On statistical Calderón problems

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 $γ$ determining solutions $u=u_{γ, f}$ of the partial differential equation \begin{equation*} \begin{split} \nabla \cdot(γ\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 $f \mapsto Λ_γ(f) = {γ\frac{\partial u_{γ,f}}{\partial ν}}\Big|_{\partial D},$ with $\partial/\partial ν$ denoting the outward normal derivative, is considered. The data $Y$ consists of $Λ_γ$ corrupted by additive Gaussian noise at noise level $\varepsilon>0$, and a statistical algorithm $\hat γ(Y)$ is constructed which is shown to recover $γ$ in supremum-norm loss at a statistical convergence rate of the order $\log(1/\varepsilon)^{-δ}$ as $\varepsilon \to 0$. It is further shown that this convergence rate is optimal, up to the precise value of the exponent $δ>0$, in an information theoretic sense. The estimator $\hat γ(Y)$ has a Bayesian interpretation in terms of the posterior mean of a suitable Gaussian process prior and can be computed by MCMC methods.

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