Variational Methods for 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，其动机是成像和材料科学中的当代问题，这些问题是高端技术进步的核心和国家科学重要性。该项目的两个主要课题是：(1)现代半导体和纳米结构的数学研究，这在微电和光电子技术中具有关键意义，例如光学的反射或抗反射涂层，集成电路的绝缘体和半导体层的制造，量子阱激光器；(2)图像分割和彩色图像的涂漆和重新着色的分析研究。对计算机视觉、医学成像、胶片修复和扫描探针显微镜的进步至关重要。博士后和研究生在项目的过程中进行培训。这些项目的共同特点包括处理涉及不同维度的能量。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底层场。这样的特征阻碍了对功能分析框架的使用，它们脱离了传统的数学理论，它们需要最先进的技术、创造性的想法和创新数学工具的引入。研究者和她的合作者使用新的和最近开发的方法，并在变分法，几何测量理论和非线性偏微分方程中深入阐述思想，以解决包括主题(1)外延和量子点的形成，位错的发生和传播，复合材料的均质化以及主题(2)信号去噪和去纹理，去抖动，涂漆和重新着色的问题。这些主题为应用分析在研究和高级研究生和博士后教育中的整合提供了新的机会，从而允许在当代数学的前沿培养新一代应用分析人员，因为它与材料和成像科学相结合。

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.