Solving Large-Scale PDE-constrained Bayesian Inverse Problems with Riemann Manifold Hamiltonian Monte Carlo

TL;DR

This paper introduces an efficient Riemann manifold Hamiltonian Monte Carlo method tailored for large-scale PDE-constrained Bayesian inverse problems, leveraging geometric structure and low-rank approximations for improved sampling efficiency.

Contribution

It develops a novel RMHMC approach that exploits PDE geometry and uses low-rank Fisher information matrix approximations to enhance sampling efficiency in high-dimensional inverse problems.

Findings

RMHMC provides nearly independent posterior samples, increasing statistical efficiency.

Low-rank approximation reduces computational cost of Fisher matrix formation.

The method requires only two PDE solves per Hessian-vector product, improving scalability.

Abstract

We consider the Riemann manifold Hamiltonian Monte Carlo (RMHMC) method for solving statistical inverse problems governed by partial differential equations (PDEs). The power of the RMHMC method is that it exploits the geometric structure induced by the PDE constraints of the underlying inverse problem. Consequently, each RMHMC posterior sample is almost independent from the others providing statistically efficient Markov chain simulation. We reduce the cost of forming the Fisher information matrix by using a low rank approximation via a randomized singular value decomposition technique. This is efficient since a small number of Hessian-vector products are required. The Hessian-vector product in turn requires only two extra PDE solves using the adjoint technique. The results suggest RMHMC as a highly efficient simulation scheme for sampling from PDE induced posterior measures.