Special Quasirandom Structures: a selection approach for stochastic homogenization

TL;DR

This paper introduces a variance reduction method for stochastic homogenization of elliptic equations, selecting realizations that better satisfy statistical properties to improve efficiency and accuracy.

Contribution

It adapts a selection approach from materials science to homogenization, providing theoretical analysis and numerical validation of its effectiveness.

Findings

The method reduces variance in homogenization estimates.

The approach is effective in simplified and more general settings.

Numerical results demonstrate improved efficiency over traditional methods.

Abstract

We adapt and study a variance reduction approach for the homogenization of elliptic equations in divergence form. The approach, borrowed from atomistic simulations and solid-state science [von Pezold et al, Physical Review B 2010; Wei et al, Physical Review B 1990; Zunger et al, Physical Review Letters 1990], consists in selecting random realizations that best satisfy some statistical properties (such as the volume fraction of each phase in a composite material) usually only obtained asymptotically. We study the approach theoretically in some simplified settings (one-dimensional setting, perturbative setting in higher dimensions), and numerically demonstrate its efficiency in more general cases.