Variational Methods for Materials Science and Mathematical Imaging

材料科学和数学成像的变分方法

基本信息

批准号：
1906238
负责人：
Irene Fonseca
金额：
$ 68.42万
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：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906238&HistoricalAwards=false
关键词：
Variational Methods Materials Science Mathematical

项目摘要

The mathematical theories and techniques developed in this project provide a foundation for understanding aspects of imaging and of the properties of materials. The use of self-assembly processes to manufacture modern semiconductor nanostructures, quantum wires, and quantum dots is 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 the processing of nanoscale materials. We address the variational study of relevant observed phenomena in nanowires and in the epitaxial deposition of a thin film onto a substrate. We study phase nucleation in Lithium-Ion batteries, which are central to advances in portable electronic devices, electric vehicles, and renewable energy storage; phase nucleation is important in understanding charge-discharge dynamics (poor cycle life) and other material limitations of these batteries. In what concerns the mathematics of imaging, we pursue the analytical investigation of image processing, restoration, and registration, which are fundamental to the advance of computer vision, medical imaging, film restoration, and scanning probe microscopy. These projects offer opportunities for integrating research in applied analysis with the education of advanced graduate students at the interface between mathematics and the physical sciences and engineering. Graduate students participate in the research of the project.What unifies these topics is that underlying energies involve higher order derivatives in spaces with discontinuous admissible fields, multiple scales interact, bulk and surface energies compete, and degeneracy of usually expected properties prevails. Together, these difficulties prevent the use of well-understood mathematical theories, and require new ideas and the introduction of innovative mathematical tools. Contemporary methods in the calculus of variations and nonlinear partial differential equations are used to study quasi-static (elliptic) and evolution (parabolic) systems of equations in a range of problems arising in materials science that span epitaxy, batteries, nanowires, and phase transitions. These methods are combined in novel ways with multi-level (machine learning) training schemes to study models for edge detection, image segmentation, signal denoising and detexturing, joint image segmentation, and image registration. Graduate students participate in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目中开发的数学理论和技术为理解成像和材料特性方面的基础。使用自组装过程来制造现代的半导体纳米结构，量子线和量子点在微电和光电技术中至关重要，例如光学的反射性或抗反射涂层，用于光学材料的反射性或抗反射涂层，用于集成型物质和量子的材料的制造，用于整体循环和量子的构造和量子。我们解决了相关观察到的纳米线现象以及薄膜外延沉积中相关现象的变异研究。我们研究锂离子电池中的相成核，这对于便携式电子设备，电动汽车和可再生能源存储的进步至关重要；相核对于理解电荷释放动力学（循环寿命差）和这些电池的其他物质局限性很重要。在与成像的数学有关的情况下，我们对图像处理，恢复和注册进行分析研究，这是计算机视觉，医学成像，胶片修复和扫描探针显微镜的基础。这些项目为在数学与物理科学与工程学之间的界面上与高级研究生的教育提供了将研究中的研究与高级研究生的教育相结合的机会。研究生参与了项目的研究。这些主题统一的是，潜在的能量涉及在不连续可允许的领域的空间中，多个尺度相互作用，散装和表面能量竞争的高阶衍生物，并且通常预期的性能的脱落。这些困难一起阻止了使用良好的数学理论，并需要新的想法并引入创新的数学工具。在变化和非线性偏微分方程的计算中的当代方法用于研究材料科学中的一系列问题中的准静态（椭圆形）和进化（抛物线）系统，这些问题涵盖了外观，电池，电池，纳米线和相变的材料科学。这些方法以新颖的方式与多层次（机器学习）训练方案相结合，以研究用于边缘检测，图像分割，信号降低和尺寸的模型，关节图像分割和图像注册。研究生参加了该项目的研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。