Semi-parametric Bernstein-von Mises Theorem in a Parabolic PDE Problem

TL;DR

This paper establishes a Bayesian Bernstein-von Mises theorem for estimating diffusivity in a heat equation with unknown absorption, demonstrating asymptotic normality of the posterior under certain conditions.

Contribution

It provides the first semi-parametric Bernstein-von Mises result for a PDE inverse problem with a Gaussian process prior on the absorption function.

Findings

Posterior distribution of diffusivity converges to a normal distribution.

Conditions on prior and smoothness are specified for the theorem.

The approach enables uncertainty quantification in PDE inverse problems.

Abstract

We consider the heat equation with absorption in a bounded domain of $\mathbb{R}^d$, where both the scalar diffusivity and the absorption function are unknown. We investigate a Bayesian approach for recovering the diffusivity from a noisy observation of the solution to the PDE over the domain. Given a Gaussian process prior on the absorption function, we derive a Bernstein-von Mises theorem for the marginal posterior distribution of the diffusivity under assumptions on the prior and on smoothness properties of the absorption.