Sequential Implicit Sampling Methods for Bayesian Inverse Problems

Abstract

The solutionto the inverse problems under the Bayesian framework is given by a posterior probability density. For large-scale problems, sampling the posterior can be an extremely challenging task. Markov chain Monte Carlo provides a general way for sampling but it can be computationally expensive. Gaussian type methods, such as the ensemble Kalman filter (EnKF), make Gaussian assumptions at some point(s) in the algorithms, even for the possible non-Gaussian densities, which may lead to inaccuracy. In this paper, the implicit sampling method, one of the importance sampling methods, and the newly proposed sequential implicit sampling method are investigated for the inverse problem involving time-dependent partial differential equations. The sequential implicit sampling method combines the idea of the EnKF and implicit sampling and it is particularly suitable for time-dependent problems. Moreover, the new method is capable of reducing the computational cost in the optimization, which is a necessary and the most expensive step in the implicit sampling method. The sequential implicit sampling method has been tested on a seismic wave inversion. The numerical experiments show its accuracy and efficiency by comparing it with some popular Gaussian approximation methods and importance sampling methods.