Convergence of Multilevel MCMC methods on path spaces

Abstract

Within this work, the efficiency of Markov Chain Monte Carlo methods on infinite dimensional spaces, such as function spaces, is analyzed. We study two aspects in this respect:<br /> The first aspect is a Multilevel Markov Chain Monte Carlo algorithm. It extends a Multilevel Monte Carlo method introduced by Giles to Markov Chains, and overcomes the need for a trade-off between discretization error and Monte Carlo error. We develop the Multilevel algorithm, state and prove its order of convergence and show results of numerical simulations.<br /> The second part of this work deals with the analysis of the speed of convergence of the Metropolis Adjusted Langevin Algorithm (MALA). Controlling the speed of convergence is an important tool for bounding the error of Markov Chain Monte Carlo methods. It is also a crucial ingredient for bounding the order of convergence of the Multilevel algorithm. We apply a method of Eberle to the Hilbert space case and obtain a subexponential bound on the distance of the distribution of the MALA-process to its invariant measure.<br /> Both aspects are illustrated by an application from molecular dynamics called Transition Path Sampling. In this example, Markov Chain Monte Carlo methods on path spaces are used to simulate the properties of transitions from one metastable state of a molecule to another. We present this application and apply the results on the Multilevel estimator and the MALA-process in this context.