Localization for MCMC: sampling high-dimensional posterior distributions with local structure

TL;DR

This paper explores how covariance localization techniques from weather prediction can improve MCMC sampling efficiency in high-dimensional Bayesian inverse problems by enforcing local structure and demonstrating dimension-independent convergence.

Contribution

It introduces a localization approach for MCMC in inverse problems, showing conditions for accurate posterior moments and efficient sampling in high dimensions.

Findings

Gibbs sampler convergence is dimension-independent for localized linear problems.

Localization reduces computational complexity in high-dimensional inverse problems.

Numerical examples demonstrate improved sampling efficiency with localization.

Abstract

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated "local" structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems.