Multilevel Monte Carlo methods

Multilevel Monte Carlo Methods

We study Monte Carlo approximations to high dimensional parameter dependent integrals. We survey the multilevel variance reduction technique introduced by the author in [4] and present extensions and new developments of it. The tools needed for the convergence analysis of vector-valued Monte Carlo methods are discussed, as well. Applications to stochastic solution of integral equations are give...

Massively parallel Multilevel Monte Carlo methods

Multilevel Monte Carlo methods and uncertainty quantification

Attention is drawn to the fact that copyright of this thesis rests with its author. This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the prior written consent of the author. This thesis may be ...

Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems∗

We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes’s rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo, or multilevel Mon...

Multilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEs

In this paper Monte Carlo Finite Element (MC FE) approximations for elliptic homogenization problems with random coefficients which oscillate on n ∈ 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 finest physical length scale, bu...