Fast Algorithms for Bayesian Inversion

Abstract

In this article, we review a few fast algorithms for solving large-scale stochastic inverse problems using Bayesian methods. After a brief introduction to the Bayesian stochastic inverse methodology, we review the following computational techniques, to solve large scale problems: the fast Fourier transform, the fast multipole method (classical and a black-box version), and finallym the hierarchical matrix approach. We emphasize that this is mainly a survey paper presenting a few fast algorithms applicable to large-scale Bayesian inversion techniques, applicable to applications arising from geostatistics. The article is presented at a level accessible to graduate students and computational engineers. Hence, we mainly present the algorithmic ideas and theoretical results.