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This article is concerned with numerical methods to approximate effectivecoefficients in stochastic homogenization of discrete linear ellipticequations, and their numerical analysis --- which has been made possible byrecent contributions on quantitative stochastic homogenization theory by two ofus and by Otto. This article makes the connection between our theoreticalresults and computations. We give a complete picture of the numerical methodsfound in the literature, compare them in terms of known (or expected)convergence rates, and study them numerically. Two types of methods arepresented: methods based on the corrector equation, and methods based on randomwalks in random environments. The numerical study confirms the sharpness of theanalysis (which it completes by making precise the prefactors, next to theconvergence rates), supports some of our conjectures, and calls for newtheoretical developments.

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