Variational Methods for Materials and Imaging Sciences

材料和成像科学的变分方法

基本信息

批准号：
1411646
负责人：
Irene Fonseca
金额：
$ 122.23万
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：
2014
资助国家：
美国
起止时间：
2014-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411646&HistoricalAwards=false
关键词：
Variational Methods Materials Imaging Sciences

项目摘要

The objectives of this project are the identification and pursuit of emerging areas of applied analysis, motivated by contemporary issues in imaging and materials science at the core of advances in high-end technology and of national scientific importance. The two main topics of the project are(1) the mathematical study of modern semiconductors and nano structures, of pivotal importance in microelectric and optoelectronic technologies, such as reflective or anti-reflective coatings for optics, the fabrication of layers of insulators and semiconductors for integrated circuits, quantum well lasers, and(2) the analytical investigation of image segmentation and inpainting and recolorization for color images, fundamental to the advance of computer vision, medical imaging, film restoration, and scanning probe microscopy. Postdocs and graduate students are trained in the course of the project. Common features of the projects include the treatment of energies that involve terms of different dimensionality. These often exhibit a large range of length and time scales, higher order derivatives, and discontinuous underlying fields. Such features prevent the use of well understood functional analytic frameworks, they escape traditional mathematical theories, and they require state-of-the-art techniques, creative ideas, and the introduction of innovative mathematical tools. The investigator and her collaborators use new and recently developed methods and a deep articulation of ideas in the calculus of variations, geometric measure theory, and nonlinear partial differential equations, to address problems that include in topic (1) epitaxy and the formation of quantum dots, the onset and propagation of dislocations, homogenization of composite materials, and in topic (2) signal denoising and detexturing, dejittering, inpainting, and recolorization. These topics offer new opportunities for the integration of applied analysis in research and in the education of advanced graduate students and postdoctoral fellows, thus allowing for the training of a new generation of applied analysts at the forefront of contemporary mathematics as it interfaces with materials and imaging sciences.

该项目的目标是识别和追求应用分析的新兴领域，其动力来自成像和材料科学领域的当代问题，这些问题是高端技术进步的核心，具有国家科学的重要性。该项目的两个主要主题是（1）现代半导体和纳米结构的数学研究，在微电子和光电技术中至关重要，例如光学的反射或抗反射涂层、用于光学器件的绝缘体和半导体层的制造集成电路、量子阱激光器，以及（2）图像分割、彩色图像修复和重新着色的分析研究，这是计算机视觉、医学成像、胶片修复和扫描探针显微镜进步的基础。博士后和研究生在项目过程中接受培训。这些项目的共同特征包括涉及不同维度的能量处理。这些通常表现出大范围的长度和时间尺度、高阶导数和不连续的基础场。这些特征阻碍了人们充分理解的函数分析框架的使用，它们逃避了传统的数学理论，并且需要最先进的技术、创造性的想法以及创新数学工具的引入。研究人员和她的合作者使用新的和最近开发的方法以及对变分计算、几何测度理论和非线性偏微分方程思想的深入阐述，来解决主题（1）外延和量子点形成中包括的问题、位错的发生和传播、复合材料的均质化，以及主题 (2) 中的信号去噪和去纹理、去抖动、修复和重新着色。这些主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了新的机会，从而可以在当代数学前沿培养新一代应用分析师，因为它与材料和成像相结合科学。