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We investigate how ideas from covariance localization in numerical weatherprediction can be used in Markov chain Monte Carlo (MCMC) sampling ofhigh-dimensional posterior distributions arising in Bayesian inverse problems.To localize an inverse problem is to enforce an anticipated "local" structureby (i) neglecting small off-diagonal elements of the prior precision andcovariance matrices; and (ii) restricting the influence of observations totheir neighborhood. For linear problems we can specify the conditions underwhich posterior moments of the localized problem are close to those of theoriginal problem. We explain physical interpretations of our assumptions aboutlocal structure and discuss the notion of high dimensionality in localproblems, which is different from the usual notion of high dimensionality infunction space MCMC. The Gibbs sampler is a natural choice of MCMC algorithmfor localized inverse problems and we demonstrate that its convergence rate isindependent of dimension for localized linear problems. Nonlinear problems canalso be tackled efficiently by localization and, as a simple illustration ofthese ideas, we present a localized Metropolis-within-Gibbs sampler. Severallinear and nonlinear numerical examples illustrate localization in the contextof MCMC samplers for inverse problems.

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