Spectral measure and approximation of homogenized coefficients

TL;DR

This paper introduces a spectral approach to numerically approximate homogenized coefficients in stochastic homogenization of discrete elliptic equations, providing explicit strategies with convergence rates linked to the spectral properties of the environment.

Contribution

It applies a spectral representation formula to develop and analyze computable approximations of homogenized coefficients, connecting spectral data to approximation errors.

Findings

Spectral bounds relate to approximation accuracy.

Explicit numerical strategies achieve CLT convergence rates.

Applicable across all spatial dimensions.

Abstract

This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.