Classification of Methods for Bayesian Inverse Problems Governed by Partial Differential Equations

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Inverse problems emerge in all areas of science, engineering, technology, and medicine. They provide a systematic and rigorous way to extract knowledge and insight from observational data. When this data corresponds to observations of natural or engineered systems that can be described by mathematical models, the properties and structure of the inverse problem depend on the properties of these models, which commonly involve partial differential equations (PDEs). It is crucial that efficient inverse problem solution methods exploit these properties. This is in particular the case when the inversion parameters are high (or infinite) dimensional, when the mathematical models are given by PDEs, and when one is interested in quantifying the uncertainty in the parameters, as is important in many applications.This project will systematically study properties and develop algorithms for three inverse problems that are representative of a wide class of Bayesian inverse problems governed by PDEs: (1) a parabolic inverse problem with spatially (and temporally) well-separated parameter and observation locations, (2) an elliptic Stokes flow problem for which a rich set of measurement data are available and the locations corresponding to parameters and observations are not well-separated, and (3) a hyperbolic problem with sparse point measurements. The PI will study these problems theoretically, develop and classify structure-exploiting methods to approximate their solutions, and implement these methods in an open-source software library. All three prototype problems have important and societally relevant real-world, large-scale analogues. Thus, any algorithmic or theoretical findings obtained for the three model problems will have immediate benefit for these grand challenge inverse problems.