Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem
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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 on O, u=0 on ∂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.
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