Séminaire de mathématiques appliquées (archives)

The motivation of this work is the detection of cerebrovascular accidents by microwave tomographic imaging. This requires the solution of an inverse problem based on a minimization algorithm with successive solutions of a direct problem such as the modeling of a whole-microwave measurement system. The feasibility of this type of imaging is conditioned on one hand by an accurate reconstruction of the material properties of the propagation medium and on the other hand by a considerable reduction of simulation time. Fulfilling these two requirements will enable a very rapid and accurate diagnosis. From the mathematical and numerical point of view, this means solving Maxwell's equations in time-harmonic regime using high-order finite elements, domain decomposition method and parallel computing.

A new numerical scheme to compute isothermal and unsteady flow of an incompressible viscoplastic Bingham medium will be presented. The main difficulty, for both theoretical and numerical approaches, is due to the non-differentiability of the plastic part of the stress tensor in regions where the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms of a projection. A new time scheme, based on the classical incremental projection method for the Newtonian Navier-Stokes equations, is proposed. The plastic tensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-time relaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. Stability and error analyses of the numerical scheme will be shown. Numerical results, obtained on the well-known two-dimensional lid-driven cavity test case, will be detailed for Reynolds number up to 10 000 and Bingham number up to 100.

During this talk, we will focus on the problem of testing the equality of metric measure spaces (mm-spaces) up to an isomorphism (a measure-preserving isometry), giving samples on these spaces. For this purpose, we introduce a new shape signature, the distance-to-a-measure signature, which is a probability measure on R+ built from the mm-space of interest. To reach our goal, we use bootstrap methods, involving Wasserstein metrics.

Le résumé est visible ici.

Les methodes dite hypocoercives permettent de montrer le retour exponentiel en temps vers l'equilibre Maxwellien pour une grande classe d'equations cinétiques inhomogènes. Elles sont basées sur des estimations de type commutateurs d'inspiration microlocale. Nous montrerons dans cet exposé que ces methodes permettent egalement de montrer le retour vers l'equilibre pour des nouveaux schemas cinetiques semi-discrets ou discrets, meme s'il n'y plus de commutateurs au sens continu du terme et que la notion meme d'equilibre est ambigue. Nous en profiterons pour parler egalement du cas coercif (homogene) qui est une des premieres etapes non triviale de l'analyse discrete. Il s'agit d'un travail en collaboration avec Pauline Laffite (Centrale-Supelec) et Guillaume Dujardin (Université de Lille).

Euler equations for compressible flows treats pressure as a scalar quantity. However, for several applications this description of pressure is not suitable. Many extended model based on the higher moments of Boltzmann equations are considered to overcome this issue. One such model is Ten-moment Gaussian closure equations, which treats pressure as symmetric tensor.

In this work, we develop a higher-order, positivity preserving Discontinuous Galerkin (DG) scheme for Ten-moment Gaussian closure equations. The key challenge is to preserve positivity of density and symmetric pressure tensor. This is achieved by constructing a positivity limiter. In addition to preserve positivity, the scheme also ensures the accuracy of the approximation for smooth solutions. The theoretical results are then verified using several numerical experiments. This is a joint work with Dr. Praveen Chandrashekar (TIFR-CAM, Bangalore) and Ms. Asha Meena (IIT Delhi).

Dans cet exposé, nous nous intéresserons à la résolution numérique d'équations de convection-diffusion non-linéaires munies de conditions de flux nuls ou/et Dirichlet non-homogènes au bord. Pour une certaine classe de non-linéarités, ces équations admettent une large gamme d'entropies relatives assurant la convergence de la solution en temps long vers l'état stationnaire.

Nous proposerons un schéma volumes finis construit à partir des flux discrets de l'équation stationnaire et préservant toutes ces fonctionnelles de Lyapunov. Cela nous permettra d'obtenir un comportement en temps long précis au niveau discret ainsi que les estimations nécessaires pour assurer la convergence du schéma. Nous illustrerons par des simulations numériques le bon comportement en temps long du schéma sur différents modèles, tels que l'équation de Fokker-Planck avec champ magnétique ou l'équation des milieux poreux. Ce travail a été réalisé en collaboration avec Francis Filbet.