Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem

For O a bounded domain in R-d and a given smooth function g : O -> R, we consider the statistical nonlinear inverse problem of recovering the conductivity f > 0 in the divergence form equation. del center dot (f del u) = g on O, u = 0 on partial derivative O, from N discrete noisy point evaluations of the solution u = u(f) on 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(O)-distance.