On statistical Calder\'on problems

TL;DR

This paper develops a statistical method for recovering the unknown electrical conductivity inside a domain from noisy boundary measurements, achieving near-optimal convergence rates and providing a Bayesian interpretation for practical computation.

Contribution

It introduces a Bayesian-based estimator for the Calderón problem with Gaussian noise, establishing optimal convergence rates and practical MCMC computation methods.

Findings

Achieves logarithmic convergence rate in noise level

Proves the optimality of the convergence rate

Provides a Bayesian interpretation and MCMC implementation

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 \begin{equation*} \begin{split} \nabla \cdot(\gamma \nabla u)&=0 \quad \text{ in }D, \\ u&=f \quad \text { on } \partial D, \end{split} \end{equation*} from noisy observations $Y$ of the Dirichlet-to-Neumann map \[f \mapsto \Lambda_\gamma(f) = {\gamma \frac{\partial u_{\gamma,f}}{\partial \nu}}\Big|_{\partial D},\] 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 $\varepsilon>0$, 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 $\log(1/\varepsilon)^{-\delta}$ as $\varepsilon \to 0$. 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.