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A study on consistent preconditioners for iterative solutions of large-scale linear systems

Research Project

Project/Area Number 25390145
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Computational science
Research InstitutionTokyo Denki University (2015-2016)
The University of Tokyo (2013-2014)

Principal Investigator

Itoh Shoji  東京電機大学, 理工学部, 研究員 (70333482)

Co-Investigator(Renkei-kenkyūsha) SUGIHARA Masaaki  青山学院大学, 理工学部, 教授 (80154483)
Project Period (FY) 2013-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2013: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Keywords大規模行列計算 / クリロフ部分空間法 / 前処理系 / 双ランチョス / 数理工学 / 応用数学 / 解析・評価 / クリロフ部分空間
Outline of Final Research Achievements

In this study, we analyzed preconditioned bi-Lanczos iterative algorithms, which assume the existence of a dual system. Various iterative methods are often used in conjunction with preconditioning that improve the properties of linear equations; such typical algorithms are preconditioned CGS (PCGS) or preconditioned BiCG (PBiCG). Especially, we discussed a variety of PCGS algorithms including improved PCGS, by comparing two typical PCGS, and we analyzed the structure of the solution vector for each Krylov subspace. Further, we analyzed the structures on the polynomials of the PBiCG algorithms that correspond to the above PCGS. The improved PCGS has been verified as a coordinative to the left-PCGS, and it has the advantages of both the conventional and the left-PCGS. And, we showed that the direction of the preconditioned system of the bi-Lanczos algorithms can be switched by the construction and setting of the initial residual vector of the dual system.

Report

(5 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (11 results)

All 2016 2015 2014 Other

All Journal Article (3 results) (of which Open Access: 3 results,  Peer Reviewed: 1 results,  Acknowledgement Compliant: 1 results) Presentation (5 results) (of which Invited: 1 results) Remarks (3 results)

  • [Journal Article] The structure of the Krylov subspace in various preconditioned CGS algorithms2016

    • Author(s)
      Shoji Itoh, Masaaki Sugihara
    • Journal Title

      arXiv

      Volume: 1603 Pages: 1-21

    • Related Report
      2016 Annual Research Report
    • Open Access
  • [Journal Article] The structure of the polynomials in preconditioned BiCG algorithms and the switching direction of preconditioned systems2016

    • Author(s)
      Shoji Itoh, Masaaki Sugihara
    • Journal Title

      arXiv

      Volume: 1603 Pages: 1-21

    • Related Report
      2016 Annual Research Report
    • Open Access
  • [Journal Article] Formulation of a preconditioned algorithm for the conjugate gradient squared method in accordance with its logical structure2015

    • Author(s)
      Shoji Itoh, Masaaki Sugihara
    • Journal Title

      Applied Mathematics

      Volume: 6 Issue: 08 Pages: 1389-1406

    • DOI

      10.4236/am.2015.68131

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] 双ランチョス解法の初期シャドウ残差ベクトルによる前処理系の切り替え2016

    • Author(s)
      伊藤祥司,杉原正顯
    • Organizer
      応用解析研究会~可積分系から計算数学まで~
    • Place of Presentation
      天満研修センター(大阪市)
    • Related Report
      2016 Annual Research Report
  • [Presentation] 大規模線形方程式に対する求解アルゴリズムの体系的性能評価手法について2015

    • Author(s)
      伊藤祥司
    • Organizer
      電気通信大学 数値解析・HPC セミナー(第1 回)
    • Place of Presentation
      電気通信大学(調布市)
    • Year and Date
      2015-07-23
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 線形方程式向き双ランチョス系統の前処理付きアルゴリズムに対する初期シャドウ残差ベクトルの構成方法について2014

    • Author(s)
      伊藤祥司,杉原正顯
    • Organizer
      日本応用数理学会2014 年度年会
    • Place of Presentation
      政策研究大学院大学(東京都港区)
    • Year and Date
      2014-09-05
    • Related Report
      2014 Research-status Report
  • [Presentation] 様々な前処理付きCGS に対する分析とそれに基づく新アルゴリズムの提案2014

    • Author(s)
      伊藤祥司,杉原正顯
    • Organizer
      2014 年並列/分散/協調処理に関する『新潟』サマー・ワークショップ(SWoPP 新潟2014)
    • Place of Presentation
      朱鷺メッセ新潟コンベンションセンター(新潟県新潟市)
    • Year and Date
      2014-07-29
    • Related Report
      2014 Research-status Report
  • [Presentation] 前処理付きBiCGに対する初期シャドウ残差ベクトルの構成方法と前処理系の切り替えについて

    • Author(s)
      伊藤祥司,杉原正顯
    • Organizer
      日本応用数理学会2013年度年会
    • Place of Presentation
      アクロス福岡(福岡市)
    • Related Report
      2013 Research-status Report
  • [Remarks] SESNA

    • URL

      http://sesna.jp

    • Related Report
      2016 Annual Research Report 2015 Research-status Report
  • [Remarks] 数値計算アルゴリズムの体系的性能評価

    • URL

      http://sesna.jp/

    • Related Report
      2014 Research-status Report
  • [Remarks] 数値計算アルゴリズムの体系的性能評価

    • URL

      http://sesna.jp/

    • Related Report
      2013 Research-status Report

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Published: 2014-07-25   Modified: 2019-07-29  

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