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Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations

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Giles, Michael B., Majka, Mateusz B., Szpruch, Lukasz, Vollmer, Sebastian and Zygalakis, Konstantinos C. (2019) Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations. Statistics and Computing . doi:10.1007/s11222-019-09890-0 ISSN 0960-3174.

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Official URL: http://dx.doi.org/10.1007/s11222-019-09890-0

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Abstract

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. https://doi.org/10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of O(ε) is achieved with O(ε−2) complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity O(ε−2|logε|3) , in contrast to the complexity O(ε−3) of currently available methods. Numerical experiments confirm our theoretical findings.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Monte Carlo method, Numerical analysis, Stochastic processes
Journal or Publication Title: Statistics and Computing
Publisher: Springer
ISSN: 0960-3174
Official Date: 10 September 2019
Dates:
DateEvent
10 September 2019Available
13 August 2019Accepted
DOI: 10.1007/s11222-019-09890-0
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 30 October 2019
Date of first compliant Open Access: 30 October 2019
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
EP/P003818/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
EP/N510129/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266

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