Multi-Level Monte-Carlo Finite Element Methods for stochastic elliptic variational inequalities

Abstract

Multi-Level Monte-Carlo Finite Element (MLMC--FE) methods for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of ``mean-square'' and ``pathwise'' formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for classical Monte--Carlo and novel MLMC--FE methods. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC--FE methods then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments