Simultaneous estimation of piecewise constant coefficients in elliptic PDEs via Bayesian level-set methods

In this article, we propose a non-parametric Bayesian level-set method for simultaneous reconstruction of two different piecewise constant coefficients in an elliptic partial differential equation. We show that the Bayesian formulation of the corresponding inverse problem is well-posed and that the posterior measure as a solution to the inverse problem satisfies a Lipschitz estimate with respect to the measured data in terms of Hellinger distance. We reduce the problem to a shape-reconstruction problem and use level-set priors for the parameters of interest. We demonstrate the efficacy of the proposed method using numerical simulations by performing reconstructions of the orig-inal phantom using two reconstruction methods. Posing the inverse problem in a Bayesian paradigm allows us to do statistical inference for the parameters of interest, whereby we are able to quantify the uncertainty in the reconstructions for both methods. This illustrates a key advantage of Bayesian methods over traditional algorithms.