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

IOP Publishing https://doi.org/10.1088/1361-6420/ab7d2a Inverse Problems 2020-08-20T10:29:21Z 2020-08-20T10:29:21Z 2020 2019-10-16 2020-08-20T10:29:20Z

For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, 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 $u=u_f$ on $\mathcal O$. 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 $N^{-\lambda}, \lambda>0,$ for the reconstruction error of the associated posterior means, in $L^2(\mathcal{O})$-distance.

inverse problems; Bayesian inference; Gaussian prior; frequentist consistency; Article

Open access content. Open access content
Attribution 4.0 International (CC BY 4.0)
https://creativecommons.org/licenses/by/4.0

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HS1 oai:www.repository.cam.ac.uk:1810/309430
10.17863/CAM.56519
1488016792

UNIV OF CAMBRIDGE
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