The Library
Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations
Tools
Tools
Tools
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
|
PDF
WRAP-multi-level-Monte-Carlo-invariant-stochastic-Vollmer-2019.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons Attribution 4.0. Download (1059Kb) | Preview |
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 | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science > Mathematics | ||||||
| Journal or Publication Title: | Statistics and Computing | ||||||
| Publisher: | Springer | ||||||
| ISSN: | 0960-3174 | ||||||
| Official Date: | 10 September 2019 | ||||||
| Dates: |
|
||||||
| Date of first compliant deposit: | 30 October 2019 | ||||||
| DOI: | 10.1007/s11222-019-09890-0 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access |
Request changes or add full text files to a record
Repository staff actions (login required)
 |
View Item |
Downloads
Downloads per month over past year
