Variational Methods for Materials Science and Mathematical Imaging

在这个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提供了基础。 使用自组装工艺来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技术中具有关键重要性，例如用于光学器件的反射或抗反射涂层、用于集成电路的绝缘体和半导体层的制造、量子阱激光器以及纳米级材料的处理。 我们解决变分研究纳米线和在衬底上的薄膜外延沉积的相关观察到的现象。 我们研究锂离子电池中的相位成核，这对便携式电子设备，电动汽车和可再生能源存储的进步至关重要;相位成核对于理解这些电池的充放电动力学（循环寿命差）和其他材料限制非常重要。 在成像数学方面，我们追求图像处理，恢复和配准的分析研究，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。 这些项目提供了机会，将应用分析研究与数学与物理科学和工程之间的界面上的高级研究生教育相结合。 研究生参与了该项目的研究。将这些主题统一起来的是，潜在的能量涉及具有不连续容许场的空间中的高阶导数，多尺度相互作用，体能和表面能竞争，通常预期的性质的简并性占上风。 总之，这些困难阻止了使用很好理解的数学理论，并需要新的想法和引入创新的数学工具。 变分法和非线性偏微分方程的当代方法被用来研究准静态（椭圆）和演化（抛物）方程系统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级（机器学习）训练方案相结合，以研究边缘检测，图像分割，信号去噪和去纹理，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NSF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.