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

散射问题与非均匀介质对入射场的影响有关。由于在雷达和声纳、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，许多研究人员对散射问题进行了广泛的研究，特别是声波和电磁波的散射问题。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，是数学、物理、工程和材料科学的交叉点。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重要的潜力。拟议的研究活动的结果将通过出版物、讨论会、小型专题讨论会、会议和讲习班传播。PI将引入高级研究生课程和研究生系列研讨会。这将有助于在整个学术管道中招募和留住具有不同背景的有才华的学生。在这个项目中编写的软件代码和新的课程材料将在一个公共网站上分发，供科学界下载。研究和教育部分将整合在一起，以帮助培养新一代的研究人员，并培养研究生和博士后对应用和计算数学的更大认识和兴趣，特别是散射理论的应用。本项目概述了一项为期三年的研究计划，旨在开发有效的数学模型，研究基本数学问题，并设计有效的计算方法来解决弹性波中新的和重要的正散射和逆散射问题。拟议的研究建立在PI的基础上？在声波和电磁波散射理论方面的前期研究成果。它涉及以下三个主题：(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。本项目所开发的数学建模与分析技术和计算方法将解决弹性波正逆散射理论中的若干关键科学挑战和开放性问题，包括弹性波在非均匀介质中的传播建模与计算、弹性波方程的数值解和相关模型的适定性、随机逆源散射问题的唯一性和稳定性。所提出的计算模型和工具对于弹性力学中复杂的物理和数学问题的定量研究具有很大的前景。它们在新型弹性器件的设计和制造中具有很大的潜力，可以提供廉价且易于控制的结构虚拟原型。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。

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.