Variance Reduction in Stochastic Homogenization: The Technique of Antithetic Variables

This work is a follow up to previous articles of the same authors (Costaouec, Le Bris, and Legoll, Boletin Soc. Esp. Mat. Apl. 50:9–27, 2010; Blanc, Costaouec, Le Bris, and Legoll, Markov Processes and Related Fields, in press). It has been shown there, both numerically and theoretically, that the technique of antithetic variables successfully applies to stochastic homogenization of divergence-form linear elliptic problems and allows to reduce variance in computations. In (Costaouec, Le Bris, and Legoll, Boletin Soc. Esp. Mat. Apl. 50:9–27, 2010), variance reduction was assessed numerically for the diagonal terms of the homogenized matrix, in the case when the random field, that models uncertainty on some physical property at microscale, has a simple form. The numerical experiments have been complemented in Blanc, Costaouec, Le Bris, and Legoll (Markov Processes and Related Fields, in press) by a theoretical study. The main objective of this work is to proceed with some numerical experiments in a broader set of cases. We show the efficiency of the approach in each of the settings considered.