This article reviews the application of some advanced Monte Carlo techniques in the context of multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations, which can be biassed in some sense, for instance, by using the discretization of an associated probability law. The MLMC approach works with a hierarchy of biassed approximations, which become progressively more accurate and more expensive. Using a telescoping representation of the most accurate approximation, the method is able to reduce the computational cost for a given level of error versus i.i.d. sampling from this latter approximation. All of these ideas originated for cases where exact sampling from couples in the hierarchy is possible. This article considers the case where such exact sampling is not currently possible. We consider some Markov chain Monte Carlo and sequential Monte Carlo methods, which have been introduced in the literature, and we describe different strategies that facilitate the application of MLMC within these methods.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): CRG4
Acknowledgements: A. J. and C. S. were supported under the KAUST Competitive Research Grants Program—Round 4 (CRG4) project, advanced multilevel sampling techniques for Bayesian inverse problems with applications to subsurface, ref: 2584. K. J. H. L. was supported by Oak RidgeNational Laboratory (ORNL) Directed Research and Development Seed funding and much ofthis work was performed while he was a staff member at ORNL. We thank the editor and two reviewers whose comments have greatly improved the paper.