Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth
For researchers in stochastic homogenization, this work provides a comprehensive numerical validation and comparison of existing methods, but it is incremental as it primarily confirms known theory.
This paper numerically compares methods for computing effective coefficients in stochastic homogenization of discrete linear elliptic equations, confirming theoretical convergence rates and revealing prefactors. The study supports conjectures and identifies needs for new theory.
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.