Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals

Beskos, Alexandros; Jasra, Ajay; Law, Kody; Marzouk, Youssef; Zhou, Yan

doi:10.1137/17m1120993

Title: Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals

Abstract

In this article we develop a new sequential Monte Carlo method for multilevel Monte Carlo estimation. In particular, the method can be used to estimate expectations with respect to a target probability distribution over an infinite-dimensional and noncompact space---as produced, for example, by a Bayesian inverse problem with a Gaussian random field prior. Under suitable assumptions the MLSMC method has the optimal $$\mathcal{O}(\varepsilon^{-2})$$ bound on the cost to obtain a mean-square error of $$\mathcal{O}(\varepsilon^2)$$. The algorithm is accelerated by dimension-independent likelihood-informed proposals [T. Cui, K. J. Law, and Y. M. Marzouk, (2016), J. Comput. Phys., 304, pp. 109--137] designed for Gaussian priors, leveraging a novel variation which uses empirical covariance information in lieu of Hessian information, hence eliminating the requirement for gradient evaluations. The efficiency of the algorithm is illustrated on two examples: (i) inversion of noisy pressure measurements in a PDE model of Darcy flow to recover the posterior distribution of the permeability field and (ii) inversion of noisy measurements of the solution of an SDE to recover the posterior path measure.

Authors:

Beskos, Alexandros ^[1]; Jasra, Ajay ^[2];

^[3]; Marzouk, Youssef ^[4]; Zhou, Yan ^[2]

University College London (United Kingdom)
National Univ. of Singapore (Singapore)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

Publication Date:: Tue Mar 14 00:00:00 EDT 2017

Research Org.:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1862172

Grant/Contract Number:: AC05-00OR22725; 32112580

Resource Type:: Accepted Manuscript

Journal Name:: SIAM/ASA Journal on Uncertainty Quantification

Additional Journal Information:: Journal Volume: 6; Journal Issue: 2; Journal ID: ISSN 2166-2525

Publisher:: Society for Industrial and Applied Mathematics (SIAM)

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; multilevel Monte Carlo; sequential Monte Carlo; Bayesian inverse problem; uncertainty quantification

Citation Formats


                    Beskos, Alexandros, Jasra, Ajay, Law, Kody, Marzouk, Youssef, and Zhou, Yan. Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals.  United States: N. p., 2017. 
Web.  doi:10.1137/17m1120993.

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                    Beskos, Alexandros, Jasra, Ajay, Law, Kody, Marzouk, Youssef, & Zhou, Yan. Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals.  United States.  https://doi.org/10.1137/17m1120993

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                    Beskos, Alexandros, Jasra, Ajay, Law, Kody, Marzouk, Youssef, and Zhou, Yan. Tue .  
"Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals".  United States.  https://doi.org/10.1137/17m1120993.  https://www.osti.gov/servlets/purl/1862172.

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@article{osti_1862172,

  title        = {Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals},

  author       = {Beskos, Alexandros and Jasra, Ajay and Law, Kody and Marzouk, Youssef and Zhou, Yan},

  abstractNote = {In this article we develop a new sequential Monte Carlo method for multilevel Monte Carlo estimation. In particular, the method can be used to estimate expectations with respect to a target probability distribution over an infinite-dimensional and noncompact space---as produced, for example, by a Bayesian inverse problem with a Gaussian random field prior. Under suitable assumptions the MLSMC method has the optimal $\mathcal{O}(\varepsilon^{-2})$ bound on the cost to obtain a mean-square error of $\mathcal{O}(\varepsilon^2)$. The algorithm is accelerated by dimension-independent likelihood-informed proposals [T. Cui, K. J. Law, and Y. M. Marzouk, (2016), J. Comput. Phys., 304, pp. 109--137] designed for Gaussian priors, leveraging a novel variation which uses empirical covariance information in lieu of Hessian information, hence eliminating the requirement for gradient evaluations. The efficiency of the algorithm is illustrated on two examples: (i) inversion of noisy pressure measurements in a PDE model of Darcy flow to recover the posterior distribution of the permeability field and (ii) inversion of noisy measurements of the solution of an SDE to recover the posterior path measure.},

  doi          = {10.1137/17m1120993},

  journal      = {SIAM/ASA Journal on Uncertainty Quantification},

  number       = 2,

  volume       = 6,

  place        = {United States},

  year         = {Tue Mar 14 00:00:00 EDT 2017},

  month        = {Tue Mar 14 00:00:00 EDT 2017}

}

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Rates of convergence of the Hastings and Metropolis algorithms
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DOI: 10.1214/aos/1033066201

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Dashti, Masoumeh; Stuart, Andrew M.
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preprint, January 2013

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Works referencing / citing this record:

Transform-based particle filtering for elliptic Bayesian inverse problems
journal, October 2019

Ruchi, S.; Dubinkina, S.; Iglesias, M. A.
Inverse Problems, Vol. 35, Issue 11
DOI: 10.1088/1361-6420/ab30f3

Transform-based particle filtering for elliptic Bayesian inverse problems
text, January 2019

Ruchi, Sangeetika; Dubinkina, Svetlana; Iglesias, Marco
arXiv
DOI: 10.48550/arxiv.1901.04706

Multilevel Sequential2 Monte Carlo for Bayesian inverse problems
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Latz, Jonas; Papaioannou, Iason; Ullmann, Elisabeth
Journal of Computational Physics, Vol. 368
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Markov chain simulation for multilevel Monte Carlo
journal, January 2021

Jasra, Ajay; Law, Kody J. H.; Xu, Yaxian
Foundations of Data Science, Vol. 3, Issue 1
DOI: 10.3934/fods.2021004

Data-Free Likelihood-Informed Dimension Reduction of Bayesian Inverse Problems
text, January 2021

Cui, Tiangang; Zahm, Olivier
arXiv
DOI: 10.48550/arxiv.2102.13245

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Title: Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals

Abstract

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