The Bayesian Approach to Inverse Problems

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• Professor Andrew Stuart, University of Warwick Mathematics Institute
• Wednesday 23 April 2014, 14:00-18:00
• MR5.

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Short Course

Probabilistic thinking is of growing importance in many areas of mathematics. This short course will demonstrate 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, and the solution of the inverse problem is the distribution of the un- known 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.

The tools in mathematical analysis which you will be exposed to during the course lie at the intersection of probability and analysis, and include Gaussian measures on Hilbert space, metrics on probability measures, conditional probability and regularity estimates for elliptic PDEs and for the Navier-Stokes equation.

This talk is part of the Cambridge Centre for Analysis talks series.

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