  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.
