Bernstein-von Mises theorems for statistical inverse problems I: Schrödinger equation
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Authors
Nickl, R
Publication Date
2020Journal Title
Journal of the European Mathematical Society
ISSN
1435-9855
Publisher
European Mathematical Society
Volume
22
Issue
8
Pages
2697-2750
Type
Article
Metadata
Show full item recordCitation
Nickl, R. (2020). Bernstein-von Mises theorems for statistical inverse problems I: Schrödinger equation. Journal of the European Mathematical Society, 22 (8), 2697-2750. https://doi.org/10.4171/JEMS/975
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.
Keywords
Bayesian nonlinear inverse problems, elliptic partial differential equations, inverse scattering problem, asymptotics of nonparametric Bayes procedures
Sponsorship
European Research Council (647812)
Identifiers
External DOI: https://doi.org/10.4171/JEMS/975
This record's URL: https://www.repository.cam.ac.uk/handle/1810/279802
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