On statistical Calderón problems (Q778889)

Summary: For \(D\) a bounded domain in \(\mathbb{R}^d\), \(d \geq 2\), 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 \[ \begin{aligned} \nabla \cdot(\gamma \nabla u) &=0 \quad \text{in } D, \\ u&=f \quad \text{on } \partial D, \end{aligned} \] 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 \(\epsilon>0\), 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.