Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations (Q2842483)

This article is a continuation of the analysis begun in [\textit{A. Gloria} and \textit{F. Otto}, Ann. Probab. 39, No. 3, 779--856 (2011; Zbl 1215.35025)] and [\textit{A. Gloria} and \textit{F. Otto}, Ann. Appl. Probab. 22, No. 1, 1--28 (2012; Zbl 1387.35031)]. The conductivities \(a(e)\) on each edge of the \(d\)-dimensional Cartesian lattice \(\mathbb{Z}^d\) are assumed to be independent and identically distributed random variables with values on a bounded interval \([\alpha, \beta]\) with \(0<\alpha\leq \beta<\infty\). This set of conductivities defines a discrete elliptic differential operator. NEWLINENEWLINENEWLINEFrom the abstract: ``Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with parameter \(T\)) on some box of finite size \(L\). In this article, we replace the regularized corrector (which is the solution of a problem posed on \(\mathbb Z^d\)) by some practically computable proxy on some box of size \(R\geq L\), and quantify the associated additional error. In order to improve the convergence, one may also consider \(N\) independent realizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients. A natural optimization problem consists in properly choosing \(T,R,L\) and \(N\) in order to reduce the error at given computational complexity. Our analysis is sharp and sheds some light on this question. In particular, we propose and analyse a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errors we derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.''