Direct and Inverse Scattering Problems in Elastic Waves: Analysis and Computation

散射问题涉及到非均匀介质对入射场的影响。在雷达和声纳、地球物理勘探、无损检测、医学成像、近场光学显微镜和纳米光学等不同科学领域的重要应用的推动下，散射问题已经被许多研究人员广泛研究，特别是对于声波和电磁波。然而，由于弹性波模型方程的复杂性，许多理论分析和数值计算尚未完成。该研究本质上是多学科的，位于数学，物理，工程和材料科学的界面。这将有助于更好地理解弹性散射理论中复杂的物理和数学问题。它在推进应用数学和计算数学的前沿以及发展新的数学和科学方面具有巨大的潜力。将通过出版物、研讨会、小型专题讨论会、会议和讲习班传播拟议研究活动的成果。PI将介绍一个高级研究生课程和研究生研讨会系列。这些将有助于在整个学术管道招聘和留住具有不同背景的优秀学生。该项目开发的软件代码和新教材将在一个公共网站上传播，供科学界下载。研究和教育组成部分将被整合在一起，以帮助培养新一代的研究人员，并培养更大的认识和兴趣，在应用和计算数学，特别是应用到研究生和博士后之间的散射理论。该项目概述了一个为期三年的研究计划，用于开发有效的数学模型，研究基本的数学问题，并为新的和重要的弹性波直接和逆散射问题设计有效的计算方法。拟议的研究建立在PI？在声波和电磁波散射理论领域的研究成果。它涉及以下三个方面：（1）时域障碍物散射问题;（2）时谐介质散射问题;（3）逆随机源散射问题。本项目开发的数学建模和分析技术以及计算方法将解决弹性波直接和逆散射理论中的几个关键科学挑战和悬而未决的问题，其中包括非均匀介质中弹性波传播的建模和计算、弹性波的数值解方程和相关模型的适定性，随机源散射反问题的唯一性和稳定性。所提出的计算模型和工具是非常有前途的定量研究的复杂的物理和数学问题的弹性。他们有很大的潜力，提供廉价和易于控制的虚拟原型的结构在设计和制造的新颖的弹性device.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Scattering problems are concerned with the effect that an inhomogeneous medium has on an incident field. Driven by significant applications in diverse scientific areas such as radar and sonar, geophysical exploration, nondestructive testing, medical imaging, near-field optical microscopy, and nano-optics,the scattering problems have been extensively studied by many researchers, especially for acoustic and electromagnetic waves. However, many theoretical analysis and numerical computation are left undone for elastic waves due to the complexity of the underlying model equations. The research is multidisciplinary by nature and lies at the interface of mathematics, physics, engineering, and materials sciences. It will contribute towards better understandings of the complex physical and mathematical problems in scattering theory of elasticity. It has significant potential for advancing the frontiers of applied and computational mathematics, and for evolving new mathematics and science. The results of the proposed research activities will be disseminated through publications, seminars, minisymposia, conferences, and workshops. The PI will introduce an advanced graduate course and a graduate seminar series. These will aid in the recruitment and retention of talented students with diverse backgrounds throughout the academic pipeline. The software codes and new course materials developed in the project will be disseminated on a public website and will be available for download by the scientific community. The research and educational components will be integrated together to help to train a new generation of researchers and foster greater awareness and interests in applied and computational mathematics with particular applications to scattering theory among graduate students and postdocs. This project outlines a three-year research plan for developing effective mathematical models, examining fundamental mathematical issues, and designing efficient computational methods for new and important classes of direct and inverse scattering problems in elastic waves. The proposed research builds on the PI?s prior research accomplishments in the area of scattering theory for acoustic and electromagnetic waves. It concerns the following three topics: (1) time-domain obstacle scattering problem; (2) time-harmonic medium scattering problem; (3) inverse random source scattering problem. The mathematical modeling and analysis techniques and computational methods developed in this project will address several key scientific challenges and open problems in direct and inverse scattering theory for elastic waves, which include modeling and computation of the elastic wave propagation in an inhomogeneous medium, numerical solution of the elastic wave equations and well-posedness of the associated model, uniqueness and stability of stochastic inverse source scattering problem. The proposed computational models and tools are highly promising for quantitative study of the complex physical and mathematical problems in elasticity. They have great potentials to provide inexpensive and easily controllable virtual prototypes of the structures in the design and fabrication of novel elastic devices.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.