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

变分问题：从基础问题到图像处理和材料科学中的直接应用

• 批准号: RGPIN-2020-04911
• 负责人: Choksi, Rustum
• 金额: $ 2.7万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740289
• 关键词: Variational; Problems; Foundational; Questions; Direct

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.

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。这个建议解决了四个（类）变分问题，可以分为两个部分。第一部分从数学的角度来看是更基础的，涉及两个经典的，非局部和非凸变分问题：最佳量化（或最佳质心Voronoi镶嵌）和伽莫夫的液滴模型（及其各向异性变体）。对于这两个问题，我们将集中在两个中心目标：（a）在多大程度上可以严格地描述基态（全局极小）;（B）可以开发数值算法来探测能量景观并找到低能态。 该提案的第二部分是高度应用驱动的，涉及解决和开发两个非常不同领域的特定新变分问题：（i）图像处理中的去模糊和去噪，以及（ii）多晶材料中的晶粒生长。对于图像处理，我们将发展由申请人等人介绍的最近且非常成功的贝叶斯变分方法，该方法基于图像概率分布均值最大熵的思想。 对于晶粒生长，我们解决相场晶体模型和梯度流动动力学。我们的重点在于：在晶粒生长过程中的几何度量的普遍统计分布的理解，并解决各种扩展和减少相场晶体模型与数学和理想的物理性能之间的连接的重点。 在整个提案中，我们将采用技术（新的和旧的），这些技术涵盖了现代变分法，数值分析，凸优化和数理统计的工具。