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

Abstract

Abstract 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 ∇ ⋅ ( f ∇ u ) = g o n O , u = 0 o n ∂ 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 −λ , λ > 0, for the reconstruction error of the associated posterior means, in L 2 ( O ) -distance.