Spectral Theory and Applications for Models with Localized or Boundary Defects

该项目包括应用数学中的三个主要研究领域：(1)凝聚态和拓扑物理中模型算符的性质，(2)研究波如何传播(电磁波或流体波)的数值算法的性能，以及(3)域或图的几何形状如何影响将其“分割”成更小的有用片段的方法的研究，以确定关键社区或标签(例如从某些遗传标记中识别癌细胞)。这些听起来可能完全不相关，但该项目背后的主要思想是，当通过正确的视角来看待这些问题时，来自谐波分析和优化的类似方法可以应用于每一个问题。该项目的很大一部分将与本科生、研究生和博士后合作进行，以开发理论和计算工具，同时保持和创造与物理学家的新合作，特别是在应用方面。课程将特别注意与数值分析员合作，在每个应用领域模拟问题的前沿。将考虑解决物理、流体力学和数据分析中的复杂问题的算法，特别关注可量化的误差估计和物理上重要的例子的各种基本模式或共振的严格构造。例如，首席研究人员将扩展与合作者一起使用区域分解解决亥姆霍兹问题的工作，以进一步给出障碍散射数值格式的定量界限，类似于非均匀介质散射的情况。这将为Helmholtz方程的数值解提供增强的误差估计，Helmholtz方程在医学成像、声纳探测等反问题的各种应用中发挥着重要作用。特别是，尽管数值模拟本质上被限制在某些有界区域，但人们可以理解如何在某些点提供流体的阻尼，以允许对波在开放海洋中的传播方式进行数值模拟，而不会受到边界的影响。这是通过添加一个可能会对系统产生重大扰动的“阻尼项”来实现的。然而，在进行某种变换时，这种衰减的本质在于计算复势量子力学中出现的算符的性质。最近在偏微分方程的微局部分析、优化和椭圆理论中发展起来的复杂而丰富的工具使我们能够给出强有力的见解和建立定量界限的新方法。这些都适用于阻尼流体模型的有效性，以及与具有缺陷的晶体结构中的光波行为有关的问题。这些方法适用于不同的领域，包括但不限于成像、激光和社区探测。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project consists of three main research areas in applied mathematics: (i) properties of model operators in condensed matter and topological physics, (ii) performance of numerical algorithms for studying how waves propagate (either electromagnetic waves or fluid waves), and (iii) a study on how the geometry of a domain or graph impacts methods for “partitioning” it into smaller useful pieces for identifying key communities or labels (such as identifying cancer cells from certain genetic markers). Each of these might sound completely unrelated, but the main idea behind the project is that similar methods from harmonic analysis and optimization can be applied to each of these problems when viewed through the right lens. A substantial part of the project will be performed in collaboration with undergraduate and graduate students and postdocs, towards developing theoretical and computational tools, while at the same time maintaining and creating new collaborations with physicists, especially in terms of applications. Special attention will be paid to working with numerical analysts at the forefront of simulating problems in each application domain.Algorithms for solving complicated problems in physics, fluid mechanics and data analysis will be considered, with a special focus on quantifiable error estimates and rigorous constructions of various fundamental modes or resonances for physically important examples. For instance, the principal investigator will extend work that was performed with collaborators on solving the Helmholtz problem using domain decompositions to give further quantitative bounds on numerical schemes for obstacle scattering, in similar way to the cases of scattering by inhomogeneous media. This will give enhanced error estimates for numerical solutions to the Helmholtz equation, which play a major role in various applications for inverse problems in medical imaging, sonar detection, and more. In particular, although numerical simulations are by nature constrained to certain bounded domains, one can understand how to provide damping of a fluid at certain points to allow for a numerical simulation of the way a wave propagates in the open ocean without getting effects from the boundary. This is done by adding a “damping term” that could perturb the system in a significant way. However, upon performing a certain transformation, the essence of this damping lies in computing properties of operators that arise in quantum mechanics with complex potentials. The complex and rich tools that have been recently developed in microlocal analysis, optimization and elliptic theory of partial differential equations allow us to give strong insights and new means of establishing quantitative bounds. These apply to the efficacy of damped fluid models, as well as to questions related to the behavior of a light wave in a crystalline structure with defects. These methods apply to various fields, including but not limited to imaging, lasing and community detection.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.