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Fast solvers of PDEs on a sphere

球面上偏微分方程的快速求解器

基本信息

批准号：
14350045
负责人：
SUGIHARA Masaaki
金额：
$ 5.76万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14350045/
关键词：
surface of sphere numerical solution of partial differential equation spherical harmonic transform FMM(Fast Multipole Method)double Fourier series FFT for non-equispaced data FFT 一般化FMM 不等間隔FFT Gauss点偏微分方程式高速多重極子展開法低階数近似 2重Fo

项目摘要

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.

本项目旨在开发球面上偏微分方程的快速求解器。得到了以下结果：1。FLTSS（Fast Legendre Transform with Scalable Samling）是一套用于快速球面谐波变换的fortran例程，在网站上向公众开放。通过浅水方程的Williamson试验检验了FLTSS的性能。提出了一种基于低秩近似的广义快速多极法预处理算法，该算法可作为加速各种变换计算的基本工具。采用基于对角标度的矩阵范数一致性不等式，并引入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。它使我们能够稳定快速球谐变换。对双傅立叶级数进行了基本研究。具体地说，通过数值实验检验了用DFS逼近函数的精度和用Yea的方法求解球上泊松方程的精度。结果有力地表明，最初为径向基函数理论引入的原生空间适合于待逼近的函数类。对Dutt和Rokhlin引起的非均衡数据进行了快速傅里叶变换（FFT），并通过数值实验进行了验证。结果表明，Dutt和Rokhlin的论文中的正演FFT不能很好地作为正演傅里叶变换的离散版本，而通过简单地改变反FFT中的符号得到的算法给出了很好的结果。提出了并行FFT算法，使我们能够加速上述非均衡数据的重傅立叶级数和FFT的计算。