CAREER: Fast Algorithms for Oscillatory Integrals

许多具有挑战性的计算问题有关的高频波现象可以制定数学积分方程和变换与振荡内核。PI建议利用多方向性和蝴蝶计算的思想来开发振荡积分方程和变换的快速准确算法，并有针对性地应用于声学和电磁散射，数值波传播和反射地震学。拟议的研究活动包括：（1）高频声电磁散射的N体问题和边界积分方程的快速并行算法，（2）计算Fourier积分算子的最佳复杂度算法及其在高频波传播和地震偏移中的应用，（3）稀疏傅立叶变换的最佳复杂度算法，其中空间和傅立叶样本仅支持在低维流形上，以及（4）部分傅里叶变换的最佳复杂度算法，其中频率求和被限制到空间相关域。教育部分包括以下组成部分：（1）通过参与拟议的研究活动指导学生和博士后，（2）通过开发一门新课程来开发课程，该课程侧重于多尺度和多方向计算中的快速算法，并发表关于这些主题的调查论文，以及（3）组织暑期学校和系列讲座，旨在向学生介绍计算数学的最新发展。通过开发具有最佳性能的稳健而准确的数值算法，复杂性，这项研究将显着提高我们的理解能力的大尺度物理问题的振荡性质。在拟议的研究活动中开发的算法将直接应用于声学和电磁散射，反射地震学和医学成像。PI还将与工业和政府实验室的研究人员密切合作，传播想法，并为现实的挑战性应用提供操作软件。新一代数值算法和软件的发展要求研究人员理解计算数学的不同方面。研究和教育部分将整合在一起，以（1）帮助培养掌握算法设计，数学分析和软件开发的新一代研究人员，（2）促进本科生和代表性不足的群体（女性和少数民族学生）对计算数学的认识和兴趣。

Many challenging computational problems related to high frequency wave phenomena can be formulated mathematically as integral equations and transforms with oscillatory kernels. The PI proposes to leverage the ideas of multidirectionality and butterfly computation to develop fast and accurate algorithms for oscillatory integral equations and transforms, with targeted applications in acoustic and electromagnetic scattering, numerical wave propagation, and reflection seismology. The proposed research activities include: (1) fast and parallel algorithms for the $N$-body problems and the boundary integral equations of high frequency acoustic and electromagnetic scattering, (2) an optimal complexity algorithm for computing Fourier integral operators with applications in high frequency wave propagation and seismic migration, (3) an optimal complexity algorithm for sparse Fourier transform where the spatial and Fourier samples are supported only on a low dimensional manifold, and (4) an optimal complexity algorithm for partial Fourier transform where the frequency summation is restricted to a spatial dependent domain. The education part of includes the following components (1) mentoring students and postdocs through participating the proposed research activities, (2) curriculum development through developing a new course that focuses on the fast algorithms in multiscale and multidirectional computation and publishing survey papers on these topics, and (3) organizing summer schools and lecture series that aim to present students with recent developments in computational mathematics.Through developing robust and accurate numerical algorithms with optimal complexity, this research will significantly improve our ability of understanding large scale physical problems of oscillatory nature. The algorithms to be developed in the proposed research activities will have direct applications in acoustic and electromagnetic scattering, reflection seismology, and medical imaging. The PI will also work closely with researchers from industrial and government laboratories to disseminate ideas and deliver operational softwares for realistic challenging applications. The development of a new generation of numerical algorithms and softwares requires researchers to understand different aspects of computational mathematics. The research and educational components will integrate together to (1) help train a new generation of researchers who master algorithmic design, mathematical analysis, and software development, and (2) promote the awareness and interests in computational mathematics among undergraduates and underrepresented groups (female and minority students).