FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems

Abstract

We present a systematic construction of FEM-based dimension-independent (discretization-invariant) Markov chain Monte Carlo (MCMC) approaches to explore PDE-constrained Bayesian inverse problems in infinite dimensional parameter spaces. In particular, we consider two frameworks to achieve this goal: Metropolize-then-discretize and discretize-then-Metropolize. The former refers to the method of discretizing function-space MCMC methods. 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, perhaps ``easier'', may not be discretization-invariant using traditional approaches. This paper constructively develops finite element (FEM) discretization schemes for both frameworks and shows that both commutativity and discretization-invariance are attained. In particular, it shows how to construct discretize-then-Metropolize approaches for both Metropolis-adjusted Langevin algorithm and the 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 the gradient and the covariance matrix in discretized finite dimensional parameter spaces. The implication is that practitioners can take advantage of the developments in this paper to straightforwardly construct discretization-invariant discretize-then-Metropolize MCMC for large-scale inverse problems. Numerical results for one- and two-dimensional elliptic inverse problems with up to $17899$ parameters are presented to support the proposed developments.