Variational Problems: From Foundational Questions to Direct Applications in Image Processing and Materials Science

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Variational problems permeate diverse scientific disciplines; from pure mathematics to applied topics in image processing and materials science. This proposal addresses four (classes of) variational problems and can be separated into two parts. The first part is more foundational from the mathematical point of view and involves two classical, non-local and non-convex variational problems: optimal quantization (or optimal centrodial Voronoi tessellations) and Gamow's Liquid Drop Model (and its anisotropic variants). For both problems, we will focus on two central goals: (a) to what extent can one rigorously characterize ground states (global minimizers); (b) can one develop numerical algorithms to probe the energy landscape and find low energy states. The second part of the proposal is highly application driven, and pertains to addressing and developing specific novel variational problems in two very different fields: (i) deblurring and denoising in image processing and (ii) grain growth in polycrystalline materials. For image processing, we will develop on a recent and remarkably successful Bayesian variational method, introduced by the applicant et al, which is based upon the idea of maximum entropy on the mean via the probability distribution of the image. For grain growth, we address phase field crystal models and their gradient flow dynamics. Our focus lies on: the understanding of universal statistical distributions for geometric metrics during grain growth; and addressing various extensions and reductions of phase field crystal models with an emphasis on the connection between mathematical and desirable physical properties. Throughout this proposal, we will employ techniques (new and old) which span tools from the modern calculus of variations, numerical analysis, convex optimization, and mathematical statistics.