Multilevel Monte Carlo Methods and Uncertainty Quantication

Abstract

We consider the application of multilevel Monte Carlo methods to elliptic partialdierential equations with random coecients. Such equations arise, for example,in stochastic groundwater ow modelling. Models for random coecientsfrequently used in these applications, such as log-normal random elds with exponentialcovariance, lack uniform coercivity and boundedness with respect tothe random parameter and have only limited spatial regularity.To give a rigorous bound on the cost of the multilevel Monte Carlo estimatorto reach a desired accuracy, one needs to quantify the bias of the estimator. Thebias, in this case, is the spatial discretisation error in the numerical solution of thepartial dierential equation. This thesis is concerned with establishing boundson this discretisation error in the practically relevant and technically demandingcase of coecients which are not uniformly coercive or bounded with respect tothe random parameter.Under mild assumptions on the regularity of the coecient, we establish newresults on the regularity of the solution for a variety of model problems. Themost general case is that of a coecient which is piecewise Holder continuous withrespect to a random partitioning of the domain. The established regularity of thesolution is then combined with tools from classical discretisation error analysis toprovide a full convergence analysis of the bias of the multilevel estimator for niteelement and nite volume spatial discretisations. Our analysis covers as quantitiesof interest several spatial norms of the solution, as well as point evaluations of thesolution and its gradient and any continuously Frechet dierentiable functional.Lastly, we extend the idea of multilevel Monte Carlo estimators to the frameworkof Markov chain Monte Carlo simulations. We develop a new multilevelversion of a Metropolis Hastings algorithm, and provide a full convergence analysis.

Date of Award	19 Jun 2013
Original language	English
Awarding Institution	University of Bath
Supervisor	Robert Scheichl (Supervisor)