Gaussian surrogate models for Bayesian inverse problems

Abstract

Solving Bayesian inverse problems using Markov Chain Monte Carlo (MCMC) methods poses significant computational challenges due to the extensive numerical simulations required for each sample. To address this issue, surrogate models are often employed to approximate the complex models, thereby reducing computational costs.

This thesis focuses on the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations, particularly in scenarios when only limited training data are available. To enhance the accuracy and robustness of prediction without requiring additional observational data, we investigate the physics-informed Gaussian process regression (PI-GPR) method which provides a flexible framework for integrating physical information into the Gaussian process, and extend the method to construct different approximate posteriors for solving the Bayesian inverse problems. Benefiting from the nature of Gaussian process regression as a statistical model, the error of approximation can be quantified and integrated into the approximation of posteriors. Meanwhile, the gradient of the approximate posteriors based on Gaussian surrogate models can be analytically computed, enabling the use of gradient-based MCMC samplers like the Metropolis-adjusted Langevin algorithm (MALA) for efficient sampling. Finally, the approximate posterior can be used in the delayed-acceptance Metropolis-Hastings sampling algorithm, which helps reject unlikely candidates with a much lower cost and hence significantly reduces the overall computational cost.