title: Some variance reduction methods for numerical stochastic homogenization.
abstract: We give an overview of a series of recent studies devoted to variance reduction techniques for numerical stochastic homogenization. Numerical homogenization requires that a set of problems is solved at the microscale, the so-called corrector problems. In a random environment, these problems are stochastic and therefore need to be repeatedly solved, for several configurations of the medium considered. An empirical average over all configurations is then performed using the Monte Carlo approach, so as to approximate the effective coefficients necessary to determine the macroscopic behaviour. Variance severely affects the accuracy and the cost of such computations. Variance reduction approaches, borrowed from other contexts in the engineering sciences, can be useful. Some of these variance reduction techniques are presented, studied and tested here.
authors.affiliation: Université Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, 75205 Paris, France.
authors.forename: X
authors.initials: X
authors.lastname: Blanc
authors.affiliation: Ecole des Ponts and INRIA, 6 and 8 Avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France lebris@cermics.enpc.fr.
authors.forename: C
authors.initials: C
authors.lastname: Le Bris
authors.affiliation: Ecole des Ponts and INRIA, 6 and 8 Avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France.
authors.forename: F
authors.initials: F
authors.lastname: Legoll
keywords: elliptic partial differential equations; mathematical modelling in materials science; stochastic homogenization; variance reduction