Variational Methods for Materials Science and Mathematical Imaging

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The mathematical theories and techniques developed in this project provide a foundation for understanding aspects of imaging and of the properties of materials. The use of self-assembly processes to manufacture modern semiconductor nanostructures, quantum wires, and quantum dots is of pivotal importance in microelectric and optoelectronic technologies, such as reflective or anti-reflective coatings for optics, the fabrication of layers of insulators and semiconductors for integrated circuits, quantum well lasers, and the processing of nanoscale materials. We address the variational study of relevant observed phenomena in nanowires and in the epitaxial deposition of a thin film onto a substrate. We study phase nucleation in Lithium-Ion batteries, which are central to advances in portable electronic devices, electric vehicles, and renewable energy storage; phase nucleation is important in understanding charge-discharge dynamics (poor cycle life) and other material limitations of these batteries. In what concerns the mathematics of imaging, we pursue the analytical investigation of image processing, restoration, and registration, which are fundamental to the advance of computer vision, medical imaging, film restoration, and scanning probe microscopy. These projects offer opportunities for integrating research in applied analysis with the education of advanced graduate students at the interface between mathematics and the physical sciences and engineering. Graduate students participate in the research of the project.What unifies these topics is that underlying energies involve higher order derivatives in spaces with discontinuous admissible fields, multiple scales interact, bulk and surface energies compete, and degeneracy of usually expected properties prevails. Together, these difficulties prevent the use of well-understood mathematical theories, and require new ideas and the introduction of innovative mathematical tools. Contemporary methods in the calculus of variations and nonlinear partial differential equations are used to study quasi-static (elliptic) and evolution (parabolic) systems of equations in a range of problems arising in materials science that span epitaxy, batteries, nanowires, and phase transitions. These methods are combined in novel ways with multi-level (machine learning) training schemes to study models for edge detection, image segmentation, signal denoising and detexturing, joint image segmentation, and image registration. Graduate students participate in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.