Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth

TL;DR

This paper explores numerical methods for approximating effective coefficients in stochastic homogenization, connecting theoretical results with computational practices through analysis and numerical experiments.

Contribution

It provides a comprehensive comparison of existing numerical methods based on the corrector equation and random walks, linking theory to practical computation.

Findings

Numerical results confirm the sharpness of convergence rates.

Analysis supports some conjectures and highlights areas for further theoretical development.

The study clarifies the relationship between different numerical approaches.

Abstract

This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis --- which has been made possible by recent contributions on quantitative stochastic homogenization theory by two of us and by Otto. This article makes the connection between our theoretical results and computations. We give a complete picture of the numerical methods found in the literature, compare them in terms of known (or expected) convergence rates, and study them numerically. Two types of methods are presented: methods based on the corrector equation, and methods based on random walks in random environments. The numerical study confirms the sharpness of the analysis (which it completes by making precise the prefactors, next to the convergence rates), supports some of our conjectures, and calls for new theoretical developments.