Efficient methods for the estimation of homogenized coefficients

TL;DR

This paper introduces new, computationally efficient methods for estimating effective coefficients in homogenization problems, leveraging multiscale techniques and Green-Kubo formulas, with broad applicability and improved Monte Carlo sampling strategies.

Contribution

The paper develops novel homogenized coefficient estimation methods with optimal complexity and small constants, extending multiscale approaches and Green-Kubo formulas to general additive functionals of Markov processes.

Findings

Methods have optimal computational complexity.

Numerical results show small constant prefactors.

Extrapolation improves Monte Carlo sampling performance.

Abstract

The main goal of this paper is to define and study new methods for the computation of effective coefficients in the homogenization of divergence-form operators with random coefficients. The methods introduced here are proved to have optimal computational complexity, and are shown numerically to display small constant prefactors. In the spirit of multiscale methods, the main idea is to rely on a progressive coarsening of the problem, which we implement via a generalization of the Green-Kubo formula. The technique can be applied more generally to compute the effective diffusivity of any additive functional of a Markov process. In this broader context, we also discuss the alternative possibility of using Monte-Carlo sampling, and show how a simple one-step extrapolation can considerably improve the performance of this alternative method.