Multilevel sequential Monte Carlo samplers

Abstract
In this article we consider the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods which depend on the step-size level . hL. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretization levels . ∞>h0>h1⋯>hL. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence and a sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. It is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context. That is, relative to exact sampling and Monte Carlo for the distribution at the finest level . hL. The approach is numerically illustrated on a Bayesian inverse problem. © 2016 Elsevier B.V.

Citation
Beskos A, Jasra A, Law K, Tempone R, Zhou Y (2016) Multilevel sequential Monte Carlo samplers. Stochastic Processes and their Applications. Available: http://dx.doi.org/10.1016/j.spa.2016.08.004.

Acknowledgements
AJ, KL & YZ were supported by an AcRF tier 2 grant: R-155-000-143-112. AJ is affiliated with the Risk Management Institute and the Center for Quantitative Finance at NUS. RT, KL & AJ were additionally supported by King Abdullah University of Science and Technology (KAUST). KL was further supported by ORNLDRD Strategic Hire grant. AB was supported by the Leverhulme Trust Prize. We thank the referees for their comments which have greatly improved the article.

Publisher
Elsevier BV

Journal
Stochastic Processes and their Applications

DOI
10.1016/j.spa.2016.08.004

arXiv
1503.07259

Additional Links
http://www.sciencedirect.com/science/article/pii/S0304414916301326https://pure.manchester.ac.uk/ws/files/74306319/multi_level_smc14.pdf

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