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Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem

Published version
Peer-reviewed

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Abstract

For O a bounded domain in Rd and a given smooth function g:OR, we consider the statistical nonlinear inverse problem of recovering the conductivity f>0 in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from N discrete noisy point evaluations of the solution u=uf on O. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number N of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate Nλ,λ>0, for the reconstruction error of the associated posterior means, in L2(O)-distance.

Description

Keywords

inverse problems, Bayesian inference, Gaussian prior, frequentist consistency

Journal Title

Inverse Problems

Conference Name

Journal ISSN

0266-5611
1361-6420

Volume Title

36

Publisher

IOP Publishing
Sponsorship
European Research Council (647812)