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A control variate approach based on a defect-type theory for variance reduction in stochastic homogenization

We consider a variance reduction approach for the stochastic homogenization of divergence form linear elliptic problems. Although the exact homogenized coefficients are deterministic, their practical approximations are random. We introduce a control variate technique to reduce the variance of the computed approximations of the homogenized coefficients. Our approach is based on a surrogate model inspired by a defect-type theory, where a perfect periodic material is perturbed by rare defects. This model has been introduced in [A. Anantharaman and C. Le Bris, CRAS 2010] in the context of weakly random models. In this work, we address the fully random case, and show that the perturbative approaches proposed in [A. Anantharaman and C. Le Bris, CRAS 2010, MMS 2011] can be turned into an efficient control variable. We theoretically demonstrate the efficiency of our approach in simple cases. We next provide illustrating numerical results and compare our approach with other variance reduction strategies. We also show how to use the Reduced Basis approach proposed in [C. Le Bris and F. Thomines, Chinese Ann. Math. 2012] so that the cost of building the surrogate model remains limited.