CAREER: Direct and Inverse Scattering Problems for Wave Propagation in Complex and Random Environments

在散射理论中，由于材料性质的复杂性和物理模型和参数的不确定性，精确的建模和精确的计算提出了具有挑战性和重大的数学和计算问题。PI建议开发数学模型，检查数学问题，并设计计算方法，用于复杂和随机环境中声波和电磁波传播产生的新的重要类别的直接和逆问题。本项目开发的数学建模技术和计算方法解决了应用数学和计算数学中的几个关键科学挑战，包括：(1)非均质介质中波传播的多尺度建模和计算；(2)计算随机正反散射问题；(3)麦克斯韦方程组的数值解和相关模型的适定性；(4)不适定逆散射问题的全局唯一性、局部稳定性和数值解。这一教育计划旨在提高人们对数学广泛而重要的应用的认识，以吸引更多的学生攻读数学专业、辅修专业或研究生学位。建议的教育活动包括：(1)本科和研究生课程和课程开发；(2)指导本科生、研究生和博士后科研工作；（三）组织暑期学校、研讨会和讲习班。计算能力的急剧增长和快速算法的发展已经改变了散射理论领域的科学研究和工业应用方法。反过来，实际应用和科学发展推动了对更复杂的数学模型和数值算法的需求，以描述复杂结构的散射，并准确计算声场和电磁场，从而预测给定结构的性能，以及进行新结构的优化设计。所提出的计算模型和工具对于光学和电磁学中复杂的物理和数学问题的定性和定量研究具有很大的前景，并为光学和电磁学器件的设计和制造提供了一种廉价且易于控制的结构虚拟原型。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。此外，它具有推动应用数学和计算数学前沿的巨大潜力，甚至对其他科学分支产生影响。

In scattering theory, due to the complexity of material properties and uncertainty in physical models and parameters, precise modeling and accurate computing present challenging and significant mathematical and computational questions. The PI proposes to develop mathematical models, examine mathematical issues, and design computational methods for new and important classes of direct and inverse problems that arise from the acoustic and electromagnetic wave propagation in complex and random environments. The mathematical modeling techniques and computational methods developed in this project address several key scientific challenges in applied and computational mathematics, which include: (1) multi-scale modeling and computation of the wave propagation in a heterogeneous medium; (2) computational stochastic direct and inverse scattering problems; (3) numerical solution of Maxwell's equations and well-posedness of associated models; (4) global uniqueness, local stability, and numerical solution of the ill-posed inverse scattering problems. The educational plan is to foster greater awareness of the broad and important applications of mathematics so as to attract more students in pursuing a major, a minor, or a graduate degree in mathematics. The proposed education activities include: (1) undergraduate and graduate courses and curriculum development; (2) mentoring of undergraduate, graduate, and postdoc research; (3) organizing summer schools, seminars, and workshops. The dramatic growth of computational capability and the development of fast algorithms have transformed the methodology for scientific investigation and industrial applications in the field of scattering theory. Reciprocally, the practical applications and scientific developments have driven the need for more sophisticated mathematical models and numerical algorithms to describe the scattering of complicated structures, and to accurately compute acoustic and electromagnetic fields and thus to predict the performance of a given structure, as well as to carry out optimal design of new structures. The proposed computational models and tools are highly promising for qualitative and quantitative study of the complex physical and mathematical problems in optics and electromagnetics, and provide an inexpensive and easily controllable virtual prototype of the structures in the design and fabrication of optical and electromagnetic devices. The research is multidisciplinary by nature and lies at the interface of mathematics, physics, engineering, and materials sciences. In addition, it has significant potential to advance the frontiers of applied and computational mathematics, and even to have impact on other branches of science.