Publications

2024

• Exploring low-rank structure for an inverse scattering problem with far-field data
  
  • Zhou Yuyuan
  • Audibert Lorenzo
  • Meng Shixu
  • Zhang Bo
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024. The inverse scattering problem exhibits an inherent low-rank structure due to its ill-posed nature; however developing low-rank structures for the inverse scattering problem remains challenging. In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm-Liouville problem. We first process the far-field data to obtain a post-processed data set within a disk domain. Subsequently, the post-processed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space, by dropping higher-order terms. The low-rank structure leads to a H\"{o}lder-logarithmic type stability estimate for arbitrary unknown functions, and a Lipschitz stability estimate for unknowns belonging to a finite dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability.
• About entropy penalized stochastic control problems
  
  • Bourdais Thibaut
  , 2024. This thesis focuses on stochastic optimal control problems with constraints on the marginal laws of the state process. These problems have constituted a very active research area in the past few years since they offer many practical and theoretical perspectives. In particular, this work is motivated by demand side management applications in power systems. More specifically, we are interested in controlling the overall electrical consumption of agents on a one-day period while imposing that the terminal marginal law of the state of the agents is equal to the initial law. The interest in this formulation is to define, in a seasonal environment, a periodic control procedure which can be reconducted on each period.This thesis explores an original way of solving stochastic optimal control problems with constraints in law based on a reformulation as constrained optimization problems on the space of probability measures. The decision variable is then splitted into two decision variables, each one considering only a specific part of the constraints while the deviation between these two probability measures is penalized by adding a relative entropy term. We take advantage of this penalized version to propose an alternating minimization procedure to approximate a solution to the original problem. This procedure involves solving sequentially two simple subproblems, each one consisting in minimizing the objective function over one variable while the other is fixed. The first subproblem amounts to a pointwise minimization of a running cost function. The solution of the second subproblem can be expressed as the so called exponential twist of a reference (Markovian) probability measure whose study constitutes a substantial part of this thesis. This alternating procedure is proved to converge to an approximate solution of the original control problem under various assumptions allowing for instance non convex running costs with respect to the control variable, as well as jump diffusions dynamics.
• Viscosity solutions of centralized control problems in measure spaces
  
  • Aussedat Averil
  • Jerhaoui Othmane
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2024, 30, pp.91. This work focuses on a control problem in the Wasserstein space of probability measures over Rd. Our aim is to link this control problem to a suitable Hamilton-Jacobi-Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This regularity allows to define a notion of viscosity and a Hamiltonian function relying on directional derivatives. Using a generalization of Ekeland's principle, we show that the corresponding HJB equation admits a comparison principle, and deduce that the value function is the unique solution in this viscosity sense. The PDE tools are developed in the general framework of Measure Differential Equations. (10.1051/cocv/2024081)
  DOI : 10.1051/cocv/2024081
• Construction and analysis of spectral signatures for defects in complex media
  
  • Pourre Fabien
  , 2024. Concrete is widely used in construction, particularly in the reactor building of nuclear power plants. Monitoring its evolution and identifying any defects that could compromise its proper functioning can be achieved through non-destructive testing. Concrete is composed of aggregates, but their high concentration and proximity pose challenges for classical methods, such as the Linear Sampling Method, which fail to produce quantitative results or exploitable images.The objective of this thesis is to build an imaging algorithm that can estimatethe density of the aggregates and recover the local distribution of those smallinhomogeneities.To address this challenge, inspiration was drawn from a monotonicity property of the Transmissioneigenvalues. They correspond to the frequencies for which an incident wave exists such that the scattered field is trivial outside the scatterers.Instead of comparing the scattered field to the vacuum, this thesis introduces a new approach: comparing it, at a fixed wavenumber, to a numerical scattering problem, referred to as the background. This led to the introduction of a new class of eigenvalues known as the f-averaged Steklov eigenvalues. Each of these eigenvalues is associatedwith an artificial background problem that contains a resonator and is solution to a simple spectral problem inside the resonator. In addition, each f-averaged Steklov eigenvalue is monotonically increasing with respect to the number of inhomogeneities inside that resonator. These spectral signatures can be recovered from the data. By computing theseeigenvalues for various positions of the resonator, we can estimate the variation ofthe local density of the inhomogeneities in an unknown medium.
• Étude analytique et numérique de problèmes inverses en diffraction acoustique pour la conception de microphones spatiaux
  
  • Lerévérend Dorian
  , 2024. CONTEXTE : Ces travaux de thèse sont motivés par un besoin concret identifié par l'association Mon Cartable Connecté. Dans le cadre de l'amélioration de son dispositif de téléprésence pour aider à la scolarisation à distance d'enfants hospitalisés, l'association souhaite implémenter un son spatialisé et immersif. En raison de l'impact prouvé d'un tel dispositif sur les interactions sociales entre élèves, il faudra que la solution proposée n'évoque pas une forme humaine. Enfin, elle devra respecter des contraintes d'encombrement et de coût. Parmi les technologies existantes, nous pouvons citer les microphones ambisoniques (Zoom H3-VR, Zylia Pro, ...) et les microphones en formes de têtes humaines (Kemar, Neumann KU-100, ...). Les premiers produisent un signal spatialisé (avec de nombreux canaux) mais délivrent un son neutre, ce qui limite l'immersion. Les seconds reproduisent l'écoute naturelle humaine sans aucun traitement numérique mais génèrent seulement un son binaural (avec deux canaux). Nous proposons donc de réunir ces deux technologies dans un objet dont la forme serait différente d'une tête humaine. ENJEUX : Une première difficulté est le fait que la technologie ambisonique repose sur les fonctions harmoniques surfaciques de l'appareil d'enregistrement utilisé. En effet, ces dernières sont connues analytiquement pour des sphères et des ellipses uniquement. Nous décidons de laisser cette remarque de côté et nous nous concentrons sur la seconde difficulté : la question de l'existence d'obstacles ayant des géométries différentes et présentant la même réponse impulsionnelle est un problème ouvert. Nous pouvons tout de même définir un modèle simplifié afin de proposer des méthodes de résolution. Nous considérons trois grandeurs principales pour caractériser l'acoustique d'un objet : sa géométrie, ses propriétés d'absorption et la position de ses points d'écoute (tympans ou microphones). Il existe de nombreuses bases de données donnant libre accès à des géométries de têtes et aux réponses fréquentielles associées. Nous pouvons alors définir plusieurs problèmes d'identification des valeurs d'un ou plusieurs de ces paramètres. Nous les résolvons en minimisant une fonction de coût par descente de gradient. STRUCTURE DU MANUSCRIT : Nous commençons par supposer que la tête est un obstacle impénétrable. Nous développons un solveur pour le problème direct de diffraction acoustique avec Matlab. Cet algorithme performant est basé sur l'implémentation des matrices hiérarchiques dans Gypsilab (développé en Matlab par Matthieu Aussal). Il nous permet de calculer rapidement des réponses en fréquence (HRTFs) réalistes avec différentes conditions d'impédance jusqu'à 20kHz. Voyant que l'erreur entre les résultats numériques et les mesures expérimentales peut être réduite par un choix judicieux de condition de bord, nous identifions l'impédance optimale de têtes humaines jusqu'à 15kHz à partir de données expérimentales sans phase. Se pose alors la question des formes équivalentes. Nous déterminons des triplets (forme, impédance, tympans) donnant des HRTFs assez proches de celles de têtes humaines jusqu'à 4kHz. Enfin, nous supposons que la tête est un obstacle pénétrable et changeons notre modèle. Nous transposons les développements réalisés précédemment à ce nouveau cadre. Ainsi, nous identifions les indices de réfraction de têtes humaines jusqu'à 4kHz. Nous terminons avec la détermination numérique d'obstacles équivalents définis par des triplets (forme, indice de réfraction, tympans) dont les réponses fréquentielles approchent des HRTFs mesurées expérimentalement jusqu'à 4kHz.
• Construction of transparent conditions for electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2024. We are interested in the numerical resolution of diffraction problems in closed electromagnetic waveguides by means of finite elements methods. To proceed, we need to truncate the domain and design adapted transparent conditions on the artificial boundary to avoid spurious reflections. When the guide is homogeneous in the transverse section, this can be done by writing an Electric-to-Magnetic condition based on a modal decomposition of the field. The latter takes a rather simple form thanks to the orthogonality of transverse modes. For guides that are heterogeneous in the transverse section, the transverse modes are no longer orthogonal but satisfy bi-orthogonality relations linked to the Poynting energy flux. Modal decompositions are more delicate to derive and it may happen that certain modes have phase and group velocities of different sign, which prevents the use of Perfectly Matched Layers. Adapting techniques already developed in elasticity, we derive a new transparent condition based on a Poynting-to-Magnetic operator with overlap. To illustrate the method, we present numerical results obtained with Nédélec finite elements using the XLiFE++ library.
• Notes de cours sur les équations de Maxwell et leur approximation
  
  • Ciarlet Patrick
  , 2024, pp.151.
• SpinDoctor-IVIM: A virtual imaging framework for intravoxel incoherent motion MRI
  
  • Lashgari Mojtaba
  • Yang Zheyi
  • Bernabeu Miguel O
  • Li Jing-Rebecca
  • Frangi Alejandro F
  Medical Image Analysis, Elsevier, 2024, 99, pp.103369. <div><p>Intravoxel incoherent motion (IVIM) imaging is increasingly recognised as an important tool in clinical MRI, where tissue perfusion and diffusion information can aid disease diagnosis, monitoring of patient recovery, and treatment outcome assessment. Currently, the discovery of biomarkers based on IVIM imaging, similar to other medical imaging modalities, is dependent on long preclinical and clinical validation pathways to link observable markers derived from images with the underlying pathophysiological mechanisms. To speed up this process, virtual IVIM imaging is proposed. This approach provides an efficient virtual imaging tool to design, evaluate, and optimise novel approaches for IVIM imaging. In this work, virtual IVIM imaging is developed through a new finite element solver, SpinDoctor-IVIM, which extends SpinDoctor, a diffusion MRI simulation toolbox. SpinDoctor-IVIM simulates IVIM imaging signals by solving the generalised Bloch-Torrey partial differential equation. The input velocity to SpinDoctor-IVIM is computed using HemeLB, an established Lattice Boltzmann blood flow simulator. Contrary to previous approaches, SpinDoctor-IVIM accounts for volumetric microvasculature during blood flow simulations, incorporates diffusion phenomena in the intravascular space, and accounts for the permeability between the intravascular and extravascular spaces. The above-mentioned features of the proposed framework are illustrated with simulations on a realistic microvasculature model.</p></div> (10.1016/j.media.2024.103369)
  DOI : 10.1016/j.media.2024.103369
• Fading regularization method for an inverse boundary value problem associated with the biharmonic equation
  
  • Boukraa Mohamed Aziz
  • Caillé Laëtitia
  • Delvare Franck
  Journal of Computational and Applied Mathematics, Elsevier, 2024, 457, pp.116285. In this paper, we propose a numerical algorithm that combines the fading regularization method with the method of fundamental solutions (MFS) to solve a Cauchy problem associated with the biharmonic equation. We introduce a new stopping criterion for the iterative process and compare its performance with previous criteria. Numerical simulations using MFS validate the accuracy of this stopping criterion for both compatible and noisy data and demonstrate the convergence, stability, and efficiency of the proposed algorithm, as well as its ability to deblur noisy data. (10.1016/j.cam.2024.116285)
  DOI : 10.1016/j.cam.2024.116285
• High-order numerical integration on self-affine sets
  
  • Joly Patrick
  • Kachanovska Maryna
  • Moitier Zoïs
  , 2024. We construct an interpolatory high-order cubature rule to compute integrals of smooth functions over self-affine sets with respect to an invariant measure. The main difficulty is the computation of the cubature weights, which we characterize algebraically, by exploiting a self-similarity property of the integral. We propose an \( h \)-version and a \( p \)-version of the cubature, present an error analysis and conduct numerical experiments.
• Propagation of ultrasounds in random multi-scale media and effecitve speed of sound estimation
  
  • Goepfert Quentin
  , 2024. Ultrasounds are widely used in medical imaging modalities. Originally, the ultrasound devices were built to image the internal structure of the tissues. In recent years, a change of paradigm operated and the goal is now also to assess physical parameters that can be used for medical diagnosis.The speed of acoustic waves inside soft tissues can be used for diagnosis of breast cancers or hepatic steatosis. Moreover, it determines the quality of the tomographic reconstruction of the tissues. Indeed, the images are usually computed by backpropagating the measured echoes at the speed of sound in water. However, the discrepancy between the speed of sound in water and the actual speed of sound inside the tissues results in nonphysical artifacts on the image.In order to establish a quantitative estimator of the propagation speed of sound inside the soft tissues, it is necessary to deeply understand the scattering of the medium. It is commonly admitted that the backscattered echoes are produced by numerous unresolved scatterers inside the medium (cell nuclei, mitochondria...). The scattering is then often modeled by the Born approximation. However, this model does not capture the variation of the effective speed of sound inside the tissue due to the unresolved scatterers. The goal of this thesis is thus to establish a propagation model that takes into account the variations of the effective speed of sound inside the tissues. Then, we will theoretically study the estimators previously introduced by Alexandre Aubry in his work.The tissue is here modeled as a bounded homogeneous mediumin which lie unresolved scatterers. As their distribution is unknown and inaccessible, their number and position is modeled as a random process. To obtain a simple form of the backscattered field, the techniques and tools developed for the quantitative stochastic homogenization theory will be used and a high-order asymptotic expansion will be proven.An asymptotic analysis of the imaging functional is carried out by using the high-order asymptotic expansion. Furthermore, the theoretical study of the estimators introduced by Alexandre Aubry and his team confirms and justifies some of the experimental results. In particular, it is possible to recover the effective speed of sound by a local spatial average of the imaging function.Numerical simulation supports each and every major result proven in this thesis.
• Guided modes in a hexagonal periodic graph like domain
  
  • Delourme Bérangère
  • Fliss Sonia
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2024, 22 (3), pp.1196-1245. This paper deals with the existence of guided waves and edge states in particular two-dimensional media obtained by perturbing a reference periodic medium with honeycomb symmetry. This reference medium is a thin periodic domain (the thickness is denoted δ &gt; 0) with an hexagonal structure, which is close to an honeycomb quantum graph. In a first step, we show the existence of Dirac points (conical crossings) at arbitrarily large frequencies if δ is chosen small enough. We then perturbe the domain by cutting the perfectly periodic medium along the so-called zigzag direction, and we consider either Dirichlet or Neumann boundary conditions on the cut edge. In the two cases, we prove the existence of edges modes as well as their robustness with respect to some perturbations, namely the location of the cut and the thickness of the perturbed edge. In particular, we show that different locations of the cut lead to almost-non dispersive edge states, the number of locations increasing with the frequency. All the results are obtained via asymptotic analysis and semi-explicit computations done on the limit quantum graph. Numerical simulations illustrate the theoretical results. (10.1137/23M1600177)
  DOI : 10.1137/23M1600177
• Notes de cours sur les méthodes variationnelles pour l'analyse et la résolution de problèmes non coercifs
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  , 2024.
• The Sphericity Paradox and the Role of Hoop Stresses in Free Subduction on a Sphere
  
  • Chaillat Stéphanie
  • Gerardi Gianluca
  • Li Yida
  • Chamolly Alexander
  • Li Zhong‐hai
  • Ribe Neil M.
  Journal of Geophysical Research : Solid Earth, American Geophysical Union, 2024, 129 (9), pp.e2024JB029500. Oceanic plates are doubly curved spherical shells, which influences how they respond to loading during subduction. Here we study a viscous fluid model for gravity‐driven subduction of a shell comprising a spherical plate and an attached slab. The shell is 100–1,000 times more viscous than the upper mantle. We use the boundary‐element method to solve for the flow. Solutions of an axisymmetric model show that the effect of sphericity on the flexure of shells is greater for smaller shells that are more nearly flat (the “sphericity paradox”). Both axisymmetric and three‐dimensional models predict that the deviatoric membrane stress in the slab should be dominated by the longitudinal normal stress (hoop stress), which is typically about twice as large as the downdip stress and of opposite sign. Our models also predict that concave‐landward slabs can exhibit both compressive and tensile hoop stress depending on the depth, whereas the hoop stress in convex slabs is always compressive. We test these two predictions against slab shape and earthquake focal mechanism data from the Mariana subduction zone, assuming that the deviatoric stress in our flow models corresponds to that implied by centroid moment tensors. The magnitude of the hoop stress exceeds that of the downdip stress for about half the earthquakes surveyed, partially verifying our first prediction. Our second prediction is supported by the near‐absence of earthquakes under tensile hoop stress in the portion of the slab having convex geometry. (10.1029/2024JB029500)
  DOI : 10.1029/2024JB029500
• Accelerated iterative DG finite element solvers for large-scale time-harmonic acoustic problems
  
  • Modave Axel
  , 2024. Finite element methods are widely used to solve time-harmonic wave propagation problems, but solving large cases can be extremely difficult even with the computational power of parallel computers. In this work, the linear system resulting from the finite element discretization is solved with iterative solution methods, which are efficient in parallel but can require a large number of iterations. In standard discontinuous Galerkin (DG) methods, the numerical solution is discontinuous at the interfaces between the elements. In hybridizable DG methods, additional unknowns are introduced at the interfaces between the finite elements, and the physical unknowns are eliminated from the global system, resulting in a hybridized system. We have recently proposed a new strategy, called CHDG, where the additional unknowns correspond to transmission variables, whereas in the standard approach they are numerical fluxes. This strategy improves the properties of the hybridized system for faster iterative solution procedures. In this talk, we present and study a 3D CHDG implementation with nodal finite element basis functions. The resulting scheme has properties amenable to efficient parallel computing. Numerical results are presented to validate the method, and preliminary 3D computational results are proposed. (10.3397/IN_2024_2877)
  DOI : 10.3397/IN_2024_2877
• Far-field sound field estimation using robotized measurements and the boundary elements method
  
  • Pascal Caroline
  • Marchand Pierre
  • Chapoutot Alexandre
  • Doaré Olivier
  , 2024, 270 (11), pp.816-827. Sound Field Estimation (SFE) is a numerical technique widely used to identify and reconstruct the acoustic fields radiated by unknown structures. In particular, SFE proves to be useful when data is only available close to the source, but information in the whole space is required. However, the practical implementation of this method is still hindered by two major drawbacks: the lack of efficient implementation of existing numerical methodologies, and the time-consuming and tedious roll-out of acoustic measurements. This paper aims to provide a solution to both issues. First, the measurements step is fully automated by using a robotic arm, able to accurately gather geometric and acoustic data without any human assistance. In this matter, a particular attention has been paid to the impact of the robot on the acoustic pressure measurements. The sound field prediction is then tackled using the Boundary Element Method (BEM), and implemented using the FreeFEM++ BEM library. Numerically simulated measurements have allowed us to assess the method accuracy, and the overall solution has been successfully tested using actual robotized measurements of an unknown loudspeaker (10.3397/IN_2024_2661)
  DOI : 10.3397/IN_2024_2661
• Computation of Green's functions for the acoustic scattering by an elastic structure excited by a turbulent flow in water
  
  • Pacaut Louise
  • Serre Gilles
  • Mercier Jean-François
  • Chaillat Stéphanie
  • Cotté Benjamin
  , 2024, 270 (5), pp.5995-6006. To model the hydrodynamic noise produced by an elastic ship hull or propeller excited by a turbulent boundary layer, we need an efficient method to compute the acoustic scattering by an elastic body surrounded by a fluid. In 3D, Boundary Element Methods (BEM) are used to reduce the computational costs, for both the fluid and the elastic body. A natural way to compute the boundary integral representation (BIR) of the sound pressure is to use formulations based on the free space acoustic and elastic Green's functions. However, since the turbulent flow along the elastic body is known only statistically, the use of these Green's functions would be too expensive. A remedy is to compute a Green's function adapted to the physical problem, thus satisfying the transmission conditions of the fluid-structure problem. This so-called "tailored Green's function" is determined by solving a coupled acoustic-elastic problem with the BEM, and leads to a simplified BIR of the sound pressure compatible with a stochastic source term. We first validate the computation of the tailored Green's function over a classic spherical geometry. Then we compare the scattering of multiple quadrupoles by elastic or rigid NACA0012 profiles. (10.3397/IN_2024_3671)
  DOI : 10.3397/IN_2024_3671
• Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator
  
  • Audibert Lorenzo
  • Meng Shixu
  Inverse Problems, IOP Publishing, 2024, 40 (9), pp.095007. In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter. (10.1088/1361-6420/ad5e18)
  DOI : 10.1088/1361-6420/ad5e18
• Multiscale modeling for a class of high-contrast heterogeneous sign-changing problems
  
  • Ye Changqing
  • Jin Xingguang
  • Ciarlet Patrick
  • Chung Eric T.
  , 2024. The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These problems find their physical background in negative-index metamaterials, either as inclusions embedded into common materials as the matrix or vice versa. In this paper, we propose a numerical method based on the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) specifically designed for sign-changing problems. The construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. The numerical results demonstrate the effectiveness of the proposed method in handling sophisticated coefficient profiles and the robustness of coefficient contrast ratios. Under several technical assumptions and by applying the T-coercivity theory, we establish the inf-sup stability and provide an a priori error estimate for the proposed method.
• Modeling fluid injection effects in dynamic fault rupture using Fast Boundary Element Methods
  
  • Bagur Laura
  , 2024. Earthquakes due to either natural or anthropogenic sources cause important human and material damage. In both cases, the presence of pore fluids influences the triggering of seismic instabilities.A new and timely question in the community is to show that the earthquake instability could be mitigated by active control of the fluid pressure. In this work, we study the ability of Fast Boundary Element Methods (Fast BEMs) to provide a multi-physic large-scale robust solver required for modeling earthquake processes, human induced seismicity and their mitigation.In a first part, a Fast BEM solver with different temporal integration algorithms is used. We assess the performances of various possible adaptive time-step methods on the basis of 2D seismic cycle benchmarks available for planar faults. We design an analytical aseismic solution to perform convergence studies and provide a rigorous comparison of the capacities of the different solving methods in addition to the seismic cycles benchmarks tested. We show that a hybrid prediction-correction / adaptive time-step Runge-Kutta method allows not only for an accurate solving but also to incorporate both inertial effects and hydro-mechanical couplings in dynamic fault rupture simulations.In a second part, once the numerical tools are developed for standard fault configurations, our objective is to take into account fluid injection effects on the seismic slip. We choose the poroelastodynamic framework to incorporate injection effects on the earthquake instability. A complete poroelastodynamic model would require non-negligible computational costs or approximations. We justify rigorously which predominant fluid effects are at stake during an earthquake or a seismic cycle. To this aim, we perform a dimensional analysis of the equations, and illustrate the results using a simplified 1D poroelastodynamic problem. We formally show that at the timescale of the earthquake instability, inertial effects are predominant whereas a combination of diffusion and elastic deformation due to pore pressure change should be privileged at the timescale of the seismic cycle, instead of the diffusion model mainly used in the literature.
• Méthodes d'inversion de type one-shot et décomposition de domaine
  
  • Vu Tuan Anh
  , 2024. Notre objectif principal est d’analyser la convergence d’une méthode d’optimisation basée sur le gradient, pour résoudre des problèmes inverses d’identification de paramètres, dans laquelle les problèmes directs et adjoints correspondants sont résolus par un solveur itératif. Le couplage des itérations pour les trois inconnues (le paramètre du problème inverse, la solution du problème direct et la solution du problème adjoint) donne ce que l’on appelle les méthodes d’inversion de type one-shot. De nombreux tests numériques ont montré que l’utilisation de très peu d’itérations internes pour les problèmes directs et adjoints peut néanmoins conduire à une bonne convergence pour le problème inverse. Cela nous motive à développer une théorie de convergence rigoureuse pour les méthodes de type one-shot en utilisant un petit nombre fixe d’itérations internes, avec un schéma semi-implicite pour la mise à jour du paramètre et une fonction de coût régularisée. Notre théorie couvre une classe générale de problèmes inverses linéaires dans le cadre discret de dimension finie, pour lesquels les problèmes directs et adjoints sont résolus par des méthodes génériques d’itération de point fixe. En étudiant le rayon spectral de la matrice par blocs des itérations couplées, nous prouvons que pour des pas de descente suffisamment petits, les méthodes de type one-shot (semi-implicites) convergent. En particulier, dans le cas scalaire, où les inconnues appartiennent à des espaces à une dimension, nous établissons des conditions de convergence suffisantes et même nécessaires sur le pas de descente. Ensuite, nous appliquons des méthodes de type one-shot aux problèmes inverses de conductivité (linéarisés et puis non linéaires), et résolvons les problèmes directs et adjoints par des méthodes de décomposition de domaines, plus spécifiquement des méthodes de Schwarz optimisées sans recouvrement. Nous analysons un algorithme de décomposition de domaine qui calcule simultanément les solutions directe et adjointe pour une conductivité donnée. En combinant cet algorithme avec la mise à jour du paramètre par descente de gradient, nous obtenons une méthode one-shot de décomposition de domaine qui résout le problème inverse. Nous proposons deux versions discrétisées de l’algorithme couplé, dont la seconde (dans le cas du problème inverse de conductivité linéarisé) s’inscrit dans le cadre abstrait de notre théorie de convergence. Enfin, plusieurs expériences numériques sont fournies pour illustrer les performances des méthodes de type one-shot, en comparaison avec la méthode de descente de gradient classique dans laquelle les problèmes directs et adjoints sont résolus par des solveurs directs. En particulier, nous observons que, même dans le cas de données bruitées, très peu d’itérations internes peuvent toujours garantir une bonne convergence des méthodes de type one-shot.
• Substructuring based FEM-BEM coupling for Helmholtz problems
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2024. This talk concerns the solution of the Helmholtz equation in a medium composed of a bounded heterogeneous domain and an unbounded homogeneous one. Such problems can be expressed using classical FEM-BEM coupling techniques. We solve these coupled formulations using iterative solvers based on substructuring Domain Decomposition Methods (DDM), and aim to develop a convergence theory, with fast and guaranteed convergence. A recent article of Xavier Claeys proposed a substructuring Optimized Schwarz Method, with a nonlocal exchange operator, for Helmholtz problems on a bounded domain with classical conditions on its boundary (Dirichlet, Neumann, Robin). The variational formulation of the problem can be written as a bilinear application associated with the volume and another with the surface, for which, under certain sufficient assumptions, convergence of the DDM strategy is guaranteed. In this presentation we show how some specific FEM-BEM coupling methods fit, or not, the previous framework, in which we consider Boundary Integral Equations (BIEs) instead of classical boundary conditions. In particular, we prove that the symmetric Costabel coupling satisfies the framework assumptions, implying that the convergence is guaranteed. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Fast and accurate boundary integral equation methods for the multi-layer transmission problem
  
  • Cortes Elsie A
  • Carvalho Camille
  • Chaillat Stéphanie
  • Tsogka Chrysoula
  , 2024. We consider a multi-layer transmission problem, which can be used for example to describe the light scattering in meta-materials (assemblings of various concentric penetrable materials). Our goal is to solve the multi-layer problem accurately with optimal discretization. Generally, the costs to solve this problem grow as more layers are introduced - solving this problem is thus particularly challenging for 3D models. For this reason, we use boundary integral equation (BIE) methods: they reduce the dimensionality of the problem and can provide high order accuracy. However, BIE methods suffer from the so-called close evaluation problem. We address it using modified representations. We further examine how to improve the speed of our method by optimizing the accuracy over number of discretization points ratio. In particular, we investigate whether the usual rule of thumb to mesh interfaces, based on the most constraining material, is necessary for the multi-layer transmission problem. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Construction of transparent conditions for electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2024. We are interested in the numerical resolution of diffraction problems in closed electromagnetic waveguides by means of finite elements methods. To proceed, we need to truncate the domain and design adapted transparent conditions on the artificial boundary to avoid spurious reflections. When the guide is homogeneous in the transverse section, this can be done by writing an Electric-to-Magnetic condition based on a modal decomposition of the field. The latter takes a rather simple form thanks to the orthogonality of transverse modes. For guides that are heterogeneous in the transverse section, the transverse modes are no longer orthogonal but satisfy bi-orthogonality relations linked to the Poynting energy flux. Modal decompositions are more delicate to derive and it may happen that certain modes have phase and group velocities of different sign, which prevents the use of Perfectly Matched Layers. Adapting techniques already developed in elasticity, we derive a new transparent condition based on a Poynting-to-Magnetic operator with overlap. To illustrate the method, we present numerical results obtained with Nédélec finite elements using the XLiFE++ library.
• Computation of a fluid-structure Green's function using a BEM-BEM coupling
  
  • Pacaut Louise
  • Mercier Jean-François
  • Chaillat Stéphanie
  • Serre Gilles
  , 2024. In order to determine the elasto-acoustic noise produced by a boat hull excited by a turbulent boundary layer, we propose a numerical method to compute the acoustic scattering by an elastic body surrounded by a fluid. To reduce the computational costs a Boundary Element Method (BEM) is used. Since the turbulent flow along the hull is known only statistically, a formulation combining the free field acoustic and elastic Green's functions is not adequate. A better suited choice is to determine a global Green's function satisfying the transmission conditions of the fluid-structure problem. The boundary integral representation of the scattered pressure is then simplified. This so-called tailored Green's function is determined by solving an acoustic/elastic coupled problem with a BEM. Here we focus on a particular difficulty: when the source is close to the surface, the numerical accuracy of the Green's function deteriorates. We describe a method to regularize our BEM scheme in this context. We validate the method for the problem of an elastic sphere in water.