Functional Analysis and Computational Methods in Imaging, Materials, and Atmospheric Sciences

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The investigators, their students and collaborators study mathematical formulations and efficient computational techniques for applications arising in image analysis, materials science, and atmospheric and climate modeling. This multidisciplinary research combines areas of computational mathematics, inverse problems, image analysis, interfaces and free boundaries, and atmospheric sciences. They study image restoration using cartoon and texture decompositions, restoration of images in the presence of a stochastic point spread function, implicit open curve evolution and applications to free boundary problems in materials sciences, and variational data fusion of atmospheric records acquired by multiple instruments. The research team develops novel variational approaches, iterative and numerical analysis techniques for solving these inverse problems.The proposed activity provides the link between efficient mathematical formulations, imaging approaches, and applications in climate and materials sciences, where similar approaches have not yet been attempted. In particular, capability for merging data acquired by multiple sensors is a key part to our understanding of the Earth's climate system, and therefore, is of importance when making projections about climate change and climate impacts. Current atmospheric data fusion methods are largely ad hoc and establishing a firm mathematical foundation and computational methods for combining important records enhances their scientific credibility and further a wide range of scientific analyses. The investigators promote multidisciplinary teaching, training and learning. Mathematics students are exposed to a broad range of topics and techniques: (i) in applied and computational mathematics, image processing and analysis; and (ii) topics outside mathematics, including materials and atmospheric sciences.