Approximation of effective coefficients for the stochastic homogenization of discrete elliptic equations in high dimension.

In a recent work, Felix Otto and the author have introduced and analyzed a numerical method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We have shown that the overall error is the sum of three terms: a stochastic error of variance type, a systematic error related to some regularizing parameter, and an error related to the approximation on a finite box. The stochastic error, which has the scaling of the central limit theorem, is the dominant error up to dimension 8. Then, the systematic error saturates and dominates the overall error --- this analysis is optimal. In this talk, I will quickly recall the framework and the results above, and then introduce a general class of formulas for the approximation of homogenized coefficients. These formulas have the advantage that the associated systematic error can be made of higher order than the stochastic error in any dimension, thus generalizing the results by Felix Otto and the author to d>8.

This is joint work with Jean-Christophe Mourrat, Université de Provence.