Quantitative version of Kipnis-Varadhan's theorem and Monte-Carlo approximation of homogenized coefficients
Résumé
This article is devoted to the analysis of a Monte-Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time $t>0$, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions rescaled by $t$ of $n$ independent random walks in $n$ independent environments. Relying on a new quantitative version of Kipnis-Varadhan's theorem (which is of independent interest), we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the rescaled final position of the random walk in terms of $t$. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of $n$ and $t$, and prove a large-deviation estimate. Compared to other numerical strategies, this Monte-Carlo approach has the advantage to be dimension-independent in terms of convergence rate and computational cost.
Domaines
Analyse numérique [math.NA]
Origine : Fichiers produits par l'(les) auteur(s)