Bayesian approach to an elliptic inverse problem

Abstract: We consider the inverse problem of finding the diffusion coefficient of a linear uniformly elliptic partial differential equation in divergence form, from noisy measurements of the forward solution in the interior. We adopt a Bayesian approach to the problem. We consider the prior measure on the diffusion coefficient to be either a Besov or Gaussian measure. We show that if the functions drawn from the prior are regular enough, the posterior measure is well-defined and Lipschitz continuous with respect to the data in the Hellinger metric. We also quantify the errors incurred by approximating the posterior measure in a finite dimensional space. This is joint work with Stephen Harris and Andrew Stuart.