A transport-based multifidelity preconditioner for Markov chain Monte Carlo

Benjamin Peherstorfer, Youssef Marzouk

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
JournalAdvances in Computational Mathematics
DOIs
StatePublished - Jan 1 2019

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Markov Chain Monte Carlo
Preconditioner
Markov processes
Fidelity
Posterior distribution
Model
Sampling
Monte Carlo Sampling
Stationary Distribution
Gaussian distribution
Inverse problems
Sampling Methods
Costs
Monte Carlo method
Computational Cost
Inverse Problem

Keywords

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

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

A transport-based multifidelity preconditioner for Markov chain Monte Carlo. / Peherstorfer, Benjamin; Marzouk, Youssef.

In: Advances in Computational Mathematics, 01.01.2019.

Research output: Contribution to journalArticle

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