Bayesian Inverse Problems

Abstract

Many problems in science and engineering consist of estimating some unknown parameters of a physical system from limited and noisy observations. In practical conditions, these problems are often unstable; hence, they should be treated in an appropriate framework in the theory of inverse problems. The regularisation framework for inverse problems is concerned with the stabilisation of unstable problems. In this framework, the unknown is taken as a deterministic multi-dimensional quantity and the observation error is taken to be unknown without any probabilistic structure. In the Bayesian framework for inverse problems, which is the focus of this talk, the unknown and observation error are explicitly taken and modelled as jointly distributed random variables. One of the advantages of the Bayesian framework is that it can quantify the uncertainty of a point estimate for the unknown, unlike the deterministic framework. This framework can be used for many interesting inverse problems. A couple of them will be discussed in this short talk.