Variance reduction in random homogenization: special quasirandom structures
William Minvielle, Claude Le Bris, Frédéric Legoll · UPCommons institutional repository (Universitat Politècnica de Catalunya) · 2015
In this work, we introduce a variance reduction approach for the homogenization of a random, linear elliptic second order partial differential equation set on a bounded domain in Rd. The random diffusion coefficient matrix field A( x , ω) is assumed to be uniformly elliptic, bounded and stationary (“periodic in law”). In the limit when ε 0, the solution of the equation converges to that of a homogenized problem of the same form, the coefficient field of which is a deterministic and constant matrix A* given by an average involving the so-called corrector function that solves a random auxilliary problem set on the entire space. In practice, the corrector problem is approximated on a bounded domain QN as large as possible. A by-product of this truncation procedure is that the deterministic matrix A* is approximated by a random, apparent homoge- nized matrix A* (ω). We therefore introduce a variance reduction approach to obtain practical approximations of A* with a smaller variance in order to reduce the statistical error. We derive conditions (e.g. exact fraction in a of finite supercell environments on which we solve the corrector equation.