Emulation of Higher-Order Tensors in Manifold Monte Carlo Methods for Bayesian Inverse Problems

TL;DR

This paper introduces a Gaussian Process emulator to efficiently approximate geometric quantities in manifold Monte Carlo methods, significantly reducing computational costs for Bayesian inverse problems.

Contribution

It proposes a novel approach to emulate geometric objects in manifold MCMC using Gaussian Processes, enabling more feasible computations in high-dimensional Bayesian inverse problems.

Findings

Emulator reduces computational load in geometric MCMC methods.

Online experiment design improves emulator accuracy without affecting convergence.

Demonstrated significant efficiency gains in practical examples.

Abstract

The Bayesian approach to Inverse Problems relies predominantly on Markov Chain Monte Carlo methods for posterior inference. The typical nonlinear concentration of posterior measure observed in many such Inverse Problems presents severe challenges to existing simulation based inference methods. Motivated by these challenges the exploitation of local geometric information in the form of covariant gradients, metric tensors, Levi-Civita connections, and local geodesic flows, have been introduced to more effectively locally explore the configuration space of the posterior measure. However, obtaining such geometric quantities usually requires extensive computational effort and despite their effectiveness affect the applicability of these geometrically-based Monte Carlo methods. In this paper we explore one way to address this issue by the construction of an emulator of the model from which all geometric objects can be obtained in a much more computationally feasible manner. The main concept is to approximate the geometric quantities using a Gaussian Process emulator which is conditioned on a carefully chosen design set of configuration points, which also determines the quality of the emulator. To this end we propose the use of statistical experiment design methods to refine a potentially arbitrarily initialized design online without destroying the convergence of the resulting Markov chain to the desired invariant measure. The practical examples considered in this paper provide a demonstration of the significant improvement possible in terms of computational loading suggesting this is a promising avenue of further development.