New Directions in the Qualitative Approach to Inverse Scattering Theory

使用电磁波、声波或弹性波对材料进行成像和无损检测的能力对于国家重要的许多领域至关重要，例如特殊材料的设计和制造、公共安全、医学成像和地下勘探。不幸的是，测试复杂材料的结构缺陷或在先验信息很少的情况下识别未知目标的有效方法仍处于起步阶段。在这个由国家科学基金会资助的项目中，首席研究员和她的研究生将开发逆散射理论的全新技术，以计算有效的方式获得可靠的目标特征或有关正在检查的物体的可用信息。目标是最大限度地减少对描述未知目标的物理和/或几何形状的先验信息的依赖。这项研究将实际应用与新成像技术的数学优雅结合起来，这些技术最近导致了数学新领域的建立，称为“定性方法”。直到不久前，基本上所有现有的逆散射理论中的目标识别算法都基于弱散射近似或非线性优化技术的使用。已经开发了替代的成像方法，以避免弱散射近似中固有的错误模型假设以及对未知物体的拓扑和物理特性的先验信息的要求。此类方法属于逆散射理论中定性方法的总称，也称为非迭代方法。粗略地说，这些方法探索了由散射数据确定的线性紧算子的属性，该算子对有关散射对象的非线性信息进行编码并提供易于实现的重建算法。这些方法固有地出现了一个新的特征值问题，称为传输特征值问题。透射特征值的实际重要性在于，它们可以根据测量的散射数据来确定，并且它们携带有关散射体材料特性的信息。该提案涉及扩大定性方法的适用性，以解决逆散射问题，特别强调非均匀介质。主要关注的是无损检测和目标识别的潜在应用。研究将采取两个方向：1）针对包含不属于玻恩近似范围的小参数特征的非均匀介质开发定性反演方法，2）仅使用准后向散射和时域数据来减少定性方法所需的空间数据量。为了实现上述目标，研究者和她的研究生将需要研究与在上述两部分的研究中出现的（非自共轭和非线性）传输特征值问题相关的新理论问题。

The ability to image and perform nondestructive testing of materials using electromagnetic, sound, or elastic waves is essential in many areas of national importance, such as the design and manufacturing of exotic materials, public safety, medical imaging, and underground exploration. Unfortunately, effective methods for testing complicated materials for structural imperfections or for identifying unknown targets with little a priori information are still in a state of infancy. In this NSF-funded project, the PI and her graduate students will develop entirely new techniques in inverse scattering theory 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. This study will combine practical applications with the mathematical elegance of new imaging techniques that have recently led to the establishment of a new field in mathematics called "qualitative methods."Until not long ago, essentially all existing algorithms for target identification in inverse scattering theory were based on either a weak scattering approximation or on the use of nonlinear optimization techniques. Alternative methods for imaging have been developed to avoid incorrect model assumptions inherent in weak scattering approximations and the requirement of a priori information on the topological and physical properties of the unknown object. Such methods come under the general title of qualitative methods in inverse scattering theory, otherwise referred to as non-iterative methods. Roughly speaking, these methods explore the properties of a linear compact operator determined by the scattering data, which encodes nonlinear information about the scattering object and provides easily implementable reconstruction algorithms. Inherent to the methods appears a new eigenvalue problem, known as the transmission eigenvalue problem. The practical importance of transmission eigenvalues is that they can be determined from the measured scattering data and they carry information about the material properties of the scatterer. This proposal concerns broadening the applicability of the qualitative approach for solving inverse scattering problems with particular emphasis on inhomogeneous media. Of main concern are potential applications to nondestructive testing and target identification. The investigation will take two directions: 1) develop qualitative inversion approaches for inhomogeneous media containing small parameter features that are not in the Born approximation regime, 2) reduce the amount of spatial data needed for the qualitative approach by using only quasi-backscattering and time domain data. To accomplish the above objectives, the investigator and her graduate students will need to investigate new theoretical questions related to the (non-selfadjoint and non-linear) transmission eigenvalue problems that arise in the investigation of the above two parts.