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是必不可少的，例如外来材料的设计和制造、公共安全、医学成像和地下勘探。不幸的是，用于测试复杂材料的结构缺陷或识别未知目标的有效方法仍处于初级阶段。在这个由NSF资助的项目中，PI和她的研究生们将开发反散射理论的全新技术，以获得可靠的目标特征或关于以计算效率方式检查的对象的有用信息。目标是最大限度地减少对描述未知目标的物理和/或几何的先验信息的依赖。这项研究将把实际应用与新成像技术的数学优雅结合起来，这些新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”。直到不久前，逆散射理论中几乎所有现有的目标识别算法都是基于弱散射近似或使用非线性优化技术的。已经开发了用于成像的替代方法，以避免在弱散射近似中固有的不正确的模型假设以及对未知对象的拓扑和物理属性的先验信息的要求。这种方法一般被称为逆散射理论中的定性方法，或者称为非迭代方法。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性质，该算子编码了关于散射目标的非线性信息，并提供了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。传输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，并且它们携带有关散射体的材料属性的信息。这一建议涉及扩大定性方法的适用性，以解决反散射问题，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。研究的方向有两个：1)发展非均匀介质中含有不在Born近似范围内的小参数特征的定性反演方法；2)仅使用准后向散射和时间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，研究人员和她的研究生需要研究与上述两部分研究中出现的(非自伴和非线性)传输本征值问题有关的新的理论问题。

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.