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Developments of new linear solvers based on the induced dimension reduction theorem

Research Project

Project/Area Number 22560067
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Engineering fundamentals
Research InstitutionGifu Shotoku Gakuen University

Principal Investigator

ABE Kuniyoshi  岐阜聖徳学園大学, 経済情報学部, 教授 (10311086)

Co-Investigator(Kenkyū-buntansha) FUJINO Seiji  九州大学, 情報基盤センター, 教授 (40264965)
Co-Investigator(Renkei-kenkyūsha) ISHIWATA Emiko  東京理科大学, 理学部, 教授 (30287958)
NAKAJIMA Kengo  東京大学, 情報基盤センター, 教授 (20376528)
Research Collaborator SLEIJPEN Gerard L,G.  Utrecht University, Department of Math., Associate Professor
Project Period (FY) 2010 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2010: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywords線形方程式 / Krylov 空間法 / 帰納的次元縮小(IDR)定理 / IDR(s)法 / 一般化積型解 法 (GPBiCG) / IDRstab 法 / スムージング技法 / ブロック IDR(s)法 / Krylov空間法 / 帰納的次元縮小(IDR)定理 / 一般化積型解法 / IDRstab法 / ブロックIDR(s)法 / Krylow空間法 / 複数右辺項 / 帰納的次元縮小定理 / Induced Dimension Reduction法 / 一般化積型解法(GPBiCG)
Research Abstract

Krylov subspace methods, specifically hybrid Bi-CG methods, have been developed in the 1990s, and some advantages with the hybrid BiCG methods are well-known. A new method which is called the Induced Dimension Reduction (abbreviated to IDR) (s) has been proposed in 2007. It has recently reported that the IDR(s) method sometimes converges faster than the hybrid Bi-CG methods. Many researchers have focused on the IDR(s) method. Therefore, by combining the advantage of IDR(s) with that of the hybrid Bi-CG methods, we have proposed new solvers which have better convergence than the hybrid Bi-CG and original IDR(s) methods.

Report

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

    (25 results)

All 2013 2012 2011 2010

All Journal Article (13 results) (of which Peer Reviewed: 11 results) Presentation (12 results)

  • [Journal Article] Solving linear equations with a stabilized GPBiCG method2013

    • Author(s)
      K. Abe, G. Sleijpen
    • Journal Title

      Applied Numerical Mathematics

      Volume: 67 Pages: 4-16

    • Related Report
      2012 Annual Research Report 2012 Final Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hybrid Bi-CG methods with a Bi-CG formulation closer to the IDR2012

    • Author(s)
      K. Abe, G. Sleijpen
    • Journal Title

      approach

      Volume: 208 Pages: 10889-10899

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A BiCGStab2 variant of IDR(s) for solving linear equations2012

    • Author(s)
      K. Abe, G. Sleijpen
    • Journal Title

      AIP Conference Proceedings

      Volume: 1479 Pages: 741-744

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] IDRstab法へのスムージングの適用とその収束性2012

    • Author(s)
      相原研輔,阿部邦美,石渡恵美子
    • Journal Title

      数理解析研究所講究録

      Volume: 1791 Pages: 11-22

    • Related Report
      2012 Final Research Report
  • [Journal Article] Hybrid Bi-CG methods with a Bi-CG formulation closer to the IDR approach2012

    • Author(s)
      Kuniyoshi Abe, Gerard Sleijpen
    • Journal Title

      Applied Mathematics and Computation

      Volume: 208 Issue: 22 Pages: 10889-10899

    • DOI

      10.1016/j.amc.2012.04.049

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A BiCGStab2 variant of IDR(s) for solving linear equations2012

    • Author(s)
      Kuniyoshi Abe, Gerard Sleijpen
    • Journal Title

      ICNAAM 2012 proceedings, AIP Conference Proceedings

      Volume: 1479 Pages: 741-744

    • DOI

      10.1063/1.4756242

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] IDRstab法へのスムージングの適用とその収束性2012

    • Author(s)
      相原研輔,阿部邦美,石渡恵美子
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 1791 Pages: 11-22

    • Related Report
      2012 Annual Research Report
  • [Journal Article] An alternative implementation of the IDRstab method saving vector updates2011

    • Author(s)
      K. Aihara, K. Abe, E. Ishiwata
    • Journal Title

      JSIAM Letters

      Volume: 3 Pages: 69-72

    • NAID

      130001390409

    • Related Report
      2012 Final Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] BiCR Variants of hybrid BiCG methods for solving linear systems with nonsymmetric matrices2010

    • Author(s)
      K. Abe, G. Sleijpen
    • Journal Title

      J. of Comput. And Appl. Math.

      Volume: 234 Pages: 985-994

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A variable preconditioned GCR method using the GSOR method for singular linear systems2010

    • Author(s)
      D. Aoto, E. Ishiwata, K. Abe
    • Journal Title

      J. of Comput. And Appl. Math.

      Volume: 234 Pages: 703-712

    • NAID

      110006793758

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A strategy for reducing the inner iteration counts for the variable preconditioning GCR(m) method2010

    • Author(s)
      K. Aihara, E. Ishiwata, K. Abe
    • Journal Title

      JSIAM Letters

      Volume: 2 Pages: 77-80

    • Related Report
      2012 Final Research Report 2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] BiCR Variants of hybrid BiCG methods for solving linear systems with nonsymmetric matrices2010

    • Author(s)
      K.Abe, G.Sleijpen
    • Journal Title

      J.of Comput.And Appl.Math.

      Volume: 234 Pages: 985-994

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A variable preconditioned GCR method using the GSOR method for singular linear systems2010

    • Author(s)
      D.Aoto, E.Ishiwata, K.Abe
    • Journal Title

      J.of Comput.And Appl.Math.

      Volume: 234 Pages: 703-712

    • NAID

      110006793758

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] IDR(s) for linear equations with multiple right hand sides2012

    • Author(s)
      K. Abe
    • Organizer
      International Conference of Applied and Computational Math.
    • Place of Presentation
      Ankara, Turkey
    • Year and Date
      2012-10-05
    • Related Report
      2012 Annual Research Report 2012 Final Research Report
  • [Presentation] A BiCGStab2 variant of IDR(s) for solving linear equations2012

    • Author(s)
      K. Abe
    • Organizer
      International Conference of Numerical Analysis and Applied Math.
    • Place of Presentation
      Kos, Greece
    • Year and Date
      2012-09-20
    • Related Report
      2012 Annual Research Report 2012 Final Research Report
  • [Presentation] IDR(s) for linear equations with multiple right-hand sides2012

    • Author(s)
      Kuniyoshi Abe, Gerard Sleijpen
    • Organizer
      Numerical Linear Algebra - Algorithms, Applications, and Training
    • Place of Presentation
      Delft, The Netherlands
    • Related Report
      2012 Annual Research Report
  • [Presentation] GPBiCG 法の改良2011

    • Author(s)
      阿部邦美
    • Organizer
      大規模計算コロキウム
    • Place of Presentation
      岐阜じゅうろくプラザ
    • Year and Date
      2011-09-09
    • Related Report
      2012 Final Research Report
  • [Presentation] GPBiCG法の改良2011

    • Author(s)
      阿部邦美
    • Organizer
      大規模計算コロキウム
    • Place of Presentation
      岐阜市,岐阜じゅうろくプラザ
    • Year and Date
      2011-09-09
    • Related Report
      2011 Annual Research Report
  • [Presentation] Alternative implementations of the hybrid Bi-CG methods for linear equations2011

    • Author(s)
      K. Abe
    • Organizer
      10thIMACS International Conference on Iterative Methods in Scientific Computing
    • Place of Presentation
      Marrakech, Morocco
    • Year and Date
      2011-05-20
    • Related Report
      2012 Final Research Report
  • [Presentation] Alternative implementations of the hybrid Bi-CG methods for linear equations2011

    • Author(s)
      Kuniyoshi Abe
    • Organizer
      10^<th> IMACS International Conference on Iterative Methods in Scientific Computing
    • Place of Presentation
      Marrakech, Morocco
    • Year and Date
      2011-05-20
    • Related Report
      2011 Annual Research Report
  • [Presentation] Solving linear equations with a stabilized GPBiCG method2011

    • Author(s)
      K. Abe
    • Organizer
      湖南商学院情報学科
    • Place of Presentation
      中国湖南省湖南商学院
    • Year and Date
      2011-04-14
    • Related Report
      2012 Final Research Report 2011 Annual Research Report
  • [Presentation] Solving linear equations with a stabilized GPBiCG method2011

    • Author(s)
      K. Abe
    • Organizer
      第14回 環瀬戸内応用数理研究部会シンポジウム
    • Place of Presentation
      岡山理科大学
    • Year and Date
      2011-01-22
    • Related Report
      2012 Final Research Report
  • [Presentation] Solving linear equations with a stabilized GPBiCG method2011

    • Author(s)
      K.Abe, G.Sleijpen
    • Organizer
      第14回環瀬戸内応用数理研究部会シンポジウム
    • Place of Presentation
      岡山市, 岡山理科大学
    • Year and Date
      2011-01-22
    • Related Report
      2010 Annual Research Report
  • [Presentation] A stabilized GPBiCG method with a strategy to remedy accuracy of Bi-CG coefficients for solving linear systems2010

    • Author(s)
      K. Abe
    • Organizer
      Conference In Numerical Analysis 2010
    • Place of Presentation
      Creta, Greece
    • Year and Date
      2010-09-18
    • Related Report
      2012 Final Research Report
  • [Presentation] A stabilized GPBiCG method with a strategy to remedy accuracy of Bi-CG coefficients for solving linear systems2010

    • Author(s)
      K.Abe, G.Sleijpen
    • Organizer
      Conference In Numerical Analysis 2010
    • Place of Presentation
      Chania, Creta, Greece
    • Year and Date
      2010-09-18
    • Related Report
      2010 Annual Research Report

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Published: 2010-08-23   Modified: 2019-07-29  

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