Optimal Approximation of the First-Order Correctorin Multiscale Stochastic Elliptic PDE

This work addresses the development of an optimal computational scheme for the approximation of the first-order corrector arising in the stochastic homogenization of linear elliptic PDEs in divergence form. Equations of this type describe, for exafriple, diffusion phenomena in materials with a heterogeneous microstructure, but require enormous computational efforts in order to obtain reliable results. We derive no optimization problem for the needed computationnl work with a given error tolerance, then extract the governing parameters from numerical experiments, and finally solve the obtained optimization problem. The numerical approach investigated here is a stochastic sampling scheme for the probability space connected with a finite-element method for the discretization of the pltysical space.