Variational Methods for Materials Science and Mathematical Imaging

在这个项目中发展的数学理论和技术为理解成像和材料特性方面提供了基础。利用自组装工艺制造现代半导体纳米结构、量子线和量子点在微电子和光电子技术中具有关键意义，例如光学反射或抗反射涂层、集成电路绝缘体和半导体层的制造、量子阱激光器和纳米级材料的加工。我们讨论了纳米线和薄膜外延沉积在衬底上的相关观察现象的变分研究。我们研究锂离子电池的相成核，这是便携式电子设备，电动汽车和可再生能源存储的核心；相成核对于理解这些电池的充放电动力学（较差的循环寿命）和其他材料限制具有重要意义。在涉及成像的数学方面，我们对图像处理、恢复和配准进行分析研究，这是计算机视觉、医学成像、胶片恢复和扫描探针显微镜进步的基础。这些项目提供了将应用分析研究与高等研究生教育结合起来的机会，将数学、物理科学和工程结合起来。研究生参与该项目的研究。将这些主题统一起来的是，在具有不连续可容许场的空间中，潜在能量涉及高阶导数，多尺度相互作用，体能和表面能竞争，通常预期性质的退化普遍存在。综上所述，这些困难阻碍了人们对熟知的数学理论的运用，需要新的思想和创新的数学工具的引入。在变分法和非线性偏微分方程的现代方法被用来研究准静态（椭圆）和进化（抛物线）方程组在材料科学中出现的一系列问题，跨越外延，电池，纳米线，和相变。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，研究边缘检测、图像分割、信号去噪和去噪、联合图像分割和图像配准的模型。研究生参与该项目的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.