CAREER: Fast Algorithms for Oscillatory Integrals

许多与高频波动现象相关的具有挑战性的计算问题可以数学地表示为积分方程组和具有振荡核的变换。PI建议利用多方向性和蝶形计算的思想来开发振荡积分方程组和变换的快速而准确的算法，目标应用于声和电磁散射、数值波传播和反射地震学。所提出的研究活动包括：(1)求解N元体问题和高频声波和电磁散射的边界积分方程组的快速并行算法；(2)计算傅立叶积分算子的最优复杂性算法及其在高频波传播和地震偏移中的应用；(3)仅在低维流形上支持空间和傅立叶样本的稀疏傅里叶变换的最优复杂性算法；(4)部分傅立叶变换的最优复杂性算法，其中频率求和仅限于空间相关域。教育部分包括以下几个部分(1)通过参与建议的研究活动来指导学生和博士后；(2)通过开发一门专注于多尺度和多方向计算中的快速算法的新课程并发布关于这些主题的调查论文来进行课程开发；(3)组织暑期班和系列讲座，旨在向学生展示计算数学的最新发展。通过开发稳健和准确的具有最佳复杂性的数值算法，本研究将显著提高我们理解大规模振荡性质物理问题的能力。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还将与来自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具有挑战性的应用提供操作软件。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，以及(2)提高本科生和代表不足的群体(女性和少数族裔学生)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).