Inverse problems: A Bayesian perspective

TLDR

Inverse problems in differential equations are practically important and have spurred mathematical and computational advances, but they are ill‑posed and usually require regularization. This article reviews the Bayesian approach to regularization of inverse problems, developing a function‑space viewpoint. The authors adopt a Bayesian framework that treats regularization as a prior over function spaces, enabling probabilistic inference of solutions. The Bayesian approach fully characterizes all possible solutions and their probabilities, addresses modeling issues clearly, quantifies uncertainty and risk, is computationally feasible in many applications, and provides insight into simpler methods.

Abstract

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.