Société de Mathématiques Appliquées et Industrielles

18ème journée EDP/Probas

Homogénéisation stochastique

vendredi 4 décembre 2015

La journée sera composée de trois mini-cours :

10h - 11h30 Antoine Gloria (Université Libre de Bruxelles) : Towards a quantitative theory of stochastic homogenization.
In this talk I’ll make a survey of the results obtained in the past few years in collaboration with Felix Otto, Jean-Christophe Mourrat, Stefan Neukamm, Jim Nolen and Mitia Duerinckx. In the first part I’ll introduce a notion of vertical derivative, a measure of the sensitivity of solutions of PDEs with respect to the coefficients. As a first example, I’ll show how this can be used to prove the analyticity of homogenized coefficients with respect to Bernoulli perturbations, and recover the so-called Clausius-Mossotti formula. Then I’ll turn to quantitative results on the corrector equation, and in particular the existence of stationary correctors in dimensions d>2 when the law of the coefficicients satisfies a functional inequality (spectral gap, log-Sob) for the vertical derivatives. I’ll conclude this part with the combination with Stein’s method, which allows on to prove (and quantify) the asymptotic normality of fluctuations in stochastic homogenisation, and central limit theorems. In the second part of the talk I’ll revisit some of the results above in terms of regularity theory at large scales and shall introduce the notion of minimal radius, which drives the validity of an intrinsic regularity theory for random linear elliptic systems in divergence form. Combined with a new family of weighted functional inequalities, this yields quantitative estimates (with improved stochastic integrability) in stochastic homogenization for a class of coefficient fields that display arbitrarily slow decaying correlations. In the last part of the talk I’ll turn to the case when the coefficient field does not necessarily satisfy a functional inequality, and give nearly-optimal estimates with optimal integrability under the assumption of finite range of dependence.

13h30 - 15h00 Scott Armstrong (Université Paris-Dauphine) : Regularity theory and quantitative estimates in stochastic homogenization.
I will discuss quantitative estimates in stochastic homogenization for linear elliptic equations in divergence form. There are essentially two main ideas are : (1) it is possible to obtain a regularity theory (optimal in stochastic integrability) for equations with random coefficients by using quantitative homogenization results (which are sub-optimal in the scaling of the error) and mimicking the proof of the Schauder estimates ; and (2) this regularity theory then upgrades the quantitative estimates themselves, bypassing the need for concentration inequalities, by essentially localizing the solutions and forcing the energy to be "additive". (Joint works with C. Smart, T. Kuusi and J.-C. Mourrat.)

15h15 - 16h45 Frédéric Legoll (Ecole des Ponts) : Variance reduction strategies in stochastic homogenization.
We consider the homogenization of linear elliptic PDEs with random stationary coefficients. As is well-known, the deterministic homogenized coefficients are obtained through a corrector problem that is set on the entire space. This problem is thus intractable from the practical viewpoint. A standard approximation consists in considering large but bounded domains, and solving the corrector problem on these domains. A by-product of the truncation is that the practical approximations of the homogenized coefficients are random. It is thus natural to consider several realizations of the random environment, in a Monte Carlo fashion, and solve the truncated corrector problem for each of them. We will describe here several strategies to reduce the variance (and hence the statistical error) of the computed quantities. Some approaches are based on the a priori selection of representative microstructures. Other strategies, in the vein of the control variate method, are based on surrogate models. The approaches will be discussed both from the theoretical and numerical standpoints. (Joint works with M. Bertin, S. Brisard, C. Le Bris and W. Minvielle.)

La journée aura lieu dans l’amphithéâtre Hermite de l’Institut Henri Poincaré, 11 rue Pierre et Marie Curie, Paris 5ème (accès).

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