On statistical Calderón problems

Abstract

For $D$ a bounded domain in ${\mathbb{R}}^d,d\ge 2,$ with smooth boundary $\partial D$, the non-linear inverse problem of recovering the unknown conductivity $\gamma$ determining solutions $u=u_{\gamma ,f}$ of the partial differential equation

from noisy observations $Y$ of the Dirichlet-to-Neumann map $f\mapsto {\Lambda }_{\gamma }(f)=\gamma \frac{\partial u_{\gamma ,f}}{\partial \nu }{|}_{\partial D}$, with $\partial /\partial \nu$ denoting the outward normal derivative, is considered. The data $Y$ consists of ${\Lambda }_{\gamma }$ corrupted by additive Gaussian noise at noise level $\epsilon >0$, and a statistical algorithm $\hat{\gamma }(Y)$ is constructed which is shown to recover $\gamma$ in supremum-norm loss at a statistical convergence rate of the order $\log (1/\epsilon {)}^{-\delta }$ as $\epsilon \to 0$. It is further shown that this convergence rate is optimal, up to the precise value of the exponent $\delta >0$, in an information theoretic sense. The estimator $\hat{\gamma }(Y)$ has a Bayesian interpretation in terms of the posterior mean of a suitable Gaussian process prior and can be computed by MCMC methods.