Sampling via Measure Transport: An Introduction

Marzouk, Youssef; Moselhy, Tarek; Parno, Matthew; Spantini, Alessio

doi:10.1007/978-3-319-12385-1_23

Cited by 63 publications

(119 citation statements)

References 67 publications

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“…Rather than using importance sampling or MCMC to characterize the posterior distribution, this approach seeks a transport map that pushes forward a tractable "reference" distribution to the posterior, such that samples drawn from the reference and acted on by the map are distributed according to the posterior. Below we follow [36] to introduce the notion of transport maps.…”

Section: Transport Mapsmentioning

confidence: 99%

“…This construction leads to the notion of optimal transport ; see, e.g., [62,63,64]. Instead of introducing a cost function to regularize the problem of finding a transport map, we directly impose structure on the map T as in [40,44,36]. In particular, we will seek lower triangular maps that are monotone increasing.…”

Section: Definition Of Transport Mapsmentioning

confidence: 99%

“…. d. This condition ensures that ∇T 0 and det ∇T ≥ 0; see [36,57] for more detail. Since we assume that the reference and the target measures are absolutely continuous, existence and uniqueness of such a lower-triangular transport map is guaranteed; this map is in fact the Knothe-Rosenblatt rearrangement [6, 10, 52].…”

Section: Definition Of Transport Mapsmentioning

confidence: 99%

“…In particular, approximation errors may follow from the choice of the finite-dimensional approximation space T , the finite number of Monte Carlo samples used to discretize the expectation in the objective, and any other errors in the numerical optimization. While this error can be estimated (see [40,36]), reducing this error to an arbitrarily small threshold-e.g., by enriching T -can be computationally expensive.…”

Section: Problem Descriptionmentioning

confidence: 99%

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A transport-based multifidelity preconditioner for Markov chain Monte Carlo

Peherstorfer

Marzouk

2019

Adv Comput Math

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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 to single-fidelity MCMC sampling methods.Keywords: Bayesian inverse problems; transport maps; multifidelity; model reduction; Markov chain Monte Carlo * The second author acknowledges support of the AFOSR MURI on multi-information sources of multi-physics systems under Award Number FA9550-15-1-0038.

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Section: Transport Mapsmentioning

confidence: 99%

Section: Definition Of Transport Mapsmentioning

confidence: 99%

Section: Definition Of Transport Mapsmentioning

confidence: 99%

Section: Problem Descriptionmentioning

confidence: 99%

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A transport-based multifidelity preconditioner for Markov chain Monte Carlo

Peherstorfer

Marzouk

2019

Adv Comput Math

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“…We extend these constructions to N sto dimensions via tensorization and employ (3.2) to obtain the desired weighted (L)-Leja points. Note that T (j−1) in (3.2) can be seen as an affine transport map (see [29]).…”

mentioning

confidence: 99%

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

Farcaș¹,

Latz²,

Ullmann³

et al. 2020

SIAM J. Sci. Comput.

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Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations, deterministic quadrature is usually unable to efficiently investigate an informative and thus concentrated likelihood. This leads to a large number of required expensive evaluations of the mathematical model. To overcome these challenges, we formulate and test a multilevel adaptive sparse Leja algorithm. At each level, adaptive sparse grid interpolation and quadrature are used to approximate the posterior and perform all quadrature operations, respectively. Specifically, our algorithm uses coarse discretizations of the underlying mathematical model to investigate the parameter space and to identify areas of high posterior probability. Adaptive sparse grid algorithms are then used to place points in these areas, and ignore other areas of small posterior probability. The points are weighted Leja points. As the model discretization is coarse, the construction of the sparse grid is computationally efficient. On this sparse grid, the posterior measure can be approximated accurately with few expensive, fine model discretizations. The efficiency of the algorithm can be enhanced further by exploiting more than two discretization levels. We apply the proposed multilevel adaptive sparse Leja algorithm in numerical experiments involving elliptic inverse problems in 2D and 3D space, in which we compare it with Markov chain Monte Carlo sampling and a standard multilevel approximation.

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Sampling via Measure Transport: An Introduction

Cited by 63 publications

References 67 publications

A transport-based multifidelity preconditioner for Markov chain Monte Carlo

A transport-based multifidelity preconditioner for Markov chain Monte Carlo

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

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