Some variance reduction methods for numerical stochastic homogenization

TL;DR

This paper reviews recent variance reduction techniques applied to numerical stochastic homogenization, aiming to improve the accuracy and efficiency of computing effective coefficients in random media.

Contribution

It introduces and evaluates several variance reduction methods adapted from engineering sciences for stochastic homogenization problems.

Findings

Variance reduction improves accuracy of Monte Carlo simulations

Certain techniques significantly reduce computational cost

Methods are validated through numerical experiments

Abstract

We overview a series of recent works devoted to variance reduction techniques for numerical stochastic homogenization. Numerical homogenization requires solving a set of problems at the micro scale, the so-called corrector problems. In a random environment, these problems are stochastic and therefore need to be repeatedly solved, for several configurations of the medium considered. An empirical average over all configurations is then performed using the Monte-Carlo approach, so as to approximate the effective coefficients necessary to determine the macroscopic behavior. Variance severely affects the accuracy and the cost of such computations. Variance reduction approaches, borrowed from other contexts of the engineering sciences, can be useful. Some of these variance reduction techniques are presented, studied and tested here.