Discretization-Invariant MCMC Methods for PDE-constrained Bayesian Inverse Problems in Infinite Dimensional Parameter Spaces

In this paper we target at developing discretization-invariant, namely dimension-independent, Markov chain Monte Carlo (MCMC) methods to explore PDEconstrained Bayesian inverse problems in infinite dimensional parameter spaces. In particular, we present two frameworks to achieve this goal: Metropolize-then-discretize and discretize-then-Metropolize. The former refers to the method of first proposing a function-space MCMC method for the Bayesian inverse problem under consideration and then discretizing both of them. The latter, on the other hand, first discretizes the Bayesian inverse problem and then proposes MCMC methods for the resulting discretized posterior probability density. In general, these two frameworks do not commute, that is, the resulting finite dimensional MCMC algorithms are not identical. The discretization step of the former may not be trivial since it involves both numerical analysis and probability theory, while the latter may not be discretization-invariant using traditional approaches. This paper develops finite element (FEM) discretization schemes for both frameworks so that both commutativity and discretization-invariant are attained. In particular, it shows how to construct discretize-then-Metropolize approaches for both Metropolis-adjusted Langevin algorithm and hybrid Monte Carlo method that commute with their Metropolize-then-discretize counterparts. The key that enables this achievement is a proper FEM discretization of the prior, the likelihood, and the Bayes’ formula, together with a correct definition of quantities such as gradient and covariance matrix in discretized finite dimensional parameter spaces. Numerical results for oneand two-dimensional elliptic inverse problems with up to 17899 parameters are presented to support the proposed developments.