### 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 language | English (US) |
---|---|

Journal | Advances in Computational Mathematics |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Advances in Computational Mathematics*. https://doi.org/10.1007/s10444-019-09711-y

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

Research output: Contribution to journal › Article

*Advances in Computational Mathematics*. https://doi.org/10.1007/s10444-019-09711-y

}

TY - JOUR

T1 - A transport-based multifidelity preconditioner for Markov chain Monte Carlo

AU - Peherstorfer, Benjamin

AU - Marzouk, Youssef

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Bayesian inverse problems

KW - Markov chain Monte Carlo

KW - Model reduction

KW - Multifidelity

KW - Transport maps

UR - http://www.scopus.com/inward/record.url?scp=85068803923&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068803923&partnerID=8YFLogxK

U2 - 10.1007/s10444-019-09711-y

DO - 10.1007/s10444-019-09711-y

M3 - Article

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

ER -