Multilevel Monte Carlo methods for stochastic elliptic multiscale PDEs

Abstract

In this paper Monte Carlo Finite Element (MC FE) approximations for elliptic homogenization problems with random coefficients which oscillate on $n \in \mathbb{N}$ a-priori known, separated length scales are considered. The convergence of multilevel MC FE (MLMC FE) discretizations is analyzed. In particular, it is considered that the multilevel FE discretization resolves the nest physical length scale, but the coarsest FE mesh does not, so that the so-called "resonance" case occurs at intermediate MLMC sampling levels. It is proved that switching to an Hierarchic Multiscale Finite Element method such as the Finite Element Heterogeneous Multiscale method (FE{HMM) to compute all MLMC FE samples on meshes which under-resolve the physical length scales implies once more optimal efficiency (in terms of accuracy versus computational work) for the numerical estimates of statistical moments with first and second order FE-HMMs. Specifi cally, the method proposed here allows to obtain estimates of the expectation of the random solution, with accuracy versus work that is identical to the solution of a single deterministic problem obtained by a FE-HMM, and which is, moreover, robust with respect to the physical length scales. Numerical experiments corroborate our analytical findings.