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Special quasirandom structures: A selection approach for stochastic homogenization

Claude Le Bris, Frédéric Legoll and William Minvielle

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 [23], [24], [25], 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.

Funding source: ONR

Award Identifier / Grant number: N00014-12-1-0383

Funding source: EOARD

Award Identifier / Grant number: FA8655-13-1-3061

Funding source: ANR

Award Identifier / Grant number: Investments for the Future ANR-11-LABX-022-01

The first two authors would like to thank E. Cancès for introducing them to the SQS approach in the context of solid state physics, pointing out to them [23], [24], [25], as well as for several stimulating discussions in the early stages of this work. The authors also thank J.-D. Deuschel for his helpful comments and for pointing out [10], and X. Blanc for his remarks on a draft version of this article.

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Received: 2015-9-4
Accepted: 2016-2-5
Published Online: 2016-2-17
Published in Print: 2016-3-1

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