Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems

摘要

This paper develops meshless methods for probabilistically describingdiscretisation error in the numerical solution of partial differentialequations. This construction enables the solution of Bayesian inverse problemswhile accounting for the impact of the discretisation of the forward problem.In particular, this drives statistical inferences to be more conservative inthe presence of significant solver error. Theoretical results are presenteddescribing rates of convergence for the posteriors in both the forward andinverse problems. This method is tested on a challenging inverse problem with anonlinear forward model.