Simulation de la propagation d'ondes élastiques en domaine fréquentiel par des méthodes Galerkine discontinues

The scientific context of this thesis is seismic imaging which aims at recovering the structure of the earth. As the drilling is expensive, the petroleum industry is interested by methods able to reconstruct images of the internal structures of the earth before the drilling. The most used seismic imaging method in petroleum industry is the seismic-reflection technique which uses a wave equation model. Seismic imaging is an inverse problem which requires to solve a large number of forward problems. In this context, we are interested in this thesis in the modeling part, i.e. the resolution of the forward problem, assuming a time-harmonic regime, leading to the so-called Helmholtz equations. The main objective is to propose and develop a new finite element (FE) type solver characterized by a reduced-size discrete operator (as compared to existing such solvers) without hampering the accuracy of the numerical solution. We consider the family of discontinuous Galerkin (DG) methods. However, as classical DG methods are much more expensive than continuous FE methods when considering steady-like problems, because of an increased number of coupled degrees of freedom as a result of the discontinuity of the approximation, we develop a new form of DG method that specifically address this issue: the hybridizable DG (HDG) method. To validate the efficiency of the proposed HDG method, we compare the results that we obtain with those of a classical upwind flux-based DG method in a 2D framework. Then, as petroleum industry is interested in the treatment of real data, we develop the HDG method for the 3D elastic Helmholtz equations.Le contexte scientifique de cette thèse est l'imagerie sismique dont le but est de reconstituer la structure du sous-sol de la Terre. Comme le forage a un coût assez élevé, l'industrie pétrolière s'intéresse à des méthodes capables de reconstituer les images de la structure terrestre interne avant de le faire. La technique d'imagerie sismique la plus utilisée est la technique de sismique-réflexion qui est basée sur le modèle de l'équation d'ondes. L'imagerie sismique est un problème inverse qui requiert de résoudre un grand nombre de problèmes directs. Dans ce contexte, nous nous intéressons dans cette thèse à la résolution du problème direct en régime harmonique, soit à la résolution des équations d'Helmholtz. L'objectif principal est de proposer et de développer un nouveau type de solveur élément fini (EF) caractérisé par un opérateur discret de taille réduite (comparée à la taille des solveurs déjà existants) sans pour autant altérer la précision de la solution numérique. Nous considérons les méthodes de Galerkine discontinues (DG). Comme les méthodes DG classiques sont plus coûteuses que les méthodes EF continues si l'on considère un même problème à cause d'un grand nombre de degrés de liberté couplés, résultat des approximations discontinues, nous développons une nouvelle classe de méthode DG réduisant ce problème : la méthode DG hybride (HDG). Pour valider l'efficacité de la méthode HDG proposée, nous comparons les résultats obtenus avec ceux obtenus avec une méthode DG basée sur des flux décentrés en 2D. Comme l'industrie pétrolière s'intéresse au traitement de données réelles, nous développons ensuite la méthode HDG pour les équations élastiques d'Helmholtz 3D

Categorical Perception: A Groundwork for Deep Learning

A well-known perceptual consequence of categorization in humans and other animals, called categorical perception, is notably characterized by a within-category compression and a between-category separation: two items, close in input space, are perceived closer if they belong to the same category than if they belong to different categories. Elaborating on experimental and theoretical results in cognitive science, here we study categorical effects in artificial neural networks. We combine a theoretical analysis that makes use of mutual and Fisher information quantities, and a series of numerical simulations on networks of increasing complexity. These formal and numerical analyses provide insights into the geometry of the neural representation in deep layers, with expansion of space near category boundaries and contraction far from category boundaries. We investigate categorical representation by using two complementary approaches: one mimics experiments in psychophysics and cognitive neuroscience by means of morphed continua between stimuli of different categories, while the other introduces a categoricality index that, for each layer in the network, quantifies the separability of the categories at the neural population level. We show on both shallow and deep neural networks that category learning automatically induces categorical perception. We further show that the deeper a layer, the stronger the categorical effects. As an outcome of our study, we propose a coherent view of the efficacy of different heuristic practices of the dropout regularization technique. More generally, our view, which finds echoes in the neuroscience literature, insists on the differential impact of noise in any given layer depending on the geometry of the neural representation that is being learned, i.e. on how this geometry reflects the structure of the categories

Resolution strategy for the Hybridizable Discontinuous Galerkin system for solving Helmholtz elastic wave equations

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Hybridizable Discontinuous Galerkin Methods for modelling 3D seismic wave propagation in harmonic domain

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Discontinuous Galerkin methods for solving Helmholtz isotropic wave equations for seismic applications

International audienceSeismic applications require to solve wave equations in heterogeneous media. Thus we choose to focus on the Helmholtz wave equations resolution in isotropic heterogenous media using Galerkin discontinuous methods (DG). To do that we select three DG methods: the DG method with centered flux, the DG method with upwind flux and the hybridizable DG method in order to compare the different results. The principal issue is to obtain the better solution reducing at the maximum time and memory costs. The first step of our work was to complete a program for solving Helmholtz wave equations using DG methods with centered flux and upwind flux. To test the program and compare results the two first results, I used two test-cases: the test-case of plane wave and the one of the diffraction by a circle. The second step is currently to develop the hybridizable DG formulation for Helmholtz wave equations in time-harmonic domain

Comparison of solvers performance when solving the 3D Helmholtz elastic wave equations using the Hybridizable Discontinuous Galerkin method

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Performances analysis of a Hybridizable discontinuous Galerkin solver for the 3D Helmholtz equations in geophysical context

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Performance comparison of HDG and classical DG method for the simulation of seismic wave propagation in harmonic domain

International audienceIn the most widely used methods for Seismic Imaging, we have to solve 2N wave equations at each iteration of the selected process, where N is the number of sources and is usually large (about 1000). The efficiency of the inverse solver is thus directly related to the efficiency of the numerical method used to solve the wave equation. Seismic imaging can be performed in the time domain or in the frequency domain regime. We focus here on the second setting. The drawback of time domain is that it requires to store the solution at each time step of the forward simulation. The difficulties related to frequency domain inversion lie in the solution of huge linear systems, which cannot be achieved today when considering realistic 3D elastic media, even with the progress of high-performance computing. In this context, the goal is to develop new forward solvers that reduce the number of degrees of freedom without hampering the accuracy of the numerical solution.We consider here discontinuous Galerkin (DG) methods which are more convenient than finite difference methods to handle the topography of the subsurface. Moreover, they are more adapted than continuous Galerkin (CG) methods to deal with hp-adaptivity. This last characteristics is crucial to adapt the mesh to the different regions of the subsurface which is generally highly heterogeneous. Nevertheless, the main drawback of classical DG methods is that they are expensive because they require a large number of degrees of freedom as compared to CG methods on a given mesh. In this work we consider a new class of DG method, the hybridizable DG (HDG) method. Instead of solving a linear system involving the degrees of freedom of all volumic cells of the mesh, the principle of HDG consists in introducing a Lagrange multiplier representing the trace of the numerical solution on each face of the mesh. Hence, it reduces the number of unknowns of the global linear systems and the volumic solution is recovered thanks to a local computation on each element. We compare the performances of the HDG method with those of classical nodal DG methods and we present our first results using a HDG method for the first-order form of the elastic wave propagation equations for 2D realistic test-cases

Modelling of seismic waves propagation in harmonic domain by hybridizable discontinuous Galerkin method (HDG)

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