# Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems

## TL;DR

This paper introduces meshless probabilistic methods to quantify discretisation errors in PDE solutions, improving Bayesian inverse problem inference by accounting for solver uncertainty and demonstrating convergence and practical effectiveness.

## Contribution

It presents a novel meshless probabilistic framework for PDE discretisation error, enabling more robust Bayesian inverse inference with theoretical convergence guarantees.

## Key findings

- Convergence rates for posterior distributions are established.
- Method effectively handles nonlinear forward models.
- Provides more conservative statistical inferences under solver error.

## Abstract

This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem with a nonlinear forward model.