Adaptive multilevel Monte Carlo methods for random elliptic problems
In this thesis we introduce a novel framework for uncertainty quantification in problems with random coefficients. The developed framework utilizes the ideas of multilevel Monte Carlo (MLMC) methods and allows for exploiting the advantages of adaptive finite element techniques. In contrast to the standard MLMC method, where levels are characterized by a hierarchy of uniform meshes, we associate the MLMC levels with a chosen sequence of tolerances. Each deterministic problem corresponding to a MC sample on a given level is then approximated up to the corresponding accuracy. This can be done, for example, using pathwise a posteriori error estimation and adaptive mesh refinement techniques. We further introduce an adaptive MLMC finite element method for random linear elliptic problems based on a residual-based a posteriori error estimation technique. We provide a careful analysis of the novel method based on a generalization of existing results, for deterministic residual-based error estimation, to the random setting. We complement our theoretical results by numerical simulations illustrating the advantages of our approach compared to the standard MLMC finite element method when applied to problems with random singularities.