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以及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意义的数学和计算问题。PI建议建立数学模型，研究数学问题，并为复杂和随机环境中的声波和电磁波传播所产生的新的和重要的正反问题设计计算方法。该项目发展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)波在非均匀介质中传播的多尺度建模和计算；(2)计算随机正、逆散射问题；(3)麦克斯韦方程的数值解和相关模型的适定性；(4)病态逆散射问题的全局唯一性、局部稳定性和数值解。该教育计划旨在培养学生对数学广泛而重要的应用的更多认识，以吸引更多的学生攻读数学的主修、辅修或研究生学位。拟议的教育活动包括：(1)本科生和研究生课程和课程开发；(2)指导本科生、研究生和博士后研究；(3)组织暑期学校、研讨会和研讨会。计算能力的急剧增长和快速算法的发展改变了散射理论领域的科学研究和工业应用的方法论。反过来，实际应用和科学发展也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，准确地计算声场和电磁场，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在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中具有很高的应用前景，并为光学和电磁器件的设计和制造中的结构提供了一个廉价且易于控制的虚拟样机。这项研究本质上是多学科的，位于数学、物理、工程和材料科学的交界处。此外，它还具有极大的潜力来推进应用数学和计算数学的前沿，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.