A transport-based multifidelity preconditioner for Markov chain Monte Carlo

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 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.

Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports

“…Apart from building transport maps using TT decompositions, most of other methods approximate the transport map T by solving an optimisation problem such that T minimises some statistical divergence between the target ν π and the pushforward T μ. The mapping T often has a triangular structure, which is computationally efficient for evaluating the Jacobian and the inverse of T and can be represented using polynomials [4,43,50,51], kernel functions [15,37], invertible neural networks [7,9,10,35,49,52], etc. In this setting, the objective function has to be approximated using a Monte Carlo average and minimised by some (stochastic) gradient-based method.…”

“…-Density approximation. The work of [4,43,51] aims to minimise the Kullback-Leibler (KL) divergence of the pushforward T μ from the target ν π , in which the pushforward density naturally approximates the target density. In this case, the KL divergence is approximated using the Jacobian of T and the target density function evaluated at samples drawn from the analytically tractable reference measure.…”

“…where q k is defined in (52). Then, the unnormalised PDFπ k (51) approximates the unnormalised density function of the kth bridging measure π k with the error…”

Multilevel Adaptive Sparse Leja Approximations for Bayesian Inverse Problems

“…Alternatively, it is possible to construct a multilevel approximation without relying on such a telescoping sum. Examples are the multifidelity preconditioned MCMC method in [33], Multilevel Sequential 2 Monte Carlo [25], and our multilevel sparse Leja approximation presented in section 3.…”

“…The smoothness assumptions can be weakened by using piecewise polynomial approximations together with Voronoi tesselations of the parameter space [31]. Of course, surrogates can also be used to accelerate sampling-based approximations such as MCMC, see e.g., [26,33]. We remark that Quasi-Monte Carlo [8,35] is in principle a sampling-free method which does not rely on surrogates, however, it requires again a smooth approximand, and is often used together with randomization.…”