Claude Le Bris: Making stochastic homogenization practical

Stochastic homogenization is a well developed theory, where several important theoretical issues remain open. On the practical level, its implementation, as a computational approach to approximate problems with heterogeneities at different scales, can however be enormously expensive. This clearly limits the applicability of the theory to real world problems. Dedicated methods are thus necessary, in order to reduce the computational cost. These methods in turn require new theoretical developments. We will overview recent progress in this direction. Approaches that consider small random perturbations of classical multiscale problems will be introduced, studied and tested, as an efficient compromise between fully random problems and idealistic deterministic problems. On the other hand, for problems where randomness cannot be considered small and where variance plays a crucial practical role, several variance reduction methods will be addressed. Their efficiency will be assessed on some illustrative multiscale random problems.