Variationals Methods in Imaging and in Materials

FonsecaDMS-0905778 近年来，在对非线性现象的数学严密理解方面取得了显着进展，这是由于偏微分方程、变分法和几何测度理论的思想和技术的深刻阐述。 在这个项目中要解决的问题的共同特点是涉及不同维度的能量的处理（体积，表面项等），大范围的长度和时间尺度、高阶导数和不连续的下伏场。 主题包括：- 在成像中，使用RGB和CB模型对受损图像进行分类和重建; -在薄结构中，理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匀化，微磁学;- 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;- 在变分法中，多尺度理论和降维技术的发展，超越了传统的（高阶）基本梯度场，并且可以应用于Maxwell型系统。 上述计划的强烈动机是成像和材料科学的当代问题，这是高端技术进步的核心。 这些包括受损图像的分解，薄结构断裂的理解，用于石油回收，洗涤剂和轻质结构材料的泡沫的研究，负责重要的光学，电子和磁性的外延沉积过程中的形态和缺陷的研究，以及铁电，电磁和磁致伸缩材料和复合材料的行为的预测。 基础模型处于传统数学理论的最前沿，需要最先进的技术，新的想法和创新的数学工具的引入。 有必要通过适当的理论、数值和实验方法来弥合现有的众多尺度和其他数学上具有挑战性的特征。 该项目侧重于这一事业的理论方面，目的是帮助确定具有国家科学重要性的问题，这些问题为将应用分析纳入研究以及高等研究生和博士后研究员的教育提供了新的机会。

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.