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Multilevel sequential Monte Carlo samplers

Beskos, A; Jasra, A; Law, K; Tempone, R; Zhou, Y; (2017) Multilevel sequential Monte Carlo samplers. Stochastic Processes and their Applications , 127 (5) pp. 1417-1440. 10.1016/j.spa.2016.08.004. Green open access

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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.

Type: Article
Title: Multilevel sequential Monte Carlo samplers
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.spa.2016.08.004
Publisher version: https://doi.org/10.1016/j.spa.2016.08.004
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Science & Technology, Physical Sciences, Statistics & Probability, Mathematics, Multilevel Monte Carlo, Sequential Monte Carlo, Bayesian Inverse Problems, Inference, Models
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/1465702
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