Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

Abstract

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.

Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems

“…the twodimensional scalar obstacle problem, although the analysis for d = 2 can be inferred with similar arguments, see e.g. [3,18]. Under sufficient smoothness assumptions it holds that u(ω) ∈ W 2, p (D) for almost all ω ∈ .…”

“…Similar results can be shown for obstacle problems with random material parameters, cf. [11,18] for corresponding formulations.…”

“…This formulation can be viewed as a prototype of contact problems between deformable bodies describing many important real processes, for example, contact of car tires with various types of asphalt. We refer to [18] for convergence analysis of multilevel sample mean estimators for obstacle problems with random material parameters.…”

Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities

“…These assumptions imply Assumption 2.1 and thus existence and uniqueness of pathwise solutions upωq of (1) and u P L 2 pΩ; Hq. Note that uniform coercivity (28) can be replaced by weaker conditions (cf., e.g., [37]). On the background of the general results from Section 3 we now concentrate on MLMC finite element methods, for the numerical approximation of the expectation Erus.…”

“…For fixed ω P Ω and K " H, the quasioptimality condition (39) has been established for a variety of adaptive refinement strategies with a constant c 1 pωq (cf. e.g., [10,41,38] (37) provides an approximation Erũ L s with prescribed accuracy (34) at computational cost bounded by Cp1`d| log T ol 1 |q µ T ol´d 1 L µ`cs T ol´m axt2,du with " c s " 0, µ " 4 for d " 1, c s " 2, µ " 5 for d " 2,…”