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

For O a bounded domain in Rd 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 ∇⋅(f∇u)=gonO,u=0on∂O, from N discrete noisy point evaluations of the solution u = uf 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−λ, λ > 0, for the reconstruction error of the associated posterior means, in L2(O)-distance.