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Study on fast and robust iterative methods for solving large and sparse shifted linear systems arising from computational science

Research Project

Project/Area Number 21760058
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Engineering fundamentals
Research InstitutionAichi Prefectural University

Principal Investigator

SOGABE Tomohiro  愛知県立大学, 大学院・情報科学研究科, 准教授 (30420368)

Project Period (FY) 2009 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2009: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Keywordsシフト線形方程式 / クリロフ部分空間法 / COCR 法 / Bi-CR 法 / 数値解析 / 線形方程式
Research Abstract

We developed some efficient iterative methods for solving large and sparse shifted linear systems that arise from computational science. As a result, the shifted COCR method was proposed for solving complex symmetric case, and a variant of the shifted GMRES method was proposed for (complex) nonsymmetric case. Remarkably, the shifted COCR method was about 26 times faster than the COCR method for a problem arising from large scale electronic structure calculation. As related work, the following results are obtained: (1) An improvement of the COCR method for solving complex symmetric linear systems; (2) Improvements of the IDR method; (3) An improvement of the GMRES(m) method for solving nonsymmetric linear systems; (4) a fast solver for linear systems with a special matrix; (5) Efficient iterative method for generalized shifted linear systems with complex symmetric matrices.

Report

(5 results)
  • 2012 Annual Research Report   Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (34 results)

All 2013 2012 2011 2010 2009 Other

All Journal Article (17 results) (of which Peer Reviewed: 15 results) Presentation (16 results) (of which Invited: 1 results) Remarks (1 results)

  • [Journal Article] An efficient variant of the restarted shifted GMRES for solving shifted linear systems2013

    • Author(s)
      A. Imakura, T. Sogabe, and S.-L. Zhang
    • Journal Title

      J. Math. Res. Appl.

      Volume: 33 Pages: 127-141

    • Related Report
      2012 Annual Research Report 2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A novel algorithm and its parallelization for solving nearly penta-diagonal linear systems2013

    • Author(s)
      J. Jia and T. Sogabe
    • Journal Title

      Int. J. Comput. Math.

      Volume: 90 Pages: 435-444

    • Related Report
      2012 Final Research Report
  • [Journal Article] A novel algorithm and its parallelization for solving nearlypenta-diagonal linear systems2013

    • Author(s)
      J. Jia and T. Sogabe
    • Journal Title

      Int. J. Comput. Math.

      Volume: 90 Issue: 2 Pages: 435-444

    • DOI

      10.1080/00207160.2012.725845

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An efficient variant of the GMRES(m) method based on error equations2012

    • Author(s)
      A. Imakura, T. Sogabe, and S.-L. Zhang
    • Journal Title

      East Asia J. on Appl. Math.

      Volume: 2 Issue: 1 Pages: 19-32

    • DOI

      10.4208/eajam.280611.030911a

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Solution of generalized shifted linear systems with complex symmetric matrices2012

    • Author(s)
      T. Sogabe, T. Hoshi, S.-L. Zhang, and T. Fujiwara
    • Journal Title

      J. Comput. Phys.

      Volume: 231 Issue: 17 Pages: 5669-5684

    • DOI

      10.1016/j.jcp.2012.04.046

    • Related Report
      2012 Annual Research Report 2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] An efficient variant of the GMRES(m) method based on error equations2012

    • Author(s)
      A.Imakura, T. Sogabe, S.-L.Zhang
    • Journal Title

      East Asia J.on Appl.Math.

      Volume: 2 Pages: 19-32

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides2011

    • Author(s)
      L. Du, T. Sogabe, B. Yu, Y. Yamamoto, and S.-L. Zhang
    • Journal Title

      J. Comput. Appl. Math.

      Volume: 235 Issue: 14 Pages: 4095-4106

    • DOI

      10.1016/j.cam.2011.02.035

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] An extension of the COCR method to solving shifted linear systems with complex symmetric matrices2011

    • Author(s)
      T. Sogabe, S.-L. Zhang
    • Journal Title

      East Asia J. on Appl. Math.

      Volume: 1 Issue: 2 Pages: 97-107

    • DOI

      10.4208/eajam.260410.240510a

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A variant of the IDR(s) method with quasi-minimal residual strategy2011

    • Author(s)
      L. Du, T. Sogabe, and S.-L. Zhang
    • Journal Title

      J. Comput. Appl. Math

      Volume: 236 Issue: 5 Pages: 621-630

    • DOI

      10.1016/j.cam.2011.07.027

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] An extension of the COCR method to solving shifted linear systems with complex syimnetric matrices2011

    • Author(s)
      T.Sogabe, S.-L.Zhang
    • Journal Title

      East Asia J.on Appl.Math.

      Volume: 1 Pages: 97-107

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides2011

    • Author(s)
      L.Du, T.Sogabe, B.Yu, Y.Yamamoto, S.-L.Zhang
    • Journal Title

      J.Comput.Appl.Math.

      Volume: 235 Pages: 4095-4106

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Quasi-minimal residual smoothing technique for the IDR (s) method2011

    • Author(s)
      Lei Du, Tomohiro Sogabe, Shao-Liang Zhang
    • Journal Title

      JSIAM Letters

      Volume: 3 Pages: 13-16

    • NAID

      130000674237

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Notes on particular symmetric polynomials with applications2010

    • Author(s)
      M.E.A.El-Mikkawy, T.Sogabe
    • Journal Title

      Appl.Math.Comput.

      Volume: 215 Pages: 3311-3317

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 大規模シフト線形方程式の数値解法-クリロフ部分空間の性質に着目して-2009

    • Author(s)
      曽我部知広,張紹良
    • Journal Title

      応用数理

      Volume: 19 Pages: 27-42

    • NAID

      110007361021

    • URL

      http://ci.nii.ac.jp/naid/110007361021

    • Related Report
      2012 Final Research Report
  • [Journal Article] Lanczos- type variants of the COCR method for complex nonsymmetric linear systems2009

    • Author(s)
      Y.-F. Jing, T.-Z. Huang, Y. Zhang, L. Li, G.-H. Cheng, Z.-G. Ren, Y. Duan, T. Sogabe, and B Carpentieri
    • Journal Title

      J. Comput. Phys.

      Volume: 228 Issue: 17 Pages: 6376-6394

    • DOI

      10.1016/j.jcp.2009.05.022

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] 大規模シフト線形方程式の数値解法-クリロフ部分空間の性質に着目して-2009

    • Author(s)
      曽我部知広, 張紹良
    • Journal Title

      日本応用数理学会誌「応用数理」

      Volume: 19 Pages: 27-42

    • NAID

      110007361021

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a problem related to the Vandermonde determinant2009

    • Author(s)
      T.Sogabe, M.E.A.El-Mikkawy
    • Journal Title

      Discrete Appl.Math.

      Volume: 157 Pages: 2997-2999

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Krylov subspace methods for solving generalized shifted liner systems2012

    • Author(s)
      T.Sogabe, T.Hoshi, S.L.Zhang, T.Fujiwara
    • Organizer
      International Workshop on omputational Science and Numerical Analysis
    • Place of Presentation
      電気通信大学,東京
    • Year and Date
      2012-03-25
    • Related Report
      2011 Annual Research Report
  • [Presentation] Krylov subspace methods for solving generalized shifted linear sys tems2012

    • Author(s)
      T. Sogabe, T. Hoshi, S.-L. Zhang, and T. Fujiwara
    • Organizer
      International Workshop on Compu tational Science and Numerical Analysis, The University of Electro-Communi cations
    • Place of Presentation
      Tokyo, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] Numerical methods for genera lized shifted linear systems with comp lex symmetric matrices2012

    • Author(s)
      T. Sogabe, T. Hoshi, S.-L. Zhang, and T. Fujiwara
    • Organizer
      4th China -Japan-Korea Conference on Numerical Mathematics
    • Place of Presentation
      Otsu City, Shiga Prefec ture, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] Numerical methods for generalized shifted linear systems2012

    • Author(s)
      T. Sogabe, T. Hoshi, S.-L. Zhang and T. Fujiwara
    • Organizer
      4th China-Japan-Korea Conference on Numerical Mathematics
    • Place of Presentation
      ピアザ淡海(大津市)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 一般化シフト線形方程式に対する重み付き準最小残差法2010

    • Author(s)
      曽我部知広, 星健夫, 張紹良, 藤原毅夫
    • Organizer
      京都大学数理解析研究所 研究集会「科学技術計算アルゴリズムの数理的基盤と展開」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-10-19
    • Related Report
      2010 Annual Research Report
  • [Presentation] On a cost-efficient iteration of Ye's hybrid method for minimizing a quadratic function over a sphere constraint2010

    • Author(s)
      T.Sogabe, A.Suzuki, S.-L.Zhang
    • Organizer
      2nd IMA Conference on Numerical Linear Algebra and Optimisation
    • Place of Presentation
      バーミンガム大,英国
    • Year and Date
      2010-09-13
    • Related Report
      2009 Annual Research Report
  • [Presentation] On a cost-efficient iteration of Ye's hybrid method for minimizing a quadratic function over a sphere constraint2010

    • Author(s)
      T. Sogabe, A. Suzuki,and S.-L. Zhang
    • Organizer
      2nd IMA Conference on Numerical Linear Algebra and Optimisation
    • Place of Presentation
      University of Birmingham, UK
    • Related Report
      2012 Final Research Report
  • [Presentation] 複素対称行列を係数に持つ一般化シフト線形方程式の高速解法2009

    • Author(s)
      曽我部知広, 星健夫, 張紹良, 藤原毅夫
    • Organizer
      京都大学数理解析研究所 研究集会「数値解析と数値計算アルゴリズムの最近の展開」
    • Place of Presentation
      京大会館
    • Year and Date
      2009-12-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] 離散wavelet変換を用いた正定値対称行列のためのFSAI前処理2009

    • Author(s)
      早戸拓也, 今倉暁, 曽我部知広, 張紹良
    • Organizer
      日本応用数理学会2009年度年会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-30
    • Related Report
      2009 Annual Research Report
  • [Presentation] 球面制約付き二次計画問題の効率的解法について2009

    • Author(s)
      鈴木綾華, 曽我部知広, 張紹良
    • Organizer
      第38回数値解析シンポジウム
    • Place of Presentation
      熱川ハイツ, 静岡
    • Year and Date
      2009-06-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] An approach to fast solution of generalized shifted linear systems with complex symmetric matrices2009

    • Author(s)
      T.Sogabe, S.-L.Zhang
    • Organizer
      Numerical Analysis and Scientific Computing with Applications
    • Place of Presentation
      アガディール, モロッコ
    • Year and Date
      2009-05-21
    • Related Report
      2009 Annual Research Report
  • [Presentation] An approach to fast solution of generalized shifted linear systems with complex symmetric matrices2009

    • Author(s)
      T. Sogabe and S.-L. Zhang
    • Organizer
      Numerical Analysis and Scienti fic Computing with Applications (NASCA 2009)
    • Place of Presentation
      Agadir, Morocco
    • Related Report
      2012 Final Research Report
  • [Presentation] A fast solver for generalized shifted linear systems with complex symmetric matrices2009

    • Author(s)
      T. Sogabe and S.-L. Zhang
    • Organizer
      The 7th International Conference on Numerical Optimization and Numerical Linear Algebra
    • Place of Presentation
      Lijiang, Yunnan, China
    • Related Report
      2012 Final Research Report
  • [Presentation] An efficient method for minimizing a quad ratic function with a sphere constraint, International Symposium of Electronic Structure Calculations -Theory2009

    • Author(s)
      S. Suzuki, T. Sogabe, and S.-L. Zhang
    • Organizer
      Correla ted and Large Scale Systems and Nume rical Methods-
    • Place of Presentation
      Tokyo, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] A block IDR(s) method for nonsymmetric lin ear systems with multiple right-hand sides2009

    • Author(s)
      L. Du, T. Sogabe, and S.-L. Zhang
    • Organizer
      International Symposium of Electronic Structure Cal culations Theory, Correlated and Large Scale Sys tems and Numerical Methods-
    • Place of Presentation
      Tokyo, Japan
    • Related Report
      2012 Final Research Report
  • [Presentation] A study on the restart of the GMRES(m) method for solving nonsymmetric linear systems2009

    • Author(s)
      A. Imakura, T. Sogabe, and S.-L. Zhang
    • Organizer
      International Symposium of Electronic Structure Calculations-The ory, Correlated and Large Scale Systems and Numerical Methods-
    • Place of Presentation
      Tokyo, Japan
    • Related Report
      2012 Final Research Report
  • [Remarks]

    • URL

      http://www.ist.aichi-pu.ac.jp/person/sogabe/paper_s.html

    • Related Report
      2012 Final Research Report

URL: 

Published: 2009-04-01   Modified: 2019-07-29  

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