Sparse MCMC gpc Finite Element Methods for Bayesian Inverse Problems

Abstract

Several classes of MCMC methods for the numerical solution of Bayesian Inverse Problems for partial differential equations (PDEs) with unknown random field coefficients are considered. A general framework for their numerical analysis is presented. The complexity of MCMC sampling for the unknown fields from the posterior density, as well as the convergence of the discretization error of the PDE of interest in the forward response map, is analyzed. Particular attention is given to bounds on the overall work required by the MCMC algorithms for achieving a prescribed error level E. We show that the computational complexity of straightforward combinations of MCMC sampling strategies with standard PDE solution methods is generally excessive. Two computational strategies for substantially reducing the computational complexity of MCMC methods for Bayesian inverse problems prising in PDEs are studied: a parametric, deterministic gpc-type (generalized polynomial chaos) representation of the forward solution map of the PDE with uncertain coefficients, which has been proposed and implemented in the engineering literature (e.g. [17, 15, 16]); and a new Multi-Level Monte Carlo sampling strategy of the Markov Chain (MLMCMC) with sampling from a multilevel discretization of the posterior and a multilevel discretization of the forward PDE. We compare the computational complexity of these gpc-MCMC and MLMCMC methods to that of the plain MCMC method, and provide sufficient conditions on the regularity of the unknown coefficient for both, the gpc-MCMC and MLMCMC method, to afford substantial complexity reductions over the plain MCMC approach.