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

Abstract

© European Mathematical Society 2020 We consider the inverse problem of determining the potential f > 0 in the partial differential equation 1 2 u − fu = 0 on O, u = g on ∂O, where O is a bounded C∞-domain in Rd 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.