Variationals Methods in Imaging and in Materials

成像和材料中的变分方法

• 批准号: 0905778
• 负责人: Irene Fonseca
• 金额: $ 116.86万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905778&HistoricalAwards=false
• 关键词: Variationals; Methods; Imaging; Materials

FonsecaDMS-0905778 In recent years there has been remarkable progress in the mathematical rigorous understanding of nonlinear phenomena, resulting from a deep articulation of ideas and techniques from partial differential equations, calculus of variations, and geometric measure theory. Common features to the problems to be addressed in this project are the treatment of energies that involve terms of different dimensionality (bulk, surface terms, etc.), a large range of length and time scales, higher order derivatives, and discontinuous underlying fields. Topics include: - in imaging, the recolorization and reconstruction of damaged images using the RGB and the CB models; - in thin structures, the understanding of the interplay between rigidity and brittle features; - within the realm of singular perturbations, the study of foams, microphase separation of copolymer melts, relaxation and homogenization of multi-phase, multi-component multiple integrals, micromagnetics; - in epitaxially strained thin films, the surface morphology and diffusion; - in micromagnetics, the derivation of a model for large bodies from the small bodies model that exhibits competition between the anisotropic energy and the exchange energy terms; - in the calculus of variations, the development of multi-scale theories and dimension reduction techniques for systems that go beyond traditional (higher order) underlying gradient fields and may apply to Maxwell-type systems. The program outlined above is strongly motivated by contemporary issues in imaging and materials science at the core of advances in high-end technology. These include recolorization of damaged images, the understanding of fracture in thin structures, the study of foams used in oil recovery, detergents, and lightweight structural materials, the study of morphology and defects in the epitaxial deposition process that are responsible for important optical, electronic, and magnetic properties, and the prediction of the behavior of ferroelectric, electromagnetic, and magnetostrictive materials and composites. The underlying models are at the forefront of traditional mathematical theories, and require state-of-the-art techniques, new ideas, and the introduction of innovative mathematical tools. It is necessary to bridge the multitude of scales present, and other mathematically challenging features, by appropriate schemes of articulated theoretical, numerical, and experimental approaches. This project is focused on the theoretical side of this venture, with the aim of contributing to the identification of problems of national scientific importance that offer new opportunities for the integration of applied analysis in research and in the education of advanced graduate students and postdoctoral fellows.

FonsecaDMS-0905778 近年来，由于对偏微分方程、变分法和几何测度论的思想和技术的深入阐述，对非线性现象的数学严格理解取得了显着进展。 该项目要解决的问题的共同特征是对涉及不同维度项（体积项、表面项等）、大范围的长度和时间尺度、高阶导数和不连续基础场的能量的处理。 主题包括： - 在成像中，使用 RGB 和 CB 模型对受损图像进行重新着色和重建； - 在薄结构中，了解刚性和脆性特征之间的相互作用； - 在奇异扰动领域内，研究泡沫、共聚物熔体的微相分离、多相、多组分多重积分的松弛和均质化、微磁学； - 在外延应变薄膜中，表面形态和扩散； - 在微磁学中，从小物体模型推导出大物体模型，该模型表现出各向异性能量和交换能量项之间的竞争； - 在变分计算中，系统多尺度理论和降维技术的发展超越了传统（高阶）基础梯度场，并可能适用于麦克斯韦型系统。 上述计划受到当代成像和材料科学问题的强烈推动，这些问题是高端技术进步的核心。 这些包括受损图像的重新着色、对薄结构断裂的理解、用于采油、清洁剂和轻质结构材料的泡沫的研究、外延沉积过程中负责重要光学、电子学的形态和缺陷的研究。和磁性，以及铁电、电磁和磁致伸缩材料和复合材料行为的预测。 其底层模型处于传统数学理论的前沿，需要最先进的技术、新的思想以及创新数学工具的引入。 有必要通过适当的理论、数值和实验方法的方案来弥合现有的多种尺度和其他数学上具有挑战性的特征。 该项目的重点是该项目的理论方面，旨在帮助确定具有国家科学重要性的问题，为应用分析在研究以及高级研究生和博士后教育中的整合提供新的机会。