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在许多具有国家重要性的领域中是必不可少的，例如奇异材料的设计和制造、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识别未知目标的有效方法仍然处于起步阶段。在这个NSF资助的项目中，PI和她的研究生将开发逆散射理论中的全新技术，以获得可靠的目标特征或以计算高效的方式检查对象的可用信息。目标是尽量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。这项研究将结合联合收割机的实际应用与数学优雅的新成像技术，最近导致建立一个新的领域，在数学称为“定性方法。直到不久以前，基本上所有现有的反散射理论中的目标识别算法都是基于弱散射近似或非线性优化技术的使用。已经开发了替代成像方法，以避免弱散射近似中固有的不正确的模型假设以及关于未知物体的拓扑和物理特性的先验信息的要求。这样的方法在逆散射理论中的定性方法的总标题下，或者被称为非迭代方法。粗略地说，这些方法探索由散射数据确定的线性紧致算子的性质，该线性紧致算子编码关于散射对象的非线性信息，并提供易于实现的重建算法。固有的方法出现了一个新的特征值问题，称为传输特征值问题。透射本征值的实际重要性在于，它们可以从测量的散射数据中确定，并且它们携带关于散射体的材料特性的信息。这一建议涉及扩大适用性的定性方法解决逆散射问题，特别强调非均匀介质。主要关注的是在无损检测和目标识别方面的潜在应用。研究将采取两个方向：1）开发定性反演方法的非均匀介质包含小参数的功能，不是在玻恩近似制度，2）减少所需的空间数据量的定性方法只使用准背散射和时域数据。为了实现上述目标，调查员和她的研究生将需要调查新的理论问题有关的（非自伴和非线性）传输本征值的问题，出现在上述两个部分的调查。

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.