New Approaches to Inverse Scattering for Inhomogeneous Media

在许多具有国家重要性的领域，如特殊材料的设计和制造，公共安全，医学成像和地下勘探，必须能够使用电磁波，声波或弹性波对材料进行成像和无损检测。不幸的是，用于测试复杂材料的结构缺陷或用于以有效的方式识别未知目标的有效方法仍然处于婴儿期。在这个项目中，研究人员和她的研究生将开发逆散射理论的新技术，以获得可靠的目标签名或有关对象的有用信息，在计算效率的方式进行检查。其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，以及由宿主背景的复杂性引起的并发症。这项研究将联合收割机的实际应用与新的直接成像技术的数学优雅。 这项研究是一个多方面的努力，调查一些开放的问题与定性方法（或称为非迭代方法）解决逆散射问题的非均匀介质。中心主题是广义线性抽样方法的发展，各种反问题。这是一种新的逆散射定性方法，对于有噪数据来说是严格合理的。主要有三个项目：1）非均匀介质逆散射理论中的本征值：受透射本征值问题理论的启发，本项目提出开发一个修改散射算子的通用框架，以提供与非均匀性散射相关的新本征值问题。特别重要的是从散射数据确定这些（真实的或复杂的）本征值的问题，连同它们与不均匀性的材料性质的关系。这项工作的最终目标是使用这样的特征值的各向异性（可能吸收和色散）介质的成像。2)时域问题的定性方法：提出使用时间相关数据来解决在使用定性重建方法时需要大量空间数据的问题。时域内传输问题是发展波动方程定性方法的基本数学要素，其可解性是一个长期存在的开放性问题。本研究提出了解决这一问题的方法，从而为研究时域定性方法开辟了道路。3)周期性介质的逆散射：该项目的主要考虑是研究由于有界支撑的高振荡周期性介质的散射引起的远场行为，并应用于介质的宏观/微观结构的成像。此外，该项目将开发一种新的定性方法，用于重建周期性介质内的局部扰动，而无需计算由于周期性背景而产生的散射场，也无需恢复背景。PI将对复杂非均匀介质的逆散射理论以及时域中的新定性方法进行系统研究，如果成功，可以在数学领域之外取得进展，例如无损检测和目标识别。除了数学领域之外，本研究中出现的新的本征值问题，如吸收介质的透射本征值问题或Steklov本征值问题，在偏微分方程的非自伴本征值问题领域中引起了相当大的关注。对周期性介质散射问题的分析触及了偏微分方程中一些最活跃的当代主题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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 exotic 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. In this project, the investigator and her graduate students will develop 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 as well as of the complications arising from complexity of the hosting background. This study will combine practical applications with the mathematical elegance of new direct imaging techniques. This research is a multifaceted effort to investigate some open questions associated with the qualitative methods (otherwise referred to as non-iterative methods) for solving inverse scattering problems for inhomogeneous media. The central theme is the development of the generalized linear sampling method for a variety of inverse problems. This is a new qualitative approach to inverse scattering that is rigorously justified for noisy data. There are three main projects: 1) Eigenvalues in inverse scattering theory for inhomogeneous media: Motivated by the theory of the transmission eigenvalue problem, 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 are the issues of determining these (real or complex) eigenvalues from the scattering data, together with their relation to material properties of the inhomogeneity. The ultimate goal of this effort is to use such eigenvalues for the imaging of anisotropic (possibly absorbing and dispersive) media. 2) Qualitative methods for time domain problems: The use of time dependent data is proposed to address the issue of the need for large amount of spatial data in the use of qualitative reconstruction methods. The time domain interior transmission problem is the fundamental mathematical ingredient to develop qualitative methods for the wave equation and its solvability is a long lasting open problem. This project suggests an approach to solving this problem, hence opening the way to study time domain qualitative methods. 3) Inverse scattering for periodic media: The main consideration of this project is to investigate the far field behavior due to scattering by a highly oscillating periodic media of bounded support with application to imaging of macro/micro structure of the media. In addition, the project will develop a new qualitative method for reconstructing local perturbations inside periodic media without needing to compute the scattered field due to the periodic background nor to recover the background. The PI will undertake a systematic investigation of new qualitative methods in inverse scattering theory for complex inhomogeneous media as well as in the time domain which, if successful, can lead to progress outside the field of mathematics, such as nondestructive testing and target identification. In addition to progress outside the field of mathematics, the new eigenvalue problems that appear in this study, such as the transmission eigenvalue problem or the Steklov eigenvalue problem for absorbing media, have attracted considerable attention in the area of non-selfadjoint eigenvalue problems for partial differential equations. The analysis of scattering problems for periodic media touches some of the most active contemporary topics in PDEs.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.