New Directions in the Qualitative Approach to Inverse Scattering Theory

逆散射理论定性方法的新方向

• 批准号: 1515072
• 负责人: Fioralba Cakoni
• 金额: $ 22.02万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2016-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1515072&HistoricalAwards=false
• 关键词: New; Directions; Qualitative; Approach; Inverse

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.

在许多具有民族重要性的领域，例如设计和制造外来材料，公共安全，医学成像和地下勘探，使用电磁，声音或弹性波对材料进行图像和无损测试的能力至关重要。不幸的是，测试复杂材料的结构缺陷或识别几乎没有先验信息的未知目标的有效方法仍处于婴儿期。在这个由NSF资助的项目中，PI和她的研究生将在反散射理论中开发全新的技术，以获得可靠的目标签名或有关以计算有效方式检查对象的可用信息。目的是最大程度地减少对描述未知目标物理和/或几何形状的先验信息的依赖。这项研究将将实际应用与新成像技术的数学优雅结合起来，这些技术最近导致建立了名为“定性方法”的数学领域的新领域。直到不久前，本质上是所有现有的逆散射理论中目标算法基于弱散射理论中的所有现有算法，都是基于较弱的散射近似近似近似或对非典型优化技术的使用。已经开发了用于避免弱散射近似值固有的错误模型假设以及有关未知对象拓扑和物理特性的先验信息的要求。这种方法属于逆散射理论中定性方法的一般标题，否则称为非题方式。粗略地说，这些方法探索了由散射数据确定的线性紧凑型操作员的属性，该数据编码有关散射对象的非线性信息，并提供易于实现的重建算法。这些方法固有的是一个新的特征值问题，称为传输特征值问题。传播特征值的实际重要性是它们可以从测得的散射数据中确定，并提供有关散射器材料特性的信息。该提案涉及扩大定性方法在解决反向散射问题的适用性，并特别强调不均匀的媒体。主要关注的是潜在的应用于无损测试和目标识别。调查将采用两个方向：1）开发定性反转方法，用于不均匀的介质，其中包含不在诞生的近似制度中的小参数特征，2）减少仅通过仅使用quasi-back-back-back-back-sclasting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，研究人员和她的研究生将需要研究与（非频道和非线性）传输特征值有关的新理论问题，这些问题在研究上述两个部分时会出现。