2004 Fiscal Year Final Research Report Summary
Fast solvers of PDEs on a sphere
| Project/Area Number |
14350045
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Engineering fundamentals
|
|---|
| Research Institution | Nagoya University |
|---|
Principal Investigator |
SUGIHARA Masaaki Nagoya University, Graduate School of Information Science, Professor, 大学院・工学研究科, 教授 (80154483)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MITSUI Taketomo Nagoya University, Graduate School of Information Science, Professor, 大学院・情報科学研究科, 教授 (50027380)
MATSUO Takayasu The University of Tokyo, Graduate School of Interdisciplianry Information Studies, Assistant Professor, 大学院・情報学環, 講師 (90293670)
SUDA Reiji The University of Tokyo, Graduate School of Information Science and Technology, Associate Professor, 大学院・情報理工学系研究科, 助教授 (40251392)
TAKAHASHI Daisuke University of Tsukuba, Institute of Information Sciences and Electronics, Associate Professor, 電子・情報工学系, 助教授 (00292714)
|
|---|
| Project Period (FY) |
2002 – 2004
|
| Keywords | 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 |
|---|
| Research Abstract |
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.
|
Research Products
(12 results)
