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New Approaches to Inverse Scattering for Inhomogeneous Media

非均匀介质逆散射的新方法

基本信息

批准号：
1813492
负责人：
Fioralba Cakoni
金额：
$ 21.19万
依托单位：
Rutgers University New Brunswick
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1813492&HistoricalAwards=false
关键词：
New Approaches Inverse Scattering Inhomogeneous

项目摘要

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.

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，能够利用电磁波、声波或弹性波对材料进行成像和无损检测是必不可少的。不幸的是，对于复杂材料的结构缺陷测试或在缺乏先验信息的情况下有效识别未知目标的有效方法仍处于起步阶段。在这个项目中，研究者和她的研究生将开发逆散射理论的新技术，以获得可靠的目标特征或有关被检测对象的可用信息，以计算有效的方式。目标是最大限度地减少对描述未知目标的物理和/或几何以及由宿主背景的复杂性引起的复杂性的先验信息的依赖。这项研究将结合实际应用与新的直接成像技术的数学优雅。本研究是一个多方面的努力，探讨一些与定性方法（或称为非迭代方法）有关的开放性问题，用于解决非均匀介质的逆散射问题。中心主题是各种反问题的广义线性抽样方法的发展。这是一种新的定性方法来逆散射，是严格证明噪声数据。主要有三个项目：1)非均匀介质反散射理论中的本征值：在透射本征值问题理论的激励下，本项目提出了一个修改散射算符的一般框架，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中确定这些（实或复）特征值的问题，以及它们与材料非均匀性的关系。这项工作的最终目标是将这些特征值用于各向异性（可能是吸收和色散）介质的成像。2)时域问题的定性方法：针对定性重构方法需要大量空间数据的问题，提出了使用时间相关数据的方法。时域内透射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悬而未决的问题。本课题提出了一种解决这一问题的方法，从而开辟了研究时域定性方法的道路。3)周期介质的逆散射：本项目主要考虑的是研究高振荡周期有界支撑介质的远场散射行为，并将其应用于介质的宏观/微观结构成像。此外，该项目将开发一种新的定性方法来重建周期介质内部的局部扰动，而不需要计算周期背景引起的散射场，也不需要恢复背景。PI将对复杂非均匀介质的反散射理论以及时域中的新的定性方法进行系统的研究，如果成功，将导致数学领域以外的进步，例如无损检测和目标识别。除了在数学领域之外，本研究中出现的新特征值问题，如传输特征值问题或吸收介质的Steklov特征值问题，在偏微分方程非自伴随特征值问题领域引起了相当大的关注。周期介质散射问题的分析涉及到偏微分方程中一些最活跃的当代话题。该奖项反映了美国国家科学基金会的法定使命，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。