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Fast solvers of PDEs on a sphere

Research Project

Project/Area Number 14350045
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Engineering fundamentals
Research InstitutionNagoya 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
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥9,000,000 (Direct Cost: ¥9,000,000)
Fiscal Year 2004: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2003: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2002: ¥3,300,000 (Direct Cost: ¥3,300,000)
Keywordssurface 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Fourier級数展開 / 等間隔FFT / 二重指数関数型変数変換 / Bi-CG法 / 浅水方程式 / 乱流シミュレーション / 関数近似 / Poisson方程式
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.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (27 results)

All 2004 2003 Other

All Journal Article (14 results) Publications (13 results)

  • [Journal Article] Recent developments of the Sinc numerical method2004

    • Author(s)
      M.Sugihara, T.Matsuo
    • Journal Title

      J.Comput.Appl.Math. 164-165

      Pages: 673-689

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A variant of the Orthomin(2) method for singular linear systems2004

    • Author(s)
      K.Abe, S.-L.Zhang, T.Mitsui, C.-H.Jin
    • Journal Title

      Numerical Algorithms 36

      Pages: 189-202

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Another Preprocessing Algorithm for Generalized One-Dimensional Fast Multipole Method2004

    • Author(s)
      R.Suda, S.Kuriyama
    • Journal Title

      Journal of Computational Physics 195

      Pages: 790-803

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Recent developments of the Sine numerical method2004

    • Author(s)
      M.Sugihara, T.Matsuo
    • Journal Title

      J.Comput.Appl.Math. Vol.164-165

      Pages: 673-689

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A variant of the Orthomin(2) method for singular linear systems2004

    • Author(s)
      K.Abe, S.-L.Zhang, T.Mitsui, C.-H.Jin
    • Journal Title

      Numerical Algorithms Vol.36

      Pages: 189-202

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Another Preprocessing Algorithm for Generalized One-Dimensional Fast Multipole Method2004

    • Author(s)
      R.Suda, S.Kuriyama
    • Journal Title

      Journal of Computational Physics Vol.195

      Pages: 790-803

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Recent developments of the Sinc numerical method2004

    • Author(s)
      M.Sugihara, T.Matsuo
    • Journal Title

      J.Comput.Appl.Math. 164・165

      Pages: 673-689

    • Related Report
      2004 Annual Research Report
  • [Journal Article] High-order Schemes for Conservative or Dissipative Systems2003

    • Author(s)
      T.Matsuo
    • Journal Title

      J.Comput.Appl.Math. 152

      Pages: 305-317

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A parallel 1-D FFT algorithm for the Hitachi SR80002003

    • Author(s)
      D.Takahashi
    • Journal Title

      Parallel Computing 29

      Pages: 679-690

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] High-order Schemes for Conservative or Dissipative Systems2003

    • Author(s)
      T.Matsuo
    • Journal Title

      J.Comput.Appl.Math. Vol.152

      Pages: 305-317

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A parallel 1-D FFT algorithm for the Hitachi SR80002003

    • Author(s)
      D.Takahashi
    • Journal Title

      Parallel Computing Vol.29

      Pages: 679-690

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Fast spherical harmonic transform routine FLTSS applied to the shallow water test set

    • Author(s)
      R.Suda
    • Journal Title

      Monthly Weather Review (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Fast spherical harmonic transform routine FLTSS applied to the shallowwater test set

    • Author(s)
      R.Suda
    • Journal Title

      Monthly Weather Review (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Fast spherical harmonic transform routine FLTSS applied to the shallow water test set

    • Author(s)
      R.Suda
    • Journal Title

      Monthly Weather Review (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] M.Sugihara: "Near optimality of the sinc approximation"Mathematics of Computation. 72. 767-786 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Sugihara, T.Matsuo: "Recent developments of the Sine numerical methods"Journal of Computational and Applied Mathematics. 164-165. 673-689 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] D.Takahahi: "A parallel 1-D FFT algorithm for the Hitachi SR8000"Parallel Computing. 29. 679-690 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Suda, S.Kuriyama: "Another Preprocessing Algorithm for Generalized One-Dimensional Fast Multipole Method"Journal of Computational Physics. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Suda: "FLTSS-the Fast Legendre Transform with Stable Sampling and interpolation"Proceedings of the Estonian Academy of Science. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Esaki, T.Mitsui, F.A.Oliveira, F.Patricio: "Semi-discretization method alone the characteristics for linear advection diffusion equations"South-East Asia Bull.of Math.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] 三井斌友, 小藤俊, 齊藤善弘: "微分方程式による計算科学入門"共立出版. 214 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] D.Takahashi: "A parallel 1-D FFT algorithm for the Hitachi SR8000"Parallel Computing. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] 須田礼仁: "高速球面調和関数変換ルーチンFLTSS"情報処理学会研究報告. 2002-HPC-02. 19-24 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] R.Suda, M.Takami: "A Fast Spherical Harmonics Transform Algorithm"Mathematics of Computation. 71. 703-715 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Matsuo: "High-order schemes for conservative or dissipative systems"Journal of Computational and Applied Mathematics. 152. 305-317 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.H.Cong, T.Mitsui: "Parallel Predictor-Corrector Iteration of Pseudo Two-step RK Methods for Nonstiff IVPs"Japan Journal of Industrial and Applied Mathematics. 20. 55-70 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Sugihara: "Near optimality of the sinc approximation"Mathematics of Computation. 72. 767-786 (2003)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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