# Special Colloquium: Bayesian Inversion for Functions and Geometry

# Special Colloquium: Bayesian Inversion for Functions and Geometry

Please note special day (Thursday). This is joint with the Analysis of Fluids seminar. Many problems in the physical sciences require the determination of an unknown function from a finite set of indirect measurements. Examples include oceanography, medical imaging, oil recovery, water resource management and weather forecasting. Furthermore there are numerous inverse problems where geometric characteristics, such as interfaces, are key unknown features of the overall inversion. Applications include the determination of layers and faults within subsurface formations, and the detection of unhealthy tissue in medical imaging. We describe a theoretical and computational Bayesian framework relevant to the solution of inverse problems for functions, and for geometric features. We formulate Bayes' theorem on separable Banach spaces, a conceptual approach which leads to a form of probabilistic well-posedness and also to new and efficient MCMC algorithms which exhibit order of magnitude speed-up over standard methodologies. Furthermore the approach can be developed to apply to geometric inverse problems, where the geometry is parameterized finite-dimensionally and, via the level-set method, to infinite-dimensional parameterizations. In the latter case this leads to a well-posedness that is difficult to achieve in classical level-set inversion, but which follows naturally in the probabilistic setting.[1] A.M. Stuart. Inverse problems: a Bayesian perspective. Acta Numerica 19(2010) 451--559. http://homepages.warwick.ac.uk/~masdr/BOOKCHAPTERS/stuart15c.pdf

[2] M. Dashti and A.M. Stuart. The Bayesian approach to inverse problems.To appear in Handbook of Uncertainty Quantification, Springer, 2016. http://arxiv.org/abs/1302.6989

[3] S.L.Cotter, G.O.Roberts, A.M. Stuart and D. White, MCMC methods for functions: modifying old algorithms to make them faster. Statistical Science, 28 (2013) 424-446. http://homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart103.pdf

[4] M.A. Iglesias, K. Lin, A.M. Stuart, "Well-Posed Bayesian Geometric Inverse Problems Arising in Subsurface Flow", Inverse Problems, 30 (2014) 114001. http://arxiv.org/abs/1401.5571

[5] M.A. Iglesias, Y. Lu, A.M. Stuart, "A level-set approach to Bayesian geometric inverse problems", submitted. http://arxiv.org/abs/1504.00313