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

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Abstract

We present a systematic construction of FEM-baseddimension-independent (discretization-invariant) Markov chain MonteCarlo (MCMC) approaches to explore PDE-constrained Bayesian inverseproblems in infinite dimensional parameter spaces. In particular, weconsider two frameworks to achieve this goal:Metropolize-then-discretize and discretize-then-Metropolize. Theformer refers to the method of discretizing function-space MCMCmethods. The latter, on the other hand, first discretizes the Bayesianinverse problem and then proposes MCMC methods for the resultingdiscretized posterior probability density. In general, these twoframeworks do not commute, that is, the resulting finite dimensionalMCMC algorithms are not identical. The discretization step of theformer may not be trivial since it involves both numerical analysisand probability theory, while the latter, perhaps ``easier'', may notbe discretization-invariant using traditional approaches. This paperconstructively develops finite element (FEM) discretization schemesfor both frameworks and shows that both commutativity anddiscretization-invariance are attained. In particular, it shows how toconstruct discretize-then-Metropolize approaches for bothMetropolis-adjusted Langevin algorithm and the hybrid Monte Carlo methodthat commute with their Metropolize-then-discretize counterparts. Thekey that enables this achievement is a proper FEM discretization ofthe prior, the likelihood, and the Bayes' formula, together with acorrect definition of quantities such as the gradient and the covariancematrix in discretized finite dimensional parameter spaces. Theimplication is that practitioners can take advantage of the developmentsin this paper to straightforwardly construct discretization-invariantdiscretize-then-Metropolize MCMC for large-scale inverseproblems. Numerical results for one- and two-dimensional ellipticinverse problems with up to $17899$ parameters are presented tosupport the proposed developments.

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