Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions

Del Moral, Pierre; Jasra, Ajay; Law, Kody J. H.

doi:10.1080/07362994.2016.1272421

Title: Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions

Abstract

We consider the multilevel sequential Monte Carlo (MLSMC) method of Beskos et al. (Stoch. Proc. Appl. [to appear]). This technique is designed to approximate expectations w.r.t. probability laws associated to a discretization. For instance, in the context of inverse problems, where one discretizes the solution of a partial differential equation. The MLSMC approach is especially useful when independent, coupled sampling is not possible. Beskos et al. show that for MLSMC the computational effort to achieve a given error, can be less than independent sampling. In this article we significantly weaken the assumptions of Beskos et al., extending the proofs to non-compact state-spaces. The assumptions are based upon multiplicative drift conditions as in Kontoyiannis and Meyn (Electron. J. Probab. 10 [2005]: 61–123). The assumptions are verified for an example.

Authors:

Del Moral, Pierre ^[1]; Jasra, Ajay ^[2]; Law, Kody J. H. ^[3]

Univ. of Bordeaux (France)
National Univ. of Singapore (Singapore)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Publication Date:: Mon Jan 09 00:00:00 EST 2017

Research Org.:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Center for Nanophase Materials Sciences (CNMS)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1361332

Grant/Contract Number:: AC05-00OR22725

Resource Type:: Accepted Manuscript

Journal Name:: Stochastic Analysis and Applications

Additional Journal Information:: Journal Volume: 35; Journal Issue: 3; Journal ID: ISSN 0736-2994

Publisher:: Taylor & Francis

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING

Citation Formats


                    Del Moral, Pierre, Jasra, Ajay, and Law, Kody J. H. Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions.  United States: N. p., 2017. 
Web.  doi:10.1080/07362994.2016.1272421.

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                    Del Moral, Pierre, Jasra, Ajay, & Law, Kody J. H. Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions.  United States.  https://doi.org/10.1080/07362994.2016.1272421

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                    Del Moral, Pierre, Jasra, Ajay, and Law, Kody J. H. Mon .  
"Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions".  United States.  https://doi.org/10.1080/07362994.2016.1272421.  https://www.osti.gov/servlets/purl/1361332.

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@article{osti_1361332,

  title        = {Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions},

  author       = {Del Moral, Pierre and Jasra, Ajay and Law, Kody J. H.},

  abstractNote = {We consider the multilevel sequential Monte Carlo (MLSMC) method of Beskos et al. (Stoch. Proc. Appl. [to appear]). This technique is designed to approximate expectations w.r.t. probability laws associated to a discretization. For instance, in the context of inverse problems, where one discretizes the solution of a partial differential equation. The MLSMC approach is especially useful when independent, coupled sampling is not possible. Beskos et al. show that for MLSMC the computational effort to achieve a given error, can be less than independent sampling. In this article we significantly weaken the assumptions of Beskos et al., extending the proofs to non-compact state-spaces. The assumptions are based upon multiplicative drift conditions as in Kontoyiannis and Meyn (Electron. J. Probab. 10 [2005]: 61–123). The assumptions are verified for an example.},

  doi          = {10.1080/07362994.2016.1272421},

  journal      = {Stochastic Analysis and Applications},

  number       = 3,

  volume       = 35,

  place        = {United States},

  year         = {Mon Jan 09 00:00:00 EST 2017},

  month        = {Mon Jan 09 00:00:00 EST 2017}

}

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Works referenced in this record:

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Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay
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Whiteley, Nick
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Heinrich, S.
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Kontoyiannis, Ioannis; Meyn, Sean
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Sequential Monte Carlo Samplers: Error Bounds and Insensitivity to Initial Conditions
journal, September 2012

Whiteley, Nick
Stochastic Analysis and Applications, Vol. 30, Issue 5
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Multilevel sequential Monte Carlo samplers
journal, May 2017

Beskos, Alexandros; Jasra, Ajay; Law, Kody
Stochastic Processes and their Applications, Vol. 127, Issue 5
DOI: 10.1016/j.spa.2016.08.004

Forward and Inverse Uncertainty Quantification Using Multilevel Monte Carlo Algorithms for an Elliptic Nonlocal Equation
journal, January 2016

Jasra, Ajay; Law, Kody J. H.; Zhou, Yan
International Journal for Uncertainty Quantification, Vol. 6, Issue 6
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On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions
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Jasra, Ajay
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Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae
journal, April 2012

Whiteley, Nick; Kantas, Nikolas; Jasra, Ajay
Stochastic Processes and their Applications, Vol. 122, Issue 4
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Sequential Monte Carlo samplers
journal, June 2006

Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay
Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 68, Issue 3
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On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions
journal, June 2015

Jasra, Ajay
Journal of Applied Probability, Vol. 52, Issue 2
DOI: 10.1239/jap/1437658602

Sequential Monte Carlo samplers: error bounds and insensitivity to initial conditions
preprint, January 2011

Whiteley, Nick
arXiv
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Linear Variance Bounds for Particle Approximations of Time-Homogeneous Feynman-Kac Formulae
text, January 2011

Whiteley, Nick; Kantas, Nikolas; Jasra, Ajay
arXiv
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Stability properties of some particle filters
text, January 2011

Whiteley, Nick
arXiv
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On adaptive resampling strategies for sequential Monte Carlo methods
text, January 2012

Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay
arXiv
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Complexity Analysis of Accelerated MCMC Methods for Bayesian Inversion
preprint, January 2012

Hoang, Viet Ha; Schwab, Christoph; Stuart, Andrew M.
arXiv
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Forward and Inverse Uncertainty Quantification using Multilevel Monte Carlo Algorithms for an Elliptic Nonlocal Equation
preprint, January 2016

Jasra, Ajay; Law, Kody; Zhou, Yan
arXiv
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Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes
text, January 2005

Kontoyiannis, Ioannis; Meyn, S. P.
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Works referencing / citing this record:

Multilevel Monte Carlo in approximate Bayesian computation
journal, January 2019

Jasra, Ajay; Jo, Seongil; Nott, David
Stochastic Analysis and Applications, Vol. 37, Issue 3
DOI: 10.1080/07362994.2019.1566006

Multilevel Monte Carlo in Approximate Bayesian Computation
preprint, January 2017

Jasra, Ajay; Jo, Seongil; Nott, David
arXiv
DOI: 10.48550/arxiv.1702.03628

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Title: Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions

Abstract

Citation Formats

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Complexity analysis of accelerated MCMC methods for Bayesian inversion
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On adaptive resampling strategies for sequential Monte Carlo methods
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Complexity Analysis of Accelerated MCMC Methods for Bayesian Inversion
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