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

### Richard Nickl

University of Cambridge, UK

## Abstract

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

where $O$ is a bounded $C_{∞}$-domain in $R_{d}$ 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.

## Cite this article

Richard Nickl, Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation. J. Eur. Math. Soc. 22 (2020), no. 8, pp. 2697–2750

DOI 10.4171/JEMS/975