A Simple Collocation-Type Approach to Numerical Stochastic Homogenization

This paper proposes a novel collocation-type numerical stochastic homogenization method for prototypical stochastic homogenization problems with random coefficient fields of small correlation lengths. The presented method is based on a recently introduced localization technique that enforces a super-exponential decay of the basis functions relative to the underlying coarse mesh, resulting in considerable computational savings during the sampling phase. More generally, the collocation-type structure offers a particularly simple and computationally efficient construction in the stochastic setting with minimized communication between the patches where the basis functions of the method are computed. An error analysis that bridges numerical homogenization and the quantitative theory of stochastic homogenization is performed. In a series of numerical experiments, we study the effect of the correlation length and the discretization parameters on the approximation quality of the method.