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The aim of this article is to propose a new reduced-order modelling approach for parametric eigenvalue problems arising in electronic structure calculations. Namely, we develop nonlinear reduced basis techniques for the approximation of parametric eigenvalue problems inspired from quantum chemistry applications. More precisely, we consider here a onedimensional model which is a toy model for the computation of the electronic ground state wavefunction of a system of electrons within a molecule, solution to the many-body electronic Schrödinger equation, where the varying parameters are the positions of the nuclei in the molecule. We estimate the decay rate of the Kolmogorov n-width of the set of solutions for this parametric problem in several settings, including the standard L2-norm as well as with distances based on optimal transport. The fact that the latter decays much faster than in the traditional L2-norm setting motivates us to propose a practical nonlinear reduced basis method, which is based on an offline greedy algorithm, and an efficient stochastic energy minimization in the online phase. We finally provide numerical results illustrating the capabilities of the method and good approximation properties, both in the offline and the online phase.

In this thesis, we investigate model reduction for parameter-dependent variational inequalities. First, we consider the case where the problem is formulated by a mixed method, where the challenge is that the reduced model must satisfy an inf-sup stability condition. For this purpose, we have devised a new greedy algorithm, based on a theoretical result, which allows to efficiently build a stable reduced model. In addition, we have also devised a method to construct the dual basis leading to a larger aperture. In the case of friction, a new idea deals with the enforcement of the tangential constraints in the reduced model by means of a collocation method. Numerical results are presented for the problem of contact with and without friction. A first integration in the industrial software code_aster has been achieved in the frictionless case. In a second step, we study the reduced basis method applied to the contact problem formulated with Nitsche's method. This method, which is purely primal, leads to an inefficient reduced model owing to the nonlinearity in the formulation. To overcome this problem, we propose an model reduction procedure based on the empirical interpolation method. Numerical tests confirm the robustness and efficiency of the approach.; Dans cette thèse, nous nous intéressons à la réduction de modèle pour les inéquations variationnelles dépendant d'un paramètre. Dans un premier temps, nous considérons le cas où le problème est formulé par une méthode mixte où la difficulté est que le modèle réduit doit satisfaire une condition de stabilité inf-sup. Pour ce faire, nous avons établi un algorithme glouton, basé sur un résultat théorique, qui permet de construire efficacement un modèle réduit stable. Nous avons également développé une méthode de construction de la base duale permettant de générer un cône de plus grande ouverture. Dans le cas du frottement, des nouveaux résultats concernent l'imposition des contraintes tangentielles dans le modèle réduit par une méthode de collocation. Des résultats numériques sont présentés pour le problème de contact avec et sans frottement, et une première intégration dans le code de calcul industriel code aster a été réalisée pour le cas sans frottement. Dans un deuxième temps, nous étudions la méthode des bases réduites appliquée au problème de contact formulé avec la méthode de Nitsche. Cette méthode, purement primale, conduit à un modèle réduit inefficace en raison de la non-linéarité. Pour palier ce problème, nous proposons une procédure de réduction de modèle s'appuyant sur la méthode d'interpolation empirique. Des tests numériques confirment la robustesse et l'efficacité de l'approche.

The optimization of internal structures positions according to the internal acoustic of a cavity can lead to the study of several configurations and thus may become prohibitive in terms of computational time; for instance such as for improving the passengers comfort in an airplane. The aim of this work is to propose a numerical tool able to efficiently compute both the pressure solution and its gradients with respect to the structure position. In order to be able to compute the solution for any position of the structure in the cavity, XFEM is used in this work to arbitrarily immerse the structure within the acoustic mesh allowing to always use the same acoustic mesh [1]. The use of the XFEM approach enables to easily compute the gradient of the pressure field with respect to the design variables which govern the position of the structure in the cavity. These gradients can be used to build a more accurate surrogate model [3]. At least, the computational time is reduced by using a reduced basis based on a component mode synthesis [2] with a fixed interface for the fluid domain, which is similar to the Craig- Bampton approach where the interface nodes are the enriched nodes of the XFEM. The whole strategy is applied on 2D and 3D cavities with immersed structures. Depending on the problem complexity, the CPU time is divided by a factor from 2 to 10. REFERENCES [1] A. Legay. An extended finite element method approach for structural-acoustic prob- lems involving immersed structures at arbitrary positions. International Journal for Numerical Methods in Engineering, 93(4):376–399, 2013. [2] A. Legay. The extended finite element method combined with a modal synthesis approach for vibro-acoustic problems. International Journal for Numerical Methods in Engineering, 101(5):329–350, 2015. [3] L. Laurent, R. Le Riche, B. Soulier and P.-A. Boucard. An Overview of Gradient- Enhanced Metamodels with Applications. Archives of Computational Methods in En- gineering, 26(1):61–106, 2019.

Running a reliability analysis on engineering problems involving complex numerical models can be computationally very expensive, requiring advanced simulation methods to reduce the overall numerical cost. Gaussian process based active learning approaches for reliability analysis are thus a way to strongly decrease the computational time by reducing the number of calls to the expensive numerical model while maintaining a sufficiently accurate estimation of the probability of failure. They involve a learning phase that consists in building a Gaussian process approximation of the performance function. Then, the failure probability estimation can typically be obtained by a classification of a sample population evaluated on the final surrogate model. Hence, two sources of uncertainty that are impacting the failure probability estimator can be identified: the metamodel approximation and the sampling variability. In this PhD thesis project, we propose a methodology to quantify the sensitivity of the failure probability estimator to these two uncertainty sources. This analysis is then integrated into an active learning algorithm in order to focus the enrichment on the reduction of the major source of uncertainty. It also allows to quantify and control the global error associated to the probability estimation and thus to define a stopping criterion for the algorithm. This approach is extended to tackle rare events probability estimation problems. Another way to reduce the numerical cost of reliability analyses that is studied in this thesis is related to model order reduction techniques. We propose a coupling between active learning approaches and model order reduction. More precisely, we set an adaptive methodology that allows to choose, on the basis of a coupling criterion, if a reduced solution can be used instead of the whole numerical model to evaluate the performance function.; Les analyses de fiabilité sur des problèmes d'ingénierie complexes impliquent souvent des temps de calcul très élevés et requièrent l'utilisation de méthodes numériques avancées.L'estimation de la probabilité de défaillance par des approches d'apprentissage actif de processus gaussiens est une voie possible pour fortement réduire les temps de calcul. Elles consistent en le classement d'une population d'échantillons à partir du modèle de processus gaussien construit. Ainsi, deux sources d'incertitude ont une influence sur l'estimateur de la probabilité de défaillance : l'approximation par métamodèle et la variabilité de l'échantillonnage. Dans ces travaux de thèse, nous proposons une méthodologie pour quantifier la sensibilité de l'estimateur de la probabilité de défaillance à ces deux sources d'incertitude. Une méthodologie focalisant l'enrichissement sur la réduction de la source majeure d'incertitude ainsi qu'un critère d'arrêt basé sur l'erreur globale sont proposés. L'approche est étendue pour l'estimation d'évènements rares.Une autre voie étudiée concerne la réduction du coût numérique unitaire d'évaluation des points d'apprentissage. Ainsi, nous proposons un couplage entre les approches d'apprentissage actif et la réduction de modèle de type base réduite. Une méthodologie adaptative permettant de choisir, par le biais d'un critère de couplage, si une solution réduite peut être utilisée à la place du modèle numérique complexe est proposée.

Running a reliability analysis on engineering problems involving complex numerical models can be computationally very expensive. Hence, advanced methods are required to reduce the number of calls to the expensive computer codes. Adaptive sampling based reliability analysis methods are one promising way to reduce computational costs. Reduced order modelling is another one. In order to further reduce the numerical costs of Kriging based adaptive sampling approaches, the idea developed in this paper consists in coupling both approaches by adaptively deciding whether to use reduced-basis solutions in place of full numerical solutions whenever the performance function needs to be assessed. Thus, a method combining such adaptive sampling based reliability analyses and reduced basis modeling is proposed using on an efficient coupling criterion. The proposed method enabled significant computational cost reductions, while ensuring accurate estimations of failure probabilities.

International audience; Additive printing allows the "single step" production of virtually any complex mechanical component. However, the manufacturing process involves a layer-by-layer deposition of material, which leads to an anisotropic mechanical behavior of the whole component. This would then entail a very fine 3D model to simulate the mechanical performance accurately. This simulation would also need to be integrated within an iterative design process in order to obtain the most efficient design. Both reasons explain the prohibitive number of calculations needed, which is currently beyond the capacities of existing software and computers. Recent research papers have opened promising pathways for integrating model reduction techniques within the overall topology optimization process. However, these approaches still present challenges such as choosing the minimum number and appropriate selection of the snapshots required to get accurate simulations. In this work, we present a methodology in the combined field of reduced-order modeling and topology optimization. The key idea consists of projecting the higher dimensional system of equations onto a lower dimensional space with the reduced basis vectors constructed using Proper Orthogonal Decomposition (POD). This reduced basis is updated in an incremental "on-the-fly" manner using alternatively costly high-fidelity and more rapid lower fidelity simulation snapshots. The variable-fidelity resolutions of successive approximations to the global system of equations are then integrated into the topology optimization process. The approaches are tested and computational savings and precision are compared, using both minimum compliance and compliant mechanism design benchmark problems.

Running a reliability analysis on engineering problems involving complex numerical models can be computationally very expensive, requiring advanced simulation methods to reduce the overall numerical cost. Gaussian process based active learning approaches for reliability analysis are thus a way to strongly decrease the computational time by reducing the number of calls to the expensive numerical model while maintaining a sufficiently accurate estimation of the probability of failure. They involve a learning phase that consists in building a Gaussian process approximation of the performance function. Then, the failure probability estimation can typically be obtained by a classification of a sample population evaluated on the final surrogate model. Hence, two sources of uncertainty that are impacting the failure probability estimator can be identified: the metamodel approximation and the sampling variability. In this PhD thesis project, we propose a methodology to quantify the sensitivity of the failure probability estimator to these two uncertainty sources. This analysis is then integrated into an active learning algorithm in order to focus the enrichment on the reduction of the major source of uncertainty. It also allows to quantify and control the global error associated to the probability estimation and thus to define a stopping criterion for the algorithm. This approach is extended to tackle rare events probability estimation problems. Another way to reduce the numerical cost of reliability analyses that is studied in this thesis is related to model order reduction techniques. We propose a coupling between active learning approaches and model order reduction. More precisely, we set an adaptive methodology that allows to choose, on the basis of a coupling criterion, if a reduced solution can be used instead of the whole numerical model to evaluate the performance function., Les analyses de fiabilité sur des problèmes d'ingénierie complexes impliquent souvent des temps de calcul très élevés et requièrent l'utilisation de méthodes numériques avancées.L'estimation de la probabilité de défaillance par des approches d'apprentissage actif de processus gaussiens est une voie possible pour fortement réduire les temps de calcul. Elles consistent en le classement d'une population d'échantillons à partir du modèle de processus gaussien construit. Ainsi, deux sources d'incertitude ont une influence sur l'estimateur de la probabilité de défaillance : l'approximation par métamodèle et la variabilité de l'échantillonnage. Dans ces travaux de thèse, nous proposons une méthodologie pour quantifier la sensibilité de l'estimateur de la probabilité de défaillance à ces deux sources d'incertitude. Une méthodologie focalisant l'enrichissement sur la réduction de la source majeure d'incertitude ainsi qu'un critère d'arrêt basé sur l'erreur globale sont proposés. L'approche est étendue pour l'estimation d'évènements rares.Une autre voie étudiée concerne la réduction du coût numérique unitaire d'évaluation des points d'apprentissage. Ainsi, nous proposons un couplage entre les approches d'apprentissage actif et la réduction de modèle de type base réduite. Une méthodologie adaptative permettant de choisir, par le biais d'un critère de couplage, si une solution réduite peut être utilisée à la place du modèle numérique complexe est proposée.

International audience; Monte-Carlo methods are well suited to characterize events of which associated probabilities are not too low with respect to the simulation budget. For very seldom observed events, these approaches do not lead to accurate results. Indeed, the number of samples are often insufficient to estimate such low probabilities (at least 10 +2 samples are needed to estimate a probability of 10 − with 10% relative deviation of the Monte-Carlo estimator). Even within the framework of reduced order methods, such as a reduced basis approach, it seems difficult to accurately predict low probability events. In this paper we propose to combine a cross-entropy method with a reduced basis algorithm to compute rare events (failure) probabilities.

International audience; The Laboratoire National des Champs Magntiques Intenses (LNCMI) is a member of the European Magnetic Field Laboratory (EMFL). This is the French facility enabling researchers to perform experiments in the highest possible magnetic fields. The numerical modeling jointly developed with the Center of Modeling and Simulation of Strasbourg (Cemosis) plays an essential role in the understanding and the optimization of such magnets. In this context we have implemented a full 3D multi-physics model coupling thermoelectric, magnetostatic and mechanic dedicated to the study of LNCMI resistive high field magnets. We have proposed a parallel algorithm to efficiently compute the magnetic field in the zone of interest for researchers using Biot and Savart law. Validated by dedicated campaigns of measurements, this model has confirmed the 3D behaviour of the magnetic field which has been observed in some experiments such as magnetic levitation. In order to design, control or perform advanced analysis - e.g. uncertainty quantification, sensitivity analysis - many queries or real-time evaluations of a our 3D finite element models are required. We propose in this paper a reduced basis methodology applied to the Biot-Savart law in order to reduce by several order of magnitudes the computational cost of the 3D finite element model. Not only does it provide an invaluable tool for magnet designers but it also provides direct access to non-expert in numerical simulation for example to visualize the 3D magnetic field maps in real time while changing the current density. The description of this tool will be illustrated by numerical results on real magnet geometries.

This work focuses on providing accurate low-cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In [E. Florentin, P. Diez, Adaptive reduced basis strategy based on goal oriented error assessment for stochastic problems, Comput. Methods Appl. Mech. Engrg. 225-228 (2012) 116-127], an adaptive strategy has been introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU cost gain. Technical insight of these two topics are presented in details and numerical examples show the interest of these new developments.

International audience; A new approximation technique, called Reference Point Method (RPM), is proposed in order to reduce the computational complexity of algebraic operations for constructing reduced-order models in the case of time dependent and/or parametrized nonlinear partial differential equations. Even though model reduction techniques enable one to decrease the dimension of the initial problem in the sense that far fewer degrees of freedom are needed to represent the solution, the complexity of evaluating the nonlinear terms and assembling the low dimensional operator associated with the reduced-order model still scales with the size of the original high-dimensional model. This point can be critical, especially when the reduced-order basis changes throughout the solution strategy as it is the case for model reduction techniques based on Proper Generalized Decomposition (PGD). Based on the concept of spatial, parameter/time reference points and influence patches, the RPM defines a compressed version of the data from which an approximate low-rank separated representation by patch of the operators can be constructed by explicit formulas at low-cost without resorting to SVD-based techniques. An application of the RPM to PGD-based model reduction for a nonlinear parametrized elliptic PDE previously studied by other authors with reduced-basis method and EIM is proposed. It is shown that computational complexity to construct the reduced-order model can be divided in practice by one order of magnitude compared with the classical PGD approach.

In geosciences, applications involving model calibration require a simulator to be called severaltimes with an optimization process. However, a single simulation can take several hours and a completecalibration loop can extend over several days. The objective of this thesis is to reduce the overall simulation timeusing reduced basis (RB) techniques.More specifically, this work is devoted to applying such techniques to incompressible two-phase water-oil flowsin porous media. Despite its relative simplicity in comparison to other models used in the petroleum industry,this model is already a challenge from the standpoint of reduced order modeling. This is due to the couplingbetween its equations, the highly heterogeneous physical data, as well as the choice of reference numericalschemes.We first present the two-phase flow model, along with the finite volume (FV) scheme used for the discretizationand relevant parameterizations in reservoir simulation. Then, after having recalled the RB method, we perform areduction of the pressure equation at a fixed time step by two different approaches. In the first approach, weinterpret the FV discretization as a Ritz-Galerkin approximation, which takes us back to the standard RBframework but which is possible only under severe assumptions. The second approach frees us of theserestrictions by building the RB method directly at the discrete level.Finally, we deploy two strategies for reducing the collection in time of pressures parameterized by the variationsof the saturation. The first one simply considers time as an additional parameter. The second one attempts tobetter capture temporal causality by introducing parameterized time-trajectories., En géosciences, les applications associées au calage de modèles d'écoulement nécessitent d'appelerplusieurs fois un simulateur au cours d'un processus d'optimisation. Or, une seule simulation peut durer plusieursheures et l'exécution d'une boucle complète de calage peut s'étendre sur plusieurs jours. Diminuer le temps decalcul global à l'aide des techniques de bases réduites (RB) constitue l’objectif de la thèse.Il s'agit plus précisément dans ce travail d'appliquer ces techniques aux écoulements incompressiblesdiphasiques eau-huile en milieu poreux. Ce modèle, bien que simplifié par rapport aux modèles utilisés dansl'industrie pétrolière, constitue déjà un défi du point de vue de la pertinence de la méthode RB du fait ducouplage entre les différentes équations, de la forte hétérogénéité des données physiques, ainsi que du choix desschémas numériques de référence.Nous présentons d'abord le modèle considéré, le schéma volumes finis (VF) retenu pour l'approximationnumérique, ainsi que différentes paramétrisations pertinentes en simulation de réservoir. Ensuite, après un brefrappel de la méthode RB, nous mettons en oeuvre la réduction du problème en pression à un instant donné ensuivant deux démarches distinctes. La première consiste à interpréter la discrétisation VF comme uneapproximation de Ritz-Galerkine, ce qui permet de se ramener au cadre standard de la méthode RB mais n'estpossible que sous certaines hypothèses restrictives. La seconde démarche lève ces restrictions en construisant lemodèle réduit directement au niveau discret.Enfin, nous testons deux stratégies de réduction pour la collection en temps de pressions paramétrées par lesvariations de la saturation. La première considère le temps juste comme un paramètre supplémentaire. Laseconde tente de mieux capturer la causalité temporelle en introduisant les trajectoires en temps paramétrées.

In geosciences, applications involving model calibration require a simulator to be called severaltimes with an optimization process. However, a single simulation can take several hours and a completecalibration loop can extend over several days. The objective of this thesis is to reduce the overall simulation timeusing reduced basis (RB) techniques.More specifically, this work is devoted to applying such techniques to incompressible two-phase water-oil flowsin porous media. Despite its relative simplicity in comparison to other models used in the petroleum industry,this model is already a challenge from the standpoint of reduced order modeling. This is due to the couplingbetween its equations, the highly heterogeneous physical data, as well as the choice of reference numericalschemes.We first present the two-phase flow model, along with the finite volume (FV) scheme used for the discretizationand relevant parameterizations in reservoir simulation. Then, after having recalled the RB method, we perform areduction of the pressure equation at a fixed time step by two different approaches. In the first approach, weinterpret the FV discretization as a Ritz-Galerkin approximation, which takes us back to the standard RBframework but which is possible only under severe assumptions. The second approach frees us of theserestrictions by building the RB method directly at the discrete level.Finally, we deploy two strategies for reducing the collection in time of pressures parameterized by the variationsof the saturation. The first one simply considers time as an additional parameter. The second one attempts tobetter capture temporal causality by introducing parameterized time-trajectories., En géosciences, les applications associées au calage de modèles d'écoulement nécessitent d'appelerplusieurs fois un simulateur au cours d'un processus d'optimisation. Or, une seule simulation peut durer plusieursheures et l'exécution d'une boucle complète de calage peut s'étendre sur plusieurs jours. Diminuer le temps decalcul global à l'aide des techniques de bases réduites (RB) constitue l’objectif de la thèse.Il s'agit plus précisément dans ce travail d'appliquer ces techniques aux écoulements incompressiblesdiphasiques eau-huile en milieu poreux. Ce modèle, bien que simplifié par rapport aux modèles utilisés dansl'industrie pétrolière, constitue déjà un défi du point de vue de la pertinence de la méthode RB du fait ducouplage entre les différentes équations, de la forte hétérogénéité des données physiques, ainsi que du choix desschémas numériques de référence.Nous présentons d'abord le modèle considéré, le schéma volumes finis (VF) retenu pour l'approximationnumérique, ainsi que différentes paramétrisations pertinentes en simulation de réservoir. Ensuite, après un brefrappel de la méthode RB, nous mettons en oeuvre la réduction du problème en pression à un instant donné ensuivant deux démarches distinctes. La première consiste à interpréter la discrétisation VF comme uneapproximation de Ritz-Galerkine, ce qui permet de se ramener au cadre standard de la méthode RB mais n'estpossible que sous certaines hypothèses restrictives. La seconde démarche lève ces restrictions en construisant lemodèle réduit directement au niveau discret.Enfin, nous testons deux stratégies de réduction pour la collection en temps de pressions paramétrées par lesvariations de la saturation. La première considère le temps juste comme un paramètre supplémentaire. Laseconde tente de mieux capturer la causalité temporelle en introduisant les trajectoires en temps paramétrées.

International audience; This contribution addresses the modelling and the stochastic analysis of transient thermal processes by means of the finite element method. It focuses on the theoretical presentation as well as the application of an efficient reduced basis strategy that advantageously lowers the dimension of the investigated system. The modal content of the reduced basis is driven by the goal oriented error assessment of a user-defined quantity of interest. The first section of the article presents the stochastic system of interest: key aspects of a stochastic analysis are recalled along with the employed spatial discretization. The newly developed adaptive reduced basis strategy is then detailed in the second section before extensive numerical investigations are carried out in order to validate it in the last section of the article. A numerical benchmark allowing for the confrontation of the proposed strategy with usual Monte-Carlo simulations highlights the benefits of the method that allows for a precise control of the maximum admissible error on the quantity of interest.; Cet article porte sur la modélisation et l'analyse stochastique de phénomènes thermiques transitoires par la méthode des éléments finis. Les développements effectués portent sur la présentation théorique et l'application d'une méthodologie efficace reposant sur l'utilisation de bases réduites et permettant ainsi une diminution de la dimension du système étudié. Le contenu modal de la base réduite est relié à l'évaluation d'erreur avec objectif d'une quantité d'intérêt choisie par l'utilisateur. La première section de l'article porte sur le système stochastique étudié, et permet un rappel des notions fondamentales associées à une analyse stochastique ainsi que des bases de la procédure de discrétisation spatiale. Par la suite, la deuxième section présente en détail la méthodologie développée. Enfin, dans une dernière section, plusieurs calculs permettent de valider cette méthodologie. Un point de comparaison numérique entre la méthodologie proposée et des simulations de Monte-Carlo mettent en évidence les avantages de la méthode proposée notamment au niveau de la précision de l'erreur maximale admissible sur la quantité d'intérêt.

International audience

International audience; In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element (RVE). We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model (ROM) can be built.

Model order reduction has become an inescapable tool for the solution of high dimensional parameter-dependent equations arising in uncertainty quantification, optimization or inverse problems. In this thesis we focus on low rank approximation methods, in particular on reduced basis methods and on tensor approximation methods.The approximation obtained by Galerkin projections may be inaccurate when the operator is ill-conditioned. For projection based methods, we propose preconditioners built by interpolation of the operator inverse. We rely on randomized linear algebra for the efficient computation of these preconditioners. Adaptive interpolation strategies are proposed in order to improve either the error estimates or the projection onto reduced spaces. For tensor approximation methods, we propose a minimal residual formulation with ideal residual norms. The proposed algorithm, which can be interpreted as a gradient algorithm with an implicit preconditioner, allows obtaining a quasi-optimal approximation of the solution.Finally, we address the problem of the approximation of vector-valued or functional-valued quantities of interest. For this purpose we generalize the 'primal-dual' approaches to the non-scalar case, and we propose new methods for the projection onto reduced spaces. In the context of tensor approximation we consider a norm which depends on the error on the quantity of interest. This allows obtaining approximations of the solution that take into account the objective of the numerical simulation.; Les méthodes de réduction de modèle sont incontournables pour la résolution d'équations paramétrées de grande dimension qui apparaissent dans les problèmes de quantification d'incertitude, d'optimisation ou encore les problèmes inverses. Dans cette thèse nous nous intéressons aux méthodes d'approximation de faible rang, notamment aux méthodes de bases réduites et d'approximation de tenseur.L'approximation obtenue par projection de Galerkin peut être de mauvaise qualité lorsque l'opérateur est mal conditionné. Pour les méthodes de projection sur des espaces réduits, nous proposons des préconditionneurs construits par interpolation d'inverse d'opérateur, calculés efficacement par des outils d'algèbre linéaire "randomisée". Des stratégies d'interpolation adaptatives sont proposées pour améliorer soit les estimateurs d'erreur, soit les projections sur les espaces réduits. Pour les méthodes d'approximation de tenseur, nous proposons une formulation en minimum de résidu avec utilisation de norme idéale. L'algorithme de résolution, qui s'interprète comme un algorithme de gradient avec préconditionneur implicite, permet d'obtenir une approximation quasi-optimale de la solution.Enfin nous nous intéressons à l'approximation de quantités d'intérêt à valeur fonctionnelle ou vectorielle. Nous généralisons pour cela les approches de type "primale-duale" au cas non scalaire, et nous proposons de nouvelles méthodes de projection sur espaces réduits. Dans le cadre de l'approximation de tenseur, nous considérons une norme dépendant de l'erreur en quantité d'intérêt afin d'obtenir une approximation de la solution qui tient compte de l'objectif du calcul.

This thesis contains three different topics, all related to electronic structure problems. These three topics are presented in three independent parts.This thesis begins with a general introduction presenting the problematics and main results.The first part is concerned with Density Functional Theory (DFT), for spin-polarized models. This part is divided in two chapters. In the first of these chapters, the notion of N-representability is introduced and the characterizations of the N-representable sets of spin-density 2X2 matrices are given. In the second chapter, we show how to mathematically treat the Zeeman term in spin-polarized DFT models. The existence of minimizers that was proved in (Anantharaman, Cancès 2009) for spin-unpolarized Kohn-Sham models within the local density approximation is extended to spin-polarized models.The second part of this thesis focuses on the GW approximation. We first give a mathematical definition of the one-body Green's function, and explain why methods based on Green's functions can be used to calculate electronic-excited energies of molecules. One way to compute an approximation of the Green's function is through the self-consistent GW equations. The well-posedness of these equations is discussed, and proved in the GW0 case in a perturbative regime. This is joint work with Eric Cancès and Gabriel Stoltz.In the third and final part, numerical methods to compute band-diagrams of crystalline structure are analyzed. This part is divided in two chapters.In the first one, we consider a perfect crystal in the reduced Hartree-Fock approximation (see (Cances, Deleurence, Lewin 2008)). We prove that, if the crystal is an insulator or a semi-conductor, then supercell calculations converge to the exact solution with an exponential rate of convergence with respect to the size of the supercell. This is joint work with Salma Lahbabi. In the last chapter, we provide a new numerical method to calculate the band diagram of a crystal (which can be either an insulator or a conductor). This method, based on reduced basis techniques, speeds up traditional calculations. This is joint work with Eric Cancès, Virginie Ehrlacher, and Damiano Lombardi, Cette thèse comprend trois sujets différents, tous en rapport à des problèmes de structures électroniques. Ces trois sujets sont présentés dans trois parties indépendantes.Cette thèse commence par une introduction générale présentant les problématiques et les principaux résultats.La première partie traite de la théorie de la fonctionnelle de la densité lorsqu'elle est appliquée aux modèles d'électrons avec spins polarisés. Cette partie est divisée en deux chapitres. Dans le premier de ces chapitres, nous introduisons la notion de N-représentabilité, et nous caractérisons les ensembles de matrices de densité de spin représentables. Dans le second chapitre, nous montrons comment traiter mathématiquement le terme de Zeeman qui apparaît dans les modèles comprenant une polarisation de spin. Le résultat d'existence qui est démontré dans (Anantharaman, Cancès 2009) pour des systèmes de Kohn-Sham sans polarisation de spin est étendu au cas des systèmes avec polarisation de spin.Dans la seconde partie, nous étudions l'approximation GW. Dans un premier temps, nous donnons une définition mathématique de la fonction de Green à un corps, et nous expliquons comment les énergies d'excitation des molécules peuvent être obtenues à partir de cette fonction de Green. La fonction de Green peut être numériquement approchée par la résolution des équations GW. Nous discutons du caractère bien posé de ces équations, et nous démontrons que les équations GW0 sont bien posées dans un régime perturbatif. Ce travail a été effectué en collaboration avec Eric Cancès et Gabriel Stoltz.Dans le troisième et dernière partie, nous analysons des méthodes numériques pour calculer les diagrammes de bandes de structures cristallines. Cette partie est divisée en deux chapitres. Dans le premier, nous nous intéressons à l'approximation de Hartree-Fock réduite (voir (Cances, Deleurence, Lewin 2008)). Nous prouvons que si le cristal est un insolant ou un semi-conducteur, alors les calculs réalisés dans des supercellules convergent exponentiellement vite vers la solution exacte lorsque la taille de la supercellule tend vers l'infini. Ce travail a été réalisé en collaboration avec Salma Lahbabi. Dans le dernier chapitre, nous présentons une nouvelle méthode numérique pour le calcul des diagrammes de bandes de cristaux (qui peuvent être aussi bien isolants que conducteurs). Cette méthode utilise la technique des bases réduites, et accélère les méthodes traditionnelles. Ce travail a été fait en collaboration avec Eric Cancès, Virginie Ehrlacher et Damiano Lombardi

International audience; In this paper, we present a reduced basis method for pricing European and American options based on the Black-Scholes and Heston model. To tackle each model numerically, we formulate the problem in terms of a time dependent variational equality or inequality. We apply a suitable reduced basis approach for both types of options. The characteristic ingredients used in the method are a combined POD-Greedy and Angle-Greedy procedure for the construction of the primal and dual reduced spaces. Analytically, we prove the reproduction property of the reduced scheme and derive a posteriori error estimators. Numerical examples are provided, illustrating the approximation quality and convergence of our approach for the different option pricing models. Also, we investigate the reliability and effectivity of the error estimators.

The Generalized Empirical Interpolation Method (GEIM) is an extension first presented by Maday and Mula in Maday and Mula (2013) in 2013 of the classical empirical interpolation method (presented in 2004 by Barrault, Maday, Nguyen and Patera in Barrault et al. (2004)) where the evaluation at interpolating points is replaced by the more practical evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in Maday and Mula (2013), this allows to relax the continuity constraint in the target functions and expand both the application domain and the stability of the approach. In this paper, we present a thorough analysis of the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it to an inf–sup problem in the case of Hilbert spaces. In the second part of the paper, it will be explained how GEIM can be employed to monitor in real time physical experiments by providing an online accurate approximation of the phenomenon that is computed by combining the acquisition of a minimal number, optimally placed, measurements from the processes with their mathematical models (parameter-dependent PDEs). This idea is illustrated through a parameter dependent Stokes problem in which it is shown that the pressure and velocity fields can efficiently be reconstructed with a relatively low-dimensional interpolation space., United States. Air Force Office of Scientific Research (FA9550-09-1-0613), United States. Office of Naval Research (ONR Grant N00014-11-1-0713)

This thesis contains three different topics, all related to electronic structure problems. These three topics are presented in three independent parts.This thesis begins with a general introduction presenting the problematics and main results.The first part is concerned with Density Functional Theory (DFT), for spin-polarized models. This part is divided in two chapters. In the first of these chapters, the notion of N-representability is introduced and the characterizations of the N-representable sets of spin-density 2X2 matrices are given. In the second chapter, we show how to mathematically treat the Zeeman term in spin-polarized DFT models. The existence of minimizers that was proved in (Anantharaman, Cancès 2009) for spin-unpolarized Kohn-Sham models within the local density approximation is extended to spin-polarized models.The second part of this thesis focuses on the GW approximation. We first give a mathematical definition of the one-body Green's function, and explain why methods based on Green's functions can be used to calculate electronic-excited energies of molecules. One way to compute an approximation of the Green's function is through the self-consistent GW equations. The well-posedness of these equations is discussed, and proved in the GW0 case in a perturbative regime. This is joint work with Eric Cancès and Gabriel Stoltz.In the third and final part, numerical methods to compute band-diagrams of crystalline structure are analyzed. This part is divided in two chapters.In the first one, we consider a perfect crystal in the reduced Hartree-Fock approximation (see (Cances, Deleurence, Lewin 2008)). We prove that, if the crystal is an insulator or a semi-conductor, then supercell calculations converge to the exact solution with an exponential rate of convergence with respect to the size of the supercell. This is joint work with Salma Lahbabi. In the last chapter, we provide a new numerical method to calculate the band diagram of a crystal (which can be either an insulator or a conductor). This method, based on reduced basis techniques, speeds up traditional calculations. This is joint work with Eric Cancès, Virginie Ehrlacher, and Damiano Lombardi, Cette thèse comprend trois sujets différents, tous en rapport à des problèmes de structures électroniques. Ces trois sujets sont présentés dans trois parties indépendantes.Cette thèse commence par une introduction générale présentant les problématiques et les principaux résultats.La première partie traite de la théorie de la fonctionnelle de la densité lorsqu'elle est appliquée aux modèles d'électrons avec spins polarisés. Cette partie est divisée en deux chapitres. Dans le premier de ces chapitres, nous introduisons la notion de N-représentabilité, et nous caractérisons les ensembles de matrices de densité de spin représentables. Dans le second chapitre, nous montrons comment traiter mathématiquement le terme de Zeeman qui apparaît dans les modèles comprenant une polarisation de spin. Le résultat d'existence qui est démontré dans (Anantharaman, Cancès 2009) pour des systèmes de Kohn-Sham sans polarisation de spin est étendu au cas des systèmes avec polarisation de spin.Dans la seconde partie, nous étudions l'approximation GW. Dans un premier temps, nous donnons une définition mathématique de la fonction de Green à un corps, et nous expliquons comment les énergies d'excitation des molécules peuvent être obtenues à partir de cette fonction de Green. La fonction de Green peut être numériquement approchée par la résolution des équations GW. Nous discutons du caractère bien posé de ces équations, et nous démontrons que les équations GW0 sont bien posées dans un régime perturbatif. Ce travail a été effectué en collaboration avec Eric Cancès et Gabriel Stoltz.Dans le troisième et dernière partie, nous analysons des méthodes numériques pour calculer les diagrammes de bandes de structures cristallines. Cette partie est divisée en deux chapitres. Dans le premier, nous nous intéressons à l'approximation de Hartree-Fock réduite (voir (Cances, Deleurence, Lewin 2008)). Nous prouvons que si le cristal est un insolant ou un semi-conducteur, alors les calculs réalisés dans des supercellules convergent exponentiellement vite vers la solution exacte lorsque la taille de la supercellule tend vers l'infini. Ce travail a été réalisé en collaboration avec Salma Lahbabi. Dans le dernier chapitre, nous présentons une nouvelle méthode numérique pour le calcul des diagrammes de bandes de cristaux (qui peuvent être aussi bien isolants que conducteurs). Cette méthode utilise la technique des bases réduites, et accélère les méthodes traditionnelles. Ce travail a été fait en collaboration avec Eric Cancès, Virginie Ehrlacher et Damiano Lombardi

International audience; SUMMARYTopology optimization of large scale structures is computationally expensive, notably because of the cost of solving the equilibrium equations at each iteration. Reduced order models by projection, also known as reduced basis models, have been proposed in the past for alleviating this cost. We propose here a new method for coupling reduced basis models with topology optimization to improve the efficiency of topology optimization of large scale structures. The novel approach is based on constructing the reduced basis on the fly, using previously calculated solutions of the equilibrium equations. The reduced basis is thus adaptively constructed and enriched, based on the convergence behavior of the topology optimization. A direct approach and an approach with adjusted sensitivities are described, and their algorithms provided. The approaches are tested and compared on various 2D and 3D minimum compliance topology optimization benchmark problems. Computational cost savings by up to a factor of 12 are demonstrated using the proposed methods. Copyright \textcopyright 2014 John Wiley & Sons, Ltd.

In the search of new approaches for the efficient exploitation of large scale compu- tational platforms, the parallelization in the time direction for time dependent problems is a very promising approach. Among the existing methods in this frame, the parallel in time method, since its introduction in [10], has been developed in many ways that, altogether, allow to identify its pros and cons. Among the cons, the current approaches present in practice some efficiency limitations as regards correct scalings that “spoil” the huge potential of the idea. This article is a contribution towards overcoming this major obstruction by exploiting the idea that the numerical schemes to parallelize time could be coupled to other iterative numerical algorithms that are needed to solve the PDE. We present a parareal scheme in which these alternative iterations are truncated (i.e. not converged) during each parareal iteration but in which convergence is nevertheless achieved across the parareal iterations. In order to limit the use of too much memory necessitated by the recovery of these alternative iterations over the parareal iterations, we propose also a compression procedure via proper orthogonal decomposition. After a mathematical analysis of the convergence properties of this new approach, we present some numerical results dealing with the application of the scheme to ac- celerate the time-dependent neutron diffusion equation in a reactor core. The numerical results show a significant improvement of the performances with respect to the plain parareal algorithm, which is an important step towards making the parallelization in the time direction be a fully competitive option for the exploitation of massively parallel computers.

International audience; Cylinders made of lightweight gypsum are extracted from industrial plasterboard and then indented in-situ in an X-ray tomograph. The results from the in-situ experiment show that a compacted zone develops under the indenter, displaying a very sharp boundary with the undamaged material. Tomographic imaging during the mechanical load associated with digital volume correlation enable the displacement fields to be measured during the test. However, because of the inhomogeneous nature of the indentation test, a high spatial resolution for the displacement is called for, and because the range of displacement amplitudes is small, uncertainties on the measured displacement and strain fields are large. In this study, a new methodology is presented to address integrated digital volume correlation based on a library of fields computed from a commercial finite element software. It allows many fluctuations in the estimated displacement fields to be filtered out and the measurement to be much more robust and reliable. This opens new pathways for the identification of mechanical properties.

International audience; This work concerns the Proper Generalized Decomposition (PGD), an a priori model reduction technique used to solve problems, eventually nonlinear, defined over the time-space domain. PGD seeks the solution of a problem in a reduced-order basis generated by a dedicated algorithm. This is the LATIN method, an iterative strategy which generates the approximations of the solution over the entire time-space domain by successive enrichments. Herein an algebraic framework adapted to PGD is proposed. It defines a compressed version of the data making less expensive the elementary algebraic operations.

International audience

International audience; In an effort to extend the classical lagrangian interpolation tools, new interpolating methods that use general interpolating functions are explored. The Generalized Empirical Interpolation Method (GEIM) belongs to this class of new techniques. It generalizes the plain Empirical Interpolation Method by replacing the evaluation at interpolating points by application of a class of interpolating linear functions. Since its efficiency depends critically on the choice of the interpolating functions (that are chosen by a Greedy selection procedure), the purpose of this paper is therefore to provide a priori convergence rates for the Greedy algorithm that is used to build the GEIM interpolating spaces.

We present a novel Block-Adaptive Quantum Mechanics (BAQM) approach to interactive quantum chemistry. Although quantum chemistry models are known to be computationally demanding, we achieve interactive rates by focusing computational resources on the most active parts of the system. BAQM is based on a divide-and-conquer technique and constrains some nucleus positions and some electronic degrees of freedom on the fly to simplify the simulation. As a result, each time step may be performed significantly faster, which in turn may accelerate attraction to the neighboring local minima. By applying our approach to the nonself-consistent Atom Superposition and Electron Delocalization Molecular Orbital theory, we demonstrate interactive rates and efficient virtual prototyping for systems containing more than a thousand of atoms on a standard desktop computer. © 2012 Wiley Periodicals, Inc.

The fied of the numerical computation is widely spread in the industrial process for complex systems. From the preliminary design to the manufacturing and the maintenance and overhaul, numerical simulations are many. Instead of intensive computations, reduced order modeling offers efficient alternative methods which aim at reducing the numerical cost of nonlinear problems by projecting the partial differential equations on a reduced basis. Moreover, as the exact description of underlying phenomena leads to complex constitutive laws, a simplification is considered by exploiting a truncated spatial domain over which the problem is considered. The studies carried out aim at providing some developments to the incremental and adaptative A Priori Hyper Reduction method in the field of heterogeneous materials for parametrized analysis. Those theoretical developements were implemented in the finite element code Z-set and used for a set of numerical examples on composite structures.; Dans le milieu aéronautique, la simulation numérique s'est largement imposée dans le développement industriel des systèmes complexes. De la conception au suivi en service, les simulations numériques sont nombreuses et variées. Afin de proposer des méthodes alternatives aux méthodes de calcul intensif, la réduction de modèle permet de réduire considérablement le coût de la résolution numérique des problèmes non linéaires en projetant les équations aux dérivées partielles sur une base réduite. De plus, la description des phénomènes physiques requiert une loi de comportement élaborée dont une simplification est considérée en exploitant une partie restreinte du domaine spatial. Les études effectuées ont pour but d'apporter certains développements à la méthode incrémentale et adaptative A Priori Hyper Reduction dans le cas des matériaux hétérogènes. Ces développements sont intégrés dans le code éléments finis Z-set et utilisés pour traiter une série d'exemples académiques sur des structures composites.

This thesis has two topics : numerical methods for acoustic wave propagation in a flow and reduced order models. In the first topic, we develop a coupled finite element and boundary element method to solve the convected Helmholtz equation, when the flow is uniform outside a bounded domain. In particular, we propose a formulation that is well-posed at all the frequencies of the source. In the second topic, we propose a solution to the classical problem of round-off error accumulation that occurs when computing the a posteriori error bound in the reduced basis method. Furthermore, we propose a non intrusive method for the approximation, in a separated representation form, of linear systems resulting from the finite-dimensional approximation of boundary-value problems depending on one or several parameters, Cette thèse s'articule autour de deux thématiques : les méthodes numériques pour la propagation d'ondes acoustiques sous écoulement et les méthodes de réduction de modèles. Dans la première thématique, nous développons une méthode de couplage d'éléments finis et d'éléments de frontière pour résoudre l'équation d'Helmholtz convectée, lorsque l'écoulement est uniforme à l'extérieur d'un domaine borné. En particulier, nous proposons une formulation bien posée à toutes les fréquences de la source. Dans la deuxième thématique, nous proposons une solution au problème classique d'accumulation d'arrondis machine qui survient en calculant l'estimateur d'erreur a posteriori dans la méthode des bases réduites. Par ailleurs, nous proposons une méthode non intrusive pour calculer une approximation sous forme séparée des systèmes linéaires résultant de l'approximation en dimension finie de problèmes aux limites dépendant d'un ou plusieurs paramètres

International audience; Due to the high dimensionality of the spaces where the problems are set, adapted discretization basis are often advocated in complex physical problems (Navier-Stokes equations, solid mecanics, ab initio electronic structure computations) to express the solution in terms of solution of similar (but easier to solve) problems. However, very few mathematical studies have been undertaken to asses the numerical properties of these approximations. Within this context, we will present in this paper an overview of the tools required to develop more rigorous reduced basis approaches for quantum chemistry: a posteriori numerical analysis and fast exponential decay of the n-width of the solution set.