The Bayesian Formulation and Well-Posedness of Fractional Elliptic Inverse Problems

Understand

We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution.

• We work in a Bayesian framework and show conditions under which the posterior distribution is given by a change of measure from the prior.
• Moreover, we show well-posedness of the inverse problem, in the sense that small perturbations of the observed solution lead to small Hellinger perturbations of the associated posterior measures.
• We thus provide a mathematical foundation to the Bayesian learning of the order ---and other inputs--- of fractional models.