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A transport-based multifidelity preconditioner for Markov chain Monte Carlo

  • Benjamin PeherstorferEmail author
  • Youssef Marzouk
Article
  • 44 Downloads
Part of the following topical collections:
  1. Model reduction of parametrized Systems

Abstract

Markov chain Monte Carlo (MCMC) sampling of posterior distributions arising in Bayesian inverse problems is challenging when evaluations of the forward model are computationally expensive. Replacing the forward model with a low-cost, low-fidelity model often significantly reduces computational cost; however, employing a low-fidelity model alone means that the stationary distribution of the MCMC chain is the posterior distribution corresponding to the low-fidelity model, rather than the original posterior distribution corresponding to the high-fidelity model. We propose a multifidelity approach that combines, rather than replaces, the high-fidelity model with a low-fidelity model. First, the low-fidelity model is used to construct a transport map that deterministically couples a reference Gaussian distribution with an approximation of the low-fidelity posterior. Then, the high-fidelity posterior distribution is explored using a non-Gaussian proposal distribution derived from the transport map. This multifidelity “preconditioned” MCMC approach seeks efficient sampling via a proposal that is explicitly tailored to the posterior at hand and that is constructed efficiently with the low-fidelity model. By relying on the low-fidelity model only to construct the proposal distribution, our approach guarantees that the stationary distribution of the MCMC chain is the high-fidelity posterior. In our numerical examples, our multifidelity approach achieves significant speedups compared with single-fidelity MCMC sampling methods.

Keywords

Bayesian inverse problems Transport maps Multifidelity Model reduction Markov chain Monte Carlo 

Mathematics Subject Classification (2010)

65C05 65C40 65C60 

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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