Fast solvers of PDEs on a sphere

该项目旨在开发快速求解器的偏微分方程的一个领域。1.实现了一套快速球谐函数变换的Fortran程序FLTSS（Fast Legendre Transform with Scalable Samling）。通过对浅水方程的威廉姆森检验，检验了FLTSS的性能。本文还提出了一种基于低秩近似的广义快速多极子方法的预处理算法，该算法是加速各种变换计算的基本工具。利用基于对角尺度的矩阵范数一致性不等式，引入α-β乘积作为不稳定性指标，建立了算法误差控制和稳定性分析的一般方法.它使我们能够稳定的快速球谐变换。2.双重傅里叶级数（DFS）的基础研究。具体地说，通过数值实验，使用DFS的函数逼近和是的的方法求解泊松方程的球使用DFS的精度进行了检查。结果表明，径向基函数理论中引入的本征空间适合于这类函数的逼近。3.实现了Dutt和Rokhlin提出的非等间距数据的快速傅里叶变换（FFT），并通过数值实验进行了验证。结果表明，Dutt和Rokhlin的正向FFT不能作为正向傅里叶变换的离散形式，而通过简单地改变逆FFT中的符号得到的算法可以得到很好的结果。4.提出了并行FFT算法，这使得我们能够加速上述非等间距数据的双傅里叶级数和FFT的计算。

This project aims at developing fast solvers of PDEs on a sphere. The following results have been obtained.1.FLTSS(Fast Legendre Transform with Scalable Samling), which is a set of fortran routines for fast spherical harmonics transform, is opened to the public on website. The performance of FLTSS is examined through Williamson's tests for shallow-water equations. A preprocessing algorithm for generalized fast multipole method is also proposed, based on low-rank approximation, which should be a basic tool for accelerating a variety of transform computations. A general method for error control and stability analysis of algorithms is established by employing a consistency inequality of matrix norm based on diagonal scaling, and by introducing the alpha-beta product as an indicator of instability. It enables us to stabilize the fast spherical harmonic transform.2.A basic research for double Fourier series(DFS) is made. Specifically, the accuracy of the function approximation using DFS and that of Yea's method for solving the Poisson equation on a sphere using DFS are examined through numerical experiments. The results strongly suggest that the native space, which has been introduced originally for the theory of radial basis functions, is suited to the class of functions to be approximated.3.Fast Fourier transform(FFT) for non-equispaced data due to Dutt and Rokhlin is implemented and examined through numerical experiments. It is shown that the forward FFT in Dutt and Rokhlin's paper does not work well as a discrete version of the forward Fourier transform, while the algorithm obtained by simply changing the sign in the inverse FFT gives good results.4.Parallel FFT algorithms are proposed, which enable us to accelerate the computation of double Fourier series and FFT for non-equispaced data mentioned above.