Analytic and Geometric Inverse Problems and Related Topics

该项目的目标是开发新的数学技术来解决反问题中的核心和挑战性问题。逆问题是技术进步和现代科学研究的核心，涉及从间接、不完整或有噪声的观测中确定介质的性质。逆问题的应用范围很广，包括地球物理、声纳和医学成像技术，用于寻找地球内部石油和矿藏的位置，早期发现肺水肿，以及监测肺功能。该项目的一个主要主题是卡尔德隆问题，它构成了电阻抗断层成像的基础，这是一种应用于生物医学成像和机械部件无损检测的成像方式。卡尔德隆问题问的是介质的导电性是否可以通过在介质表面进行电压和电流测量来确定。该项目的具体重点是通过沿着边界表面的可能小部分进行测量来恢复粗糙介质的导电性，这在实践中是普遍存在的情况，以及重建各向异性介质的导电性，例如人体的肌肉组织。研究者还将研究流体动力学、弹性、各向异性线性和非线性电磁学中出现的重要逆问题。在该项目中开发的新颖数学方法可能会导致医学和地震成像方面的重要进展。该项目将集中于以下四个重要的研究课题。第一类是低正则性条件下部分数据的反边界问题。在这类问题中，人们试图从沿边界的一小部分进行的边界测量中确定代表介质特征的偏微分方程（PDE）的低正则系数。受各向异性介质中材料参数恢复的激励，第二个主题涉及几何逆边界问题，其中人们希望恢复潜在黎曼流形的拓扑和微分特征。第三个主题提出了一种广泛而系统的方法来解决高频区域椭圆偏微分方程部分数据反边界问题稳定性增加的问题，这是由于需要设计高分辨率的重建算法。研究者的目标是开发一种创新的方法，将半经典分析的强大技术应用于这类重要的问题。受物质稳定性和非厄米量子力学研究的启发，第四个主题提出了在经典Keller和Lieb-Thirring界的精神下，研究薛定谔型非自伴随算子的复特征值分布。本研究是在非紧流形的一般情况下进行的，其无穷远处的结构推广了欧几里德流形的结构，依靠强大的几何微局部分析技术。虽然看起来彼此完全独立，但项目的不同部分实际上是紧密相连的，它们之间存在协同作用，并且在一个部分开发的工具，例如Carleman和解决方案评估，可以导致其他部分的进展。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 goal of the project is to develop new mathematical techniques to attack central and challenging questions in inverse problems. Being at the core of technological advances and modern scientific investigations, inverse problems are concerned with determining properties of a medium from indirect, incomplete, or noisy observations. Applications of inverse problems range over a broad spectrum of geophysical, sonar, and medical imaging techniques, used for finding location of oil and mineral deposits in the interior of the Earth, early detection of pulmonary edema, as well as monitoring of lung function. A major topic for this project is the fundamental Calderon problem, which forms the basis of Electrical Impedance Tomography, an imaging modality with applications in biomedical imaging and nondestructive testing of mechanical parts. The Calderon problem asks whether the conductivity of a medium can be determined by performing voltage and current measurements on the surface of the medium. The specific focus of the project is on recovering the conductivity of a rough medium by performing measurements along possibly small portions of the boundary surface, a situation ubiquitous in practice, as well as on reconstructing the conductivity of an anisotropic medium, such as the muscle tissue in the human body, say. The investigator will also attack significant inverse problems arising in fluid dynamics, elasticity, and anisotropic linear and non-linear electromagnetism. The novel mathematical approaches to be developed in the project may lead to important advances in medical and seismic imaging. The project will focus on the following four significant research topics. The first one is inverse boundary problems with partial data in the low regularity setting. In this class of problems one seeks to determine low regularity coefficients of partial differential equations (PDE), representing characteristics of a medium, from boundary measurements performed along small portions of the boundary. Motivated by recovery of material parameters in anisotropic media, the second topic is concerned with geometric inverse boundary problems, where one wishes to recover also topological and differential characteristics of the underlying Riemannian manifold. The third topic proposes a broad and systematic attack on the phenomenon of increasing stability for partial data inverse boundary problems for elliptic PDE in the high frequency regime, driven by the need to design reconstruction algorithms with high resolution. The investigator aims to develop the innovative approach of bringing the powerful techniques of semiclassical analysis to bear on this important class of problems. Inspired by the study of the stability of matter and non-hermitian quantum mechanics, the fourth topic proposes to investigate the distribution of complex eigenvalues for non-self-adjoint operators of Schrodinger type, in the spirit of the classical Keller and Lieb-Thirring bounds. This research is to be carried out in the general setting of non-compact manifolds, whose structure at infinity generalizes the Euclidean one, relying upon the powerful techniques of geometric microlocal analysis. While seemingly quite independent of each other, the different parts of the project are in fact intimately connected, there being a synergy amongst them, and tools developed in one part, such as Carleman and resolvent estimates, say, can lead to progress in the others.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.