Bernstein - von Mises theorems for statistical inverse problems I: Schr\"odinger equation

The inverse problem of determining the unknown potential $f>0$ in the partial differential equation $$\frac{\mathrm{\Delta }}{2}u-fu=0\text{ on }\mathcal{O}\text{ s.t. }u=g\text{ on }\mathrm{\partial }\mathcal{O},$$ where $\mathcal{O}$ is a bounded $C^{\mathrm{\infty }}$-domain in ${\mathbb{R}}^d$ and $g>0$ is a given function prescribing boundary values, is considered. 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 $f$ in the small noise limit.