Dimension-Independent MCMC Sampling for Inverse Problems with Non-Gaussian Priors

TL;DR

This paper develops a method for designing dimension-independent MCMC algorithms for inverse problems with non-Gaussian priors, enabling efficient sampling in high or infinite-dimensional spaces.

Contribution

It extends dimension-independent MCMC bounds from Gaussian to non-Gaussian priors and demonstrates their application to groundwater flow inverse problems.

Findings

Dimension-independent bounds are achievable for non-Gaussian priors.

An efficient Metropolis-Hastings proposal is explicitly constructed.

Numerical experiments support the theoretical results.

Abstract

The computational complexity of MCMC methods for the exploration of complex probability measures is a challenging and important problem. A challenge of particular importance arises in Bayesian inverse problems where the target distribution may be supported on an infinite dimensional space. In practice this involves the approximation of measures defined on sequences of spaces of increasing dimension. Motivated by an elliptic inverse problem with non-Gaussian prior, we study the design of proposal chains for the Metropolis-Hastings algorithm with dimension independent performance. Dimension-independent bounds on the Monte-Carlo error of MCMC sampling for Gaussian prior measures have already been established. In this paper we provide a simple recipe to obtain these bounds for non-Gaussian prior measures. To illustrate the theory we consider an elliptic inverse problem arising in groundwater flow. We explicitly construct an efficient Metropolis-Hastings proposal based on local proposals, and we provide numerical evidence which supports the theory.