(弱、高阶奇异核)高频振荡边界积分方程的快速多极子与Chebyshev谱高精度算法研究

The main focus of this proposal is on fast and high accuracy numerical methods for solving some highly oscillatory boundary integral equations with weakly or hyper-singulary kernels. These equations arise in many applications of physical and engineering interest, such as retarded potential integral equations and scattering problems on acoustic waves and electromagnetic waves.. High performance computing of these equations has attracted a growing interest from scientific computation community, and is a very important, hot and difficult research topic. There are a series of challenging computational problems that will require mathematical insights to solve.. The fast multipole method (FMM), introduced by Rokhlin and Greengard, has been said to be one of the top ten algorithms of the 20th century, a numerical technique based on the multipole expansion of the Green's function, which reduces the complexity of matrix-vector multiplication involving a certain type of dense matrix from O(N^2) to O(N).. Spectral methods, along with finite difference and finite element methods, are one of the three main methodologies for solving partial differential equations on computers, often involving the use of the Fast Fourier Transform.. One feature of these highly oscillatory boundary integral equations with weakly or hyper-singulary kernels is of particular relevance. The kernel is more highly oscillatory as the frequency increases. The problems of evaluating oscillatory integrals and of solving oscillatory boundary integral equations are highly correlated. Efficient FMM is based on fast convergence of the multipole expansion of the highly oscillatory Green's function, then on efficient evaluation of a large number of (singularly) oscillatory integrals with complicated oscillators, finally on GMRES. Thus, numerical approaches for these equations by conventional FMMs are exceedingly difficult and the cost steeply increases as the frequency increases. In particular, the numerical methods based on standard numerical quadrature formulas, such as Gauss, Clenshaw-Curtis and Nyström, are not feasible for calculating these integrals.. Here, we will take advantage of FMMs, high accuracy spectral methods and efficient approximation on such highly oscillatory integrals with singularities involved in FMMs, to develop fast implementation for solving such highly oscillatory boundary integral equations with weakly or hyper-singulary kernels.. Applying orthogonal polynomial theory together with the asymptotics on highly oscillatory special Green’s function, we will consider fast convergent multipole expansions on these functions, and devote to develop new quadratures for such integrals. These are particularly challenging examples of the general problem of computing oscillatory integrals, which has a long history and has seen a great deal of research interest in recent years.. Furthermore, we will analyze good preconditioners for the GMRESs based on the asymptotics of the integrals, and design efficient spectral methods for some above boundary integral equations. The main difficulty for the spectral method is how to construct the quasi-spectral matrices for such integral equations.. Together with the above analysis and efficient methods, error bounds and convergence rates will be studied in detail.

本项目主要研究来源于物理、工程问题的(弱、高阶奇异核)高频振荡边界积分方程的快速计算与高精度算法，涉及延迟位势方程、声散射与电磁散射的数值模拟。它的高效计算被公认为是科学计算领域非常重要、难的国际热点研究课题，存在许多挑战性问题亟待解决。在此，我们将充分应用快速多极子（FMM）、Chebyshev谱方法、高频振荡积分的高性能计算等相关算法，研究高频Green函数的多极展开、针对这些方程的FMM离散格式GMRES的预处理、（一维、二维）奇异复杂振荡子高频振荡积分的高性能计算，以此，构造高频振荡边界积分方程快速算法与高精度谱方法，这些新算法的特点是算法复杂性不随着频率的增高而增加，且振荡频率越高，算法精度愈高。高频振荡边界积分方程的高效数值方法的特点，正如剑桥大学Iserles教授所言，恰当的方法可以得到很好的计算结果。

项目主要研究来源于物理、工程问题的弱、高阶奇异高频振荡边界积分方程的快速计算与高精度算法及收敛性分析，它的高效计算被公认为是科学计算领域难的研究课题，存在许多挑战性问题亟待解决。..首次从正交多项式为高频振荡函数的角度出发，建立了正交展开的谱系数与高频振荡积分的联系，针对高频贝塞尔变换发展了新型van der Corput引理，得到了新的雅可比正交多项式的Hilb估计公式，给出了雅可比谱展开系数及其导数相互间的关系式，基于达布、塞戈提出的Hilb公式，针对代数或对数奇异等有限正则函数建立了谱展开系数最优的衰减速度估计，进而得到谱展开在无穷范数或索伯列夫空间框架下的最优收敛速度，为谱元或hp等方法的收敛性分析提供了新的技巧和手段。..针对弱奇异高频振荡广义傅里叶变换提出了新型Levin方法，其可达到关于大波数的任意阶渐进精度，并具有超线性收敛性，完全克服了经典Levin方法求解奇异、高频振荡微分方程不稳定、精度低的缺点，且首次从理论上阐明了无边界约束的Levin微分方程，对含驻点情形其解均是高频振荡的，通过引入扩展系数、奇异分离变换，诱导出Levin基础微分方程组，其含有唯一非振荡解，利用稀疏谱方法、TSVD、矩阵预处理，得到高精度计算。..针对高阶奇异振荡积分提出了高精度谱方法。这些高效算法结合快速多极，应用于延迟位势方程等散射问题—高频振荡边界积分方程的快速计算和高精度数值模拟：针对大波数Burton- Miller双层边界积分方程，分析了对角FMM的高效性；针对声波散射问题中单层延迟位势大波数高频振荡边界积分方程形式，发展了傅里叶、贝塞尔双振荡子基于FFT的高精度谱方法，对时谐波等三种入射波进行了具有谱精度的反演；基于弱奇异函数的多级展开，研究了一些高频振荡积分方程的高效算法。. .“正交多项式逼近与高频振荡问题的渐进分析与高效算法”获教育部2020年度高等学校科学研究—自然科学二等奖。

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