OP: Collaborative Research: Novel Feature-Based, Randomized Methods for Large-Scale Inversion

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The desire to form an image of a region of space from externally collected data arises in applications ranging from detecting and characterizing cancers in the body, to quantifying the distribution of water, oil, or subsurface pollutants, and to the timely accurate identification of explosives in crowded venues. The physics associated with signal propagation and sensing in these problems creates substantial computational challenges for transforming raw data into useful information. The research team in this project aims to develop computational methods that greatly reduce the cost of real time imaging by providing improvements in statistical inverse theory, numerical inversion methods, simulation models, and hybrid imaging models. The main thrusts of the project will be tested on imaging applications in medical tomography, environmental remediation, and airport security imaging. The techniques form the basis for addressing analogous problems associated with inversion of optical signals across a wide range of spatial and temporal scales. As part of the project, a modular course will be developed to teach these new methods at the graduate level. The course materials will be made available over the internet.The large-scale imaging, or inverse, problems addressed by this collaborative team require the minimization of a parameter-dependent function that expresses the misfit of predicted measurements for a candidate image and actual measurement data. The potentially large number of parameters must be minimized over an ever-increasing huge number of measurements, while concurrently some unknown set of the data may be redundant. Detailed images, however, are not always needed for addressing relevant, practical questions and decision making. A combination of computational techniques will be developed to make large-scale parameter-dependent minimization computationally feasible. Furthermore, novel efficient approaches for inferring critical image features will be developed, obviating need for complete reconstruction of an image. The research builds on recent methods that exploit randomization to compute accurate estimates of solutions at greatly reduced computational cost, and on the efficient construction of smaller, approximate, reduced order numerical models that are accurate for relevant sets of parameters, and thus reduce the cost of full simulation of the sensing physics. Probabilistic approaches for inference of critical image features that guide image interpretation and decision making will be developed. The mathematics associated with this approach requires these methods to capitalize on other new tools also under development in this project.