On the contraction rate of the posterior distribution for nonlinear PDE parameter identification

TL;DR

This paper analyzes the rate at which Bayesian posterior distributions for nonlinear PDE parameters converge, relaxing previous assumptions and demonstrating results on three inverse problems.

Contribution

It introduces a novel analysis that relaxes the requirement for the true parameter to be in the Gaussian process prior's RKHS, advancing understanding of contraction rates.

Findings

Established contraction rates for posterior and variational approximations

Relaxed the assumption on the true parameter's regularity

Applied theory to three nonlinear PDE inverse problems

Abstract

In this work, we investigate the estimation of a parameter $f$ in PDEs using Bayesian procedures, and focus on posterior distributions constructed using Gaussian process priors, and its variational approximation. We establish contraction rates for the posterior distribution and the variational approximation in the regime of low-regularity parameters. The main novelty of the study lies in relaxing the condition that the ground truth parameter must lie in the reproducing kernel Hilbert space of the Gaussian process prior, which is commonly imposed in existing studies on posterior contraction rate analysis [14,40,44]. The analysis relies on a delicate approximation argument that suitably balances various error sources. We illustrate the general theory on three nonlinear inverse problems for PDEs.