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Central limit theorems for multilevel Monte Carlo methods

Journal article
Authors Håkon Hoel
S. Krumscheid
Published in Journal of Complexity
Volume 54
ISSN 0885-064X
Publication year 2019
Published at Department of Mathematical Sciences
Language en
Links dx.doi.org/10.1016/j.jco.2019.05.00...
Keywords Multilevel Monte Carlo, Central limit theorem, uncertainty quantification, Mathematics
Subject categories Mathematics

Abstract

In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo (MLMC) estimators and we provide near optimal weaker conditions under which the CLT is achieved. In particular, if the variance decay rate dominates the computational cost rate (i.e., beta > gamma), we prove that the CLT applies to the standard (variance minimizing) MLMC estimator. For other settings where the CLT may not apply to the standard MLMC estimator, we propose an alternative estimator, called the mass-shifted MLMC estimator, to which the CLT always applies. This comes at a small efficiency loss: the computational cost of achieving mean square approximation error O(epsilon(2)) is at worst a factor O(log(1/epsilon)) higher with the mass-shifted estimator than with the standard one. (C) 2019 Elsevier Inc. All rights reserved.

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