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

Abstract

We develop a novel convergence theory for the multilevel sample variance estimators in the framework of the multilevel Monte Carlo methods. We prove that, dependent on the regularity of the quantity of interest, the multilevel sample variance estimator may achieve the same asymptotic cost/error relation as the multilevel sample mean, which is superior to the standard Monte Carlo method. Weaker regularity assumptions result in reduced convergence rates, quantified in our analysis. The general convergence theory is applied to a class of scalar elliptic obstacle problems with rough random obstacle profiles, which is a simple model of contact between a deformable body with a rough uncertain substrate. Numerical experiments confirm theoretical convergence proofs.