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

变分问题渗透到不同的科学学科；从纯数学到图像处理和材料科学的应用主题。这项建议涉及四个(类)变分问题，可分为两部分。第一部分从数学的角度更具基础性，涉及两个经典的、非局部和非凸变分问题：最优量化(或最优中心Voronoi网格)和Gamow液滴模型(及其各向异性变种)。对于这两个问题，我们将集中在两个中心目标上：(A)人们可以在多大程度上严格地表征基态(全局极小化)；(B)人们能否开发数值算法来探索能量景观并找到低能态。提案的第二部分是高度应用驱动的，涉及在两个非常不同的领域中解决和开发特定的新型变分问题：(I)图像处理中的去模糊和去噪以及(Ii)多晶材料中的颗粒生长。对于图像处理，我们将开发由申请人等人介绍的最近非常成功的贝叶斯变分方法，该方法基于通过图像的概率分布的平均最大熵的思想。对于晶体生长，我们讨论了相场晶体模型及其梯度流动动力学。我们的重点在于：理解晶体生长过程中几何度量的普遍统计分布；以及处理相场晶体模型的各种扩展和简化，强调数学和所需物理性质之间的联系。在整个提案中，我们将使用技术(新的和旧的)，这些技术跨越了现代变分、数值分析、凸优化和数理统计的工具。

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.