The Bayesian formulation and well-posedness of fractional elliptic inverse problems. (24th May 2017)

• Record Type: Journal Article
• Authors: Trillos, Nicolás García; Sanz-Alonso, Daniel
• Is Part Of: Inverse problems. Volume 33:Number 6(2017:Jun.)
• Publication Date: 2017-05-24
• Subjects: Bayesian inverse problems -- extension problem -- fractional partial differential equations; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aa711e
• Languages: English
• ISSNs: 0266-5611

Abstract

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.