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

TL;DR

This paper establishes a Bayesian framework for the inverse problem of recovering the order and diffusion coefficient in fractional elliptic PDEs, proving well-posedness and mathematical foundations for Bayesian learning of fractional model parameters.

Contribution

It introduces a Bayesian formulation for fractional elliptic inverse problems and proves well-posedness, providing a rigorous foundation for Bayesian inference of fractional model parameters.

Findings

Posterior distribution is a change of measure from the prior.

Small data perturbations lead to small Hellinger distance changes in the posterior.

Provides mathematical foundation for Bayesian learning of fractional model inputs.

Abstract

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.