantly reduces the numerical diffusion. The optimal control theory is used to solve the minimization problem. The corresponding FORTRAN code is generated by a symbolic program. The validity of the method first is demonstrated with simulated data, then is tested in two laboratory experiments: one imbibition and one drainage. Good agreement is obtained for both situations.

We develop in this chapter sufficient conditions for a set \((D,\mathcal{P})\) to be s.q.c. As we have seen in Chap. 7, an s.q.c. set, which is characterized by the fact that RG(D) > 0, has necessarily a finite curvature, as R(D) ≥ RG(D) (see Proposition 7.2.9). But the condition $$R(D)\ >\ 0$$ (8.1) is not sufficient to ensure that \((D,\mathcal{P})\) is s.q.c.

The quasi-convex sets introduced in Chap. 6 do a good job in generalizing the properties of convex sets with respect to uniqueness, stability, and existence of the projection. But they miss their point on the subject of parasitic stationary points. So we shall start in this chapter from a complementary point of view, and introduce in Sect. 7.1 another family of sets, called the strictly quasi-convex sets (s.q.c. sets in short) which, almost by definition, will ensure the absence of parasitic stationary points. Note that the name “s.q.c.” has provisorily to be taken as a whole, as it will not be clear at all from the definition that s.q.c. sets are quasi-convex!

We consider in this chapter the nonlinear least squares (NLS) problem (1.10), which we recall here for convenience: $$\hat{x}\quad \mbox{ minimizes }\quad J(x) = \frac{1} {2}\|\varphi (x) - {z\|}_{F}^{2}\quad \mbox{ over }\quad C.$$ (4.1) As we have seen in Chap. 1, this inverse problem describes the identification of the parameter x ∈ C from a measurement z of φ(x) in F. We suppose that the minimum set of hypothesis (1.12) of Chap. 1 holds: $$\left \{\begin{array}{rcl} E\ & =&\ \mathrm{Banach\ space,\ with\ norm}{\quad \|\ \|}_{E}, \\ C\ & \subset &\ E\quad \mathrm{with}\ C\ \mathrm{convex\ and\ closed,} \\ F\ & =&\ \mathrm{Hilbert\ space,\ with\ norm}{\quad \|\ \|}_{F}, \\ z\ & \in &\ F \\ \varphi \ & : &\ C\ \rightsquigarrow \ F\ \mathrm{is\ differentiable\ along\ segments\ of\ C}, \\ \mathrm{and}& : &\exists \,{\alpha }_{M} \geq 0\ \mbox{ s.t. }\ \forall {x}_{0},{x}_{1} \in C,\ \forall t \in [0,1], \\ & &\|{D}_{t}\,\varphi {((1 - t){x}_{0} + t{x}_{1})\|}_{F} \leq \ {\alpha }_{M}\|{x}_{1} - {x{}_{0}\|}_{E},\end{array} \right.$$ (4.2) and we recall the definition of stationary points

We present in this chapter the nonlinear least-squares (NLS) approach to parameter estimation and inverse problems, and analyze the difficulties associated with their theoretical and numerical resolution.

We consider in this chapter various approaches for the regularization of the general NLS problem (1.10), recalled here for convenience: $$\hat{x}\quad \mbox{ minimizes }\quad J(x) = \frac{1} {2}\|\varphi (x) - {z\|}_{F}^{2}\quad \mbox{ over }\quad C. $$ (5.1) and we suppose throughout the chapter that it satisfies the minimum set of hypothesis (1.12) or (4.2).

In this chapter, we define a new class of subsets of a Hilbert space, called the quasi-convex sets to which properties (i) (uniqueness), (iii) (stability) and (iv) (existence as soon as the set is closed) of Proposition 4.1.1 can be generalized, provided they are required to hold only on some neighborhood. Technically, the whole chapter will consist in adapting the classical proofs for convex sets to the case where the segments are replaced by paths with finite curvature.

The need for more precise simulation of the transport of pollutants by underground water has drawn the attention of hydrologists to a class of approximation techniques known by the generic name of Mixed Finite Element Methods. The basic idea here is to approximate simultaneously the pressure P and its gradient, or more generally a gradient related velocity field q in such a way that both P h and q h can be proved to converge, in adequate norms, to their continuous counterparts, and that the approximated velocity field q h retains one important feature of the exact velocity field q, namely that it is continuous on each element and has a continuous normal component when passing from one element to the other

Given a closed, not necessarily convex set D of a Hilbert space, the problem of the existence of a neighborhood $\mathcal{V}$ on which the projection on D is uniquely defined and Lipschitz continuous is considered, and such that the corresponding minimization problem has no local minima. After having equipped the set D with a family $\mathcal{P}$ of paths playing for D the role the segments play for a convex set, the notion of strict quasiconvexity of $(D,\mathcal{P})$ is defined, which will ensure the existence of such a neighborhood $\mathcal{V}$. Two constructive sufficient conditions for the strict-quasiconvexity of D are given, the $R_G $-size $ \times $ curvature condition and the $\Theta $-size $ \times $ curvature condition, which both amount to checking for the strict positivity of quantities defined by simple formulas in terms of arc length, tangent vectors, and radii of curvature along all paths of $\mathcal{P}$. An application to the study of wellposedness and local minima of a nonlinear least...

We define a family of sets of a Hilbert space (“quasi-convex sets”) on which a generalization of the usual theory of projection on convex sets can be defined (existence, uniqueness, and stability of the projection of all points of some neighborhood of the set). We then give a constructive sufficient condition, called the size × curvature condition, for a setD to be quasi-convex, which involves radii of curvatures of curves lying on the setD. Finally, we use the above result for the study of nonlinear least-squares problems, as they appear in parameter estimation, for which we give a sufficient condition ensuring existence, uniqueness, and stability.

Nonlinear least squares has proven to be a very useful tool for the estimation of parameters in groundwater modeling. The reason for this success is the power of the approach: the only fundamental requirement is to posses one numerical simulator able to calculate the output ~(.,c) of the model (for example the computed pressure history) once a value x (for example the hydraulic conductivity map) has been assessed to the unknown parameter (hence cp denotes the parameter --* output or modelling operator). Given a value z of the experimental data (for example the measured pressure history), nothing can prevent you from forming the least-squares error

International audience; The estimation of distributed parameters in partial differential equations (PDE) from measures of the solution of the PDE may lead to under-determination problems. The choice of a parameterization is a usual way of adding a-priori information by reducing the number of unknowns according to the physics of the problem. The refinement indicators algorithm provides a fruitful adaptive parameterization technique that parsimoniously opens the degrees of freedom in an iterative way. We present a new general form of the refinement indicators algorithm that is applicable to the estimation of multi-dimensional parameters in any PDE. In the linear case, we state the relationship between the refinement indicator and the decrease of the usual least-squares data misfit objective function. We give numerical results in the simple case of the identity model, and this application reveals the refinement indicators algorithm as an image segmentation technique.; L'estimation de paramètres distribués dans des équations aux dérivées partielles (EDP) à partir de mesures de la solution de l'EDP peut mener à des problèmes de sous-détermination. Le choix d'une paramétrisation est un moyen usuel pour ajouter de l'information a priori en réduisant le nombre d'inconnues en relation avec la physique du problème. L'algorithme des indicateurs de raffinement fourni une technique de paramétrisation adaptative fructueuse qui ouvre parcimonieusement les degrés de liberté de façon itérative. Nous présentons une nouvelle forme générale de l'algorithme des indicateurs de raffinement qui s'applique à l'estimation des paramètres multi-dimensionnels dans toute EDP. Dans le cas linéaire, nous établissons le lien entre l'indicateur de raffinement et la décroissance de la fonction objectif des moindres carrés quantifiant l'erreur aux données. Nous donnons des résultats numériques pour le cas simple du modèle identité, et cette application permet de voir l'algorithme des indicateurs de raffinement comme une technique de segmentation d'image.

Summary The simulation of multiphase flow presents several difficulties, including (1) the occurrence of sharp moving fronts when convection is dominating, (2) the need for a good approximation of velocities to calculate the convective terms of the equation, and (3) flow singularities around wells. To handle the first difficulty, we propose a Godunov-type higher-order scheme based on a piecewise linear approximation of the saturation associated with a multidimensional slope limiter. With respect to the second, the pressure equation is approximated by means of a mixed-hybrid formulation equivalent to the classic mixed formulation but yielding a positive-definite linear system. To solve the third difficulty, we introduce macroelements around wells. Numerical experiments illustrate the capabilities of the method.