The Bayesian Approach To Inverse Problems

Many 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. The Bayesian approach to these problems is natural for many reasons, including the under-determined and ill-posed nature of the inversion, the noise in the data and the uncertainty in the differential equation models used to describe complex mutiscale physics. In this talk I will describe the advantages of formulating Bayesian inversion on function space in order to solve these problems. I will overview theoretical results concerning well-posedness of the posterior distribution, approximation theorems for the posterior distribution, and specially constructed MCMC methods to explore the posterior distribution. Special attention will be paid to various prior (regularization) strategies, including Gaussian random fields, and various geometric parameterizations such as the level set approach to piecewise constant reconstruction.

[1] M. Dashti, A.M. Stuart, "The Bayesian Approach To Inverse Problems". To appear in The Handbook of Uncertainty Quantification, Springer, 2016. arxiv.org/abs/1302.6989
[2] 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. homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart103.pdf
[3] M.A. Iglesias, K. Lin, A.M. Stuart, "Well-Posed Bayesian Geometric Inverse Problems Arising in Subsurface Flow", Inverse Problems, 30 (2014) 114001. arxiv.org/abs/1401.5571
[4] M.A. Iglesias, Y. Lu, A.M. Stuart, "A level-set approach to Bayesian geometric inverse problems", In Preparation, 2014.