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

TL;DR

This work develops posterior contraction theory for Bayesian PDE parameter identification with Gaussian-process priors, explicitly allowing true coefficients that lie outside the prior RKHS by employing a rescaled prior and a careful approximation via Karhunen–Loève truncation. The authors establish contraction rates for both the exact posterior and variational Bayes under forward-map regularity conditions, and they extend the framework to flexible priors using sieve arguments. The theory is validated on three nonlinear PDE inverse problems, including diffusion- coefficient and potential identification in elliptic and subdiffusion settings, showing convergence even for low-regularity ground truths. The results advance uncertainty quantification for nonlinear PDEs by broadening the set of truths compatible with Bayesian contraction theory and by linking performance to computable variational approximations.

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.

Theorems & Definitions (30)

• Definition 3.1
• Remark 1
• Theorem 3.1
• Theorem 3.2
• Corollary 3.3: Truncated Gaussian priors
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5