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 an elliptic divergence form partial differential equation with Dirichlet boundary conditions, 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, algebraic in N, for the reconstruction error of the associated posterior means, in L^2(O)-distance.