Blending Mathematical Models and Data: Algorithms, Analysis and Applications in the Geophysical and Medical Sciences
Many problems in the physical sciences lead to the inverse problem of determining an unknown field, which is input data for a mathematical model, from a finite set of indirect measurements of the physical system being modelled. 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 via medical imaging. We describe a theoretical and computational Bayesian framework relevant to the solution of inverse problems for fields 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. The theory underlying the subject of Bayesian inversion for fields will be developed and its implications for improved algorithms explained. Examples from oceanography, signal processing, groundwater flow and electrical impedance tomography will be used to illustrate the wide-ranging applicability of the methodology.