Bayesian inversion by parallel interacting Markov chains

Abstract

Markov chain Monte Carlo (MCMC) methods are known to produce samples of virtually any distribution. They have already been widely used in the resolution of non-linear inverse problems where no analytical expression for the forward relation between data and model parameters is available, and where linearization is unsuccessful. However, in Bayesian inversion, the total number of simulations we can afford is highly related to the computational cost of the forward model. Hence, the complete browsing of the support of the posterior distribution is hardly performed at final time, especially when the posterior is high dimensional and/or multimodal. In the latter case, the chain may stay stuck in one of the modes. Recently, the idea of making several Markov chains interact at different temperatures has been explored. These methods improve the mixing properties of classical single MCMC. Furthermore, these methods can make efficient use of large central processing unit (CPU) clusters, without increasing the global computational cost with respect to classical MCMC.

Keywords:

• inverse problem
• Bayesian inversion
• MCMC
• interacting Markov chains
• tempering
• history matching