MCMC Methods for Bayesian Inverse Problems

James Martin, University of Texas

We investigate a computational Bayesian approach to large-scale inverse problems for a 1-D seismic wave propagation model, in which the solution is given in the form of a posterior probability density over a search space of possible parameter values. Due to the inherent high dimensionality and ill-posedness of such problems, this search space can be very large, while the important regions (i.e., those with high probability density) are extremely narrow. Since traditional Markov Chain Monte Carlo (MCMC) methods behave very poorly for these types of problems, we use methods inspired from stochastic differential equations and Langevin Dynamics to steer the MCMC chain toward the important directions of the posterior probability distribution. This significantly reduces the computational complexity of the problem, drastically reducing the number of samples required for convergence of the MCMC chain. Our method is demonstrated for a two-dimensional parameterization of the search space in which the results can be readily visualized, and then extended to larger systems.

Abstract Author(s): Carsten Burstedde, Omar Ghattas, James Martin, and Lucas Wilcox