We consider the inverse problem of determining the potential in the partial differential equation
where is a bounded -domain in and 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 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 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. 2697–2750DOI 10.4171/JEMS/975