Inverse problems: A Bayesian perspective

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. We demonstrate that, when formulated in a Bayesian fashion, a wide range of inverse problems share a common mathematical framework, and we high- light a theory of well-posedness which stems from this. The well-posedness theory provides the basis for a number of stability and approximation results which we describe. We also review a range of algorithmic approaches which are used when adopting the Bayesian approach to inverse problems. These include MCMC methods, filtering and the variational approach