Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation

Abstract

We consider the inverse problem of determining the potential $f>0$ in the partial differential equation

where $\mathcal{O}$ is a bounded $C^{\infty }$-domain in ${\mathbb{R}}^d$ and $g>0$ is a given function prescribing boundary values. The data consist of the solution u corrupted by additive Gaussian noise. A nonparametric Bayesian prior for the function $f$ is devised and a Bernstein–von Mises theorem is proved which entails that the posterior distribution given the observations is approximated in a suitable function space by an infinite-dimensional Gaussian measure that has a ‘minimal’ covariance structure in an information-theoretic sense. As a consequence the posterior distribution performs valid and optimal frequentist statistical inference on various aspects of $f$ in the small noise limit.