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

Abstract

For a bounded domain in d and a given smooth function e consider the statistical nonlinear inverse problem of recovering the conductivity f > 0 in the divergence form equation (fu) = g on , u = 0 on ; from N discrete noisy point evaluations of the solution u = u f on . 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 -, > 0, for the reconstruction error of the associated posterior means, in L2()-distance.