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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
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 | |||||||||
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| 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: |
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| 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: |
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