Séminaire

14h : Edouard Oudet, Numerical approximation of 1D structures

We focus our attention on shape optimization problems in which one dimensional connected objects are involved. Very old and classical problems in calculus of variation are of this kind: euclidean Steiner’s tree problem, optimal irrigation networks, cracks propagation, etc. In a first part we quickly recall some previous work in collaboration with F. Santambrogio related to the functional relaxation of the irrigation cost. We establish a Γ-convergence of Modica and Mortola’s type and illustrate its efficiency from a numerical point of view by computing optimal networks associated to simple sources/sinks configurations. We also present more evolved situations with non Dirac sinks in which a fractal behavior of the optimal network is expected. In the second part of the talk we restrict our study to the euclidean Steiner's tree problem. We recall recent numerical approach which have been developed the last five years to approximate optimal trees and based on a recent approach obtained in collaboration with G. Orlandi and M. Bonafini, we describe the first convexification framework associated to the Euclidean Steiner tree problem which provide relevant tools from a numerical point of view.

15h30: Virginie Ehrlacher, Approximation of effective coefficients in stochastic homogenization using a boundary integral approach.

(joint work with Eric Cancès, Frédéric Legoll, Benjamin Stamm and Shuyang Xiang) A very efficient algorithm has recently been introduced in [1] in order to approximate the solution of implicit solvation models for molecules. The main ingredient of this algorithm relies in the clever use of a boundary integral formulation of the problem to solve. The aim of this talk is to present how such an algorithm can be adapted in order to compute efficiently effective coefficients in stochastic homogenization for random media with spherical inclusions. To this aim, the definition of new approximate corrector problems and approximate effective coefficients is needed and convergence results in the spirit of [2] are proved for this new formulation. Some numerical test cases will illustrate the behaviour of this method.

[1] "Domain decomposition for implicit solvation models", Eric Cancès, Yvon Maday, Benjamin Stamm, The Journal of Chemical Physics 139 (2013) 054111

[2] "Approximations of effective coefficients in stochastic homogenization", Alain Bourgeat, Andrey Piatnitski, Annales de l'institut Henri Poincaré (B) Probabilités et Statistiques 40 (2004) page 153-165