A New Approach to Imaging by Waves

在许多具有国家重要性的领域，如先进和新型材料的设计和制造、公共安全、医学成像和地下勘探，能够使用电磁波、声波或弹性波对材料进行成像和无损检测是必不可少的。不幸的是，用于测试复杂材料的结构缺陷或在几乎没有先验信息的情况下以有效方式识别未知目标的有效方法仍处于起步阶段。对于表现出方向性、与询问波的非线性相互作用或特殊几何结构的材料尤其如此，所有这些都存在于上述领域的当代应用中。在这个项目中，研究人员和她的研究生将开发可以处理上述成像问题的逆散射理论的全新技术，以便以计算高效的方式获得可靠的目标特征或关于被检查对象的有用信息。目标是最大限度地减少对描述未知目标的物理和/或几何以及由于宿主背景的复杂性而引起的数学和计算复杂性的先验信息的依赖。这项研究将把实际应用与新的成像技术的数学优雅结合起来，这些新的成像技术最近在数学中建立了一个新的领域，称为定性方法或直接成像技术。这项研究是一项多方面的努力，旨在开发基于广义线性采样方法和与非散射现象相关的光谱参数的先进和新型材料(包括非线性材料)的快速成像方法。主要有三个项目：1)波的频谱成像方法：基于传输本征值理论，该项目提出了一个修改散射算子的通用框架，以便提供与非均匀散射相关的新的特征值问题。尤其重要的是从散射数据中确定这些(实数或复数)本征值，以及它们与非均质性的材料特性的关系。研究人员的主要工作是将这一技术应用于复杂结构的成像，包括各向异性/吸收/色散介质、薄层和亚表面。2)内本征值和非散射频率：特定入射波不散射的必要条件是波数(或指定参数)是定义在散射体上的内本征值问题的特征值。另一方面，如果相应的本征函数在某一本征值处作为Helmholtz方程的解在支点外可扩展，则反之亦然。目前，人们只了解带有角点的散射体的情况。研究人员提出了一个更一般的猜想，并提出了一种证明它的方法。该方案在理论上是可行的，但如果成功，它将导致一个更普遍的唯一性结果，以支持一个入射波的散射体。3)开展非线性非均匀定性成像方法的研究。尽管定性(直接)反演方法的理论和计算方法引起了人们极大的兴趣和广泛的研究，但对于许多当代工程材料所表现出的与询问波具有非线性相互作用的成像介质，还没有做过任何工作。该项目旨在开创这一研究的先河。这一奖项反映了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。

In many areas of national importance, such as the design and manufacturing of advanced and novel materials, public safety, medical imaging, and underground exploration, it is essential to be able to image and perform nondestructive testing of materials using electromagnetic, sound, or elastic waves. Unfortunately, effective methods for testing complicated materials for structural defects or for identifying unknown targets in an efficient way with little a priori information are still in a state of infancy. This is particularly the case for material that exhibits directional properties, nonlinear interaction with interrogating waves, or peculiar geometric structures, all of which are present in contemporary applications in the above areas. In this project, the investigator and her graduate students will develop entirely new techniques in inverse scattering theory that can handle the aforementioned imaging problems, in order to obtain reliable target signatures or usable information about objects being examined in computationally efficient ways. The goal is to minimize dependence on a priori information describing the physics and/or geometry of unknown targets as well as of mathematical and computational complications arising from the complexity of the hosting background. This study will combine practical applications with the mathematical elegance of new imaging techniques that have recently led to establishing a new field in mathematics called the qualitative approach or direct imaging techniques.This research is a multifaceted effort to develop fast imaging methods of advanced and novel materials (including nonlinear materials) based on generalized linear sampling methods and spectral parameters related to non-scattering phenomena. There are three main projects:1) A Spectral Approach to Imaging with Waves: Motivated by the theory of transmission eigenvalues, this project proposes to develop a general framework for modifying the scattering operator in order to provide new eigenvalue problems associated with the scattering by an inhomogeneity. Particularly important is the determination of these (real or complex) eigenvalues from the scattering data, together with their relation to the material properties of the inhomogeneity. A major effort of the investigator is to apply this technique to image complex structures, including anisotropic/absorbing/dispersive media, thin layers, and meta-surfaces. 2) Interior Eigenvalues and Non-scattering Frequencies: A necessary condition for the non-scattering of a particular incident wave is that the wave number (or a specified parameter) is an eigenvalue of an interior eigenvalue problem defined on the support of the scatterer. On the other hand, the converse is true if at an eigenvalue the corresponding eigenfunction is extendable outside the support as a solution to the Helmholtz equation. Currently, only the case of scatterers with a corner is understood. The investigator states a more general conjecture and lays out a proposed approach to prove it. This project is theoretical in nature, but if successful, it would lead to a more general uniqueness result for the support of the scattering object with one incident wave. 3) Initiate the Development of the Qualitative Imaging Approach for Nonlinear Inhomogeneities. Despite the great interest and extensive research on the theory and computation of qualitative (direct) inversion methods, nothing has been done in connection with imaging media that exhibit nonlinear interaction with interrogating waves, which is the case for many contemporary engineered materials. This project aims to pioneer such a study.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.