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

2017/07/06 by Nickl, Richard · 5 citations
#35J10 #62G20 #65N21 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1707.01764

Abstract

The inverse problem of determining the unknown potential f>0 in the partial differential equation \fracΔ2 u - fu =0 on \mathcal O ~~s.t. u = g \text on ∂ \mathcal O, where \mathcal O is a bounded C^∞-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.

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