  For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth
function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear
inverse problem of recovering the conductivity $f>0$ in the divergence form
equation $$
  \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad
  u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point
evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical
performance of Bayesian nonparametric procedures based on a flexible class of
Gaussian (or hierarchical Gaussian) process priors, whose implementation is
feasible by MCMC methods. We show that, as the number $N$ of measurements
increases, the resulting posterior distributions concentrate around the true
parameter generating the data, and derive a convergence rate $N^{-\lambda},
\lambda>0,$ for the reconstruction error of the associated posterior means, in
$L^2(\mathcal{O})$-distance.
