Uncertainty Quantification in Bayesian In- version

Probabilistic thinking is of growing importance in many areas of mathematics. This paper highlights the beautiful mathematical framework, coupled with practical algorithms, which results from thinking probabilistically about inverse problems arising in partial differential equations. Many inverse problems in the physical sciences require the determination of an unknown field from a finite set of indirect measurements. Examples include oceanography, oil recovery, water resource management and weather forecasting. In the Bayesian approach to these problems, the unknown and the data are modelled as a jointly varying random variable, typically linked through solution of a partial differential equation, and the solution of the inverse problem is the distribution of the unknown given the data. This approach provides a natural way to provide estimates of the unknown field, together with a quantification of the uncertainty associated with the estimate. It is hence a useful practical modelling tool. However it also provides a very elegant mathematical framework for inverse problems: whilst the classical approach to inverse problems leads to ill-posedness, the Bayesian approach leads to a natural well-posedness and stability theory. Furthermore this framework provides a way of deriving and developing algorithms which are well-suited to the formidable computational challenges which arise from the conjunction of approximations arising from the numerical analysis of partial differential equations, together with approximations of central limit theorem type arising from sampling of measures. Mathematics Subject Classification (2010). Primary 35R30; Secondary 62C10.