Professor Andrew Stuart (University of Warwick)
Description: (Joint work with Marco Iglesias (Nottingham), Kui Lin (Fudan) and Yulong Lu (Warwick))
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 discuss a theoretical and computational Bayesian framework relevant to the characterization of such inverse problems.
We start with models in which the geometry is defined via a finite number of parameters, utilizing descriptions which are sufficiently general to include layered media, faults and channels, and to allow for spatial variability within the different components of the field [1].
We then proceed to demonstrate how to use level set formulations of the Bayesian inverse problem, allowing for more complex geometric interfaces which are described by an infinite set of parameters [2].
For a groundwater flow inverse problem, in which hydraulic head measurements are used to condition the prior information on the permeability, we show that the Bayesian formulation gives rise to a well-posed posterior distribution, suitable for numerical interrogation. We then describe numerical experiments which explore the posterior distribution, using state-of-the art MCMC methods.
[1] M.A. Iglesias, K. Lin, A.M. Stuart, "Well-Posed Bayesian Geometric Inverse Problems Arising in Subsurface Flow", Inverse Problems, To Appear.
http://arxiv.org/abs/1401.5571
[2] M.A. Iglesias, Y. Lu, A.M. Stuart, "A level-set approach to Bayesian geometric inverse problems", In Preparation, 2014.