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Fast, accurate and stable matrix computation algorithms based on non-orthogonal transformations

Research Project

Project/Area Number 17K19966
Research Category

Grant-in-Aid for Challenging Research (Exploratory)

Allocation TypeMulti-year Fund
Research Field Information science, computer engineering, and related fields
Research InstitutionThe University of Electro-Communications

Principal Investigator

Yamamoto Yusaku  電気通信大学, 大学院情報理工学研究科, 教授 (20362288)

Co-Investigator(Kenkyū-buntansha) 廣田 悠輔  東京電機大学, 未来科学部, 助教 (60709765)
Project Period (FY) 2017-06-30 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥6,110,000 (Direct Cost: ¥4,700,000、Indirect Cost: ¥1,410,000)
Fiscal Year 2019: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2018: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2017: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Keywords共役勾配法 / 前処理 / 不完全コレスキー分解 / 固有値問題 / ブロックヤコビ法 / 並列化 / GPU / 非直交変換 / 固有値計算 / 行列計算 / 時間依存固有値問題 / 収束性解析 / 連立1次方程式解法 / COCG法 / shifted MINRES法 / 線形計算 / 最小2乗問題 / 誤差解析
Outline of Final Research Achievements

The aim of this study is to develop new algorithms for matrix computations based on non-orthogonal transformations and analyze their convergence properties and accuracy theoretically. We chose the time-dependent eigenvalue problem and preconditioning methods for iterative solution of linear systems as examples. For the former, we developed an algorithm based on matrix multiplications, which is suited for next-generation microprocessors. For the latter, we developed a highly parallel algorithm based on the modified incomplete Cholesky factorization and the block red-block ordering.

Academic Significance and Societal Importance of the Research Achievements

従来の行列計算アルゴリズムは,計算の安定性を確保するため,直交変換を用いて構成されることが多かった。固有値計算のためのハウスホルダー法や最小2乗法のためのQR分解法はその例である。しかし,非直交変換を積極的に利用することで,計算量や精度の面でより優れたアルゴリズムが開発できる可能性がある。本研究では,時間依存固有値問題と反復法の前処理という2つの問題を取り上げ,その実証を試みた。非直交変換を利用したアルゴリズムでは,理論的解析により精度や安定性を保証することが重要であり,本研究ではその点についても解析を行った。

Report

(6 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (62 results)

All 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (10 results) Journal Article (23 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 23 results,  Open Access: 10 results) Presentation (25 results) (of which Int'l Joint Research: 7 results,  Invited: 2 results) Book (4 results)

  • [Int'l Joint Research] ザルツブルグ大学(オーストリア)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] スロバキア国立科学アカデミー(スロバキア)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] ザルツブルグ大学(オーストリア)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] スロバキア国立科学アカデミー(スロバキア)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Salzburg University(オーストリア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Slovakia Academy of Sciences(スロバキア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Salzburg University(オーストリア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Slovakia Academy of Sciences(スロバキア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Salzburg University(Austria)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Slovakia Academy of Sciences(Slovakia)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Performance prediction of massively parallel computation by Bayesian inference2022

    • Author(s)
      Kohashi Hisashi、Iwamoto Harumichi、Fukaya Takeshi、Yamamoto Yusaku、Hoshi Takeo
    • Journal Title

      JSIAM Letters

      Volume: 14 Issue: 0 Pages: 13-16

    • DOI

      10.14495/jsiaml.14.13

    • NAID

      130008164520

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Combinatorial preconditioning for accelerating the convergence of the parallel block Jacobi method for the symmetric eigenvalue problem2021

    • Author(s)
      Kugaya Masaki、Kudo Shuhei、Yamamoto Yusaku
    • Journal Title

      JSIAM Letters

      Volume: 13 Issue: 0 Pages: 56-59

    • DOI

      10.14495/jsiaml.13.56

    • NAID

      130008092825

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Acceleration and Parallelization of a Linear Equation Solver for Crack Growth Simulation Based on the Phase Field Model2021

    • Author(s)
      Ishii Gaku、Yamamoto Yusaku、Takaishi Takeshi
    • Journal Title

      Mathematics

      Volume: 9 Issue: 18 Pages: 2248-2248

    • DOI

      10.3390/math9182248

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On eigenvalues of a matrix arising in energy-preserving/dissipative continuous-stage Runge-Kutta methods2021

    • Author(s)
      Yamamoto Yusaku
    • Journal Title

      Special Matrices

      Volume: 10 Issue: 1 Pages: 34-39

    • DOI

      10.1515/spma-2021-0101

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Block red-black MILU(0) preconditioner with relaxation on GPU2021

    • Author(s)
      Shioya Akemi、Yamamoto Yusaku
    • Journal Title

      Parallel Computing

      Volume: 103 Pages: 102760-102760

    • DOI

      10.1016/j.parco.2021.102760

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] On convergence to eigenvalues and eigenvectors in the block-Jacobi EVD algorithm with dynamic ordering2021

    • Author(s)
      Yamamoto Yusaku、Oksa Gabriel、Vajtersic Marian
    • Journal Title

      Linear Algebra and its Applications

      Volume: 622 Pages: 19-45

    • DOI

      10.1016/j.laa.2021.03.027

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Convergence acceleration of the shifted LR transformations for totally nonnegative Hessenberg matrices2020

    • Author(s)
      Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura,
    • Journal Title

      Applications of Mathematics

      Volume: 65 Issue: 5 Pages: 677-702

    • DOI

      10.21136/am.2020.0378-19

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A parallelizable energy-preserving integrator MB4 and its application to quantum-mechanical wavepacket dynamics2020

    • Author(s)
      Sakai Tsubasa、Kudo Shuhei、Imachi Hiroto、Miyatake Yuto、Hoshi Takeo、Yamamoto Yusaku
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 38 Issue: 1 Pages: 105-123

    • DOI

      10.1007/s13160-020-00430-2

    • NAID

      210000183552

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Fixed-point analysis of Ogita-Aishima's symmetric eigendecomposition refinement algorithm for multiple eigenvalues2020

    • Author(s)
      Shiroma Kurumi、Yamamoto Yusaku
    • Journal Title

      JSIAM Letters

      Volume: 12 Issue: 0 Pages: 5-8

    • DOI

      10.14495/jsiaml.12.5

    • NAID

      130007805107

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Shifted Cholesky QR for Computing the QR Factorization of Ill-Conditioned Matrices2020

    • Author(s)
      Takeshi Fukaya, Ramaseshan Kannan, Yuji Nakatsukasa, Yusaku Yamamoto, Yuka Yanagisawa
    • Journal Title

      SIAM Journal on Scientific Computing

      Volume: 42 Issue: 1 Pages: A477-A503

    • DOI

      10.1137/18m1218212

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Asymptotic Quadratic Convergence of the Two-Sided Serial and Parallel Block-Jacobi SVD Algorithm2019

    • Author(s)
      Oksa Gabriel、Yamamoto Yusaku、Becka Martin、Vajtersic Marian
    • Journal Title

      SIAM Journal on Matrix Analysis and Applications

      Volume: 40 Issue: 2 Pages: 639-671

    • DOI

      10.1137/18m1222727

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Efficient implementations of the modified Gram?Schmidt orthogonalization with a non-standard inner product2019

    • Author(s)
      Imakura Akira、Yamamoto Yusaku
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 36 Issue: 2 Pages: 619-641

    • DOI

      10.1007/s13160-019-00356-4

    • NAID

      210000183505

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Asymptotic Quadratic Convergence of the Two-Sided Serial and Parallel Block-Jacobi SVD Algorithm2019

    • Author(s)
      Gabriel Oksa・Yusaku Yamamoto・Martin Becka・Marian Vajtersic
    • Journal Title

      SIAM Journal on Matrix Analysis and Applications

      Volume: to appear

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Efficient Implementations of the Modified Gram-Schmidt Orthogonalization with a Non-standard Inner Product2019

    • Author(s)
      Akira Imakura・Yusaku Yamamoto
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: to appear

    • NAID

      210000183505

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] A Middleware for Parallel Generalized Eigenvalue Solvers to Attain High Scalability and Usability2019

    • Author(s)
      Kazuyuki Tanaka・Hiroto Imachi・Tomoya Fukumoto・Akiyoshi Kuwata・Yuki Harada・Takeshi Fukaya・Yusaku Yamamoto・Takeo Hoshi
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: to appear

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On using the shifted minimal residual method for quantum-mechanical wave packet simulation2019

    • Author(s)
      Hiroaki Seito・Takeo Hoshi・Yusaku Yamamoto
    • Journal Title

      JSIAM Letters

      Volume: 11 Issue: 0 Pages: 13-16

    • DOI

      10.14495/jsiaml.11.13

    • NAID

      130007607760

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] 荻田・相島の固有ベクトル反復改良法に基づく実対称行列の固有値分解追跡手法2019

    • Author(s)
      白間久瑠美・工藤周平・山本有作
    • Journal Title

      日本応用数理学会論文誌

      Volume: 29 Pages: 78-120

    • NAID

      130007618979

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Asymptotic Quadratic Convergence of the Parallel Block-Jacobi EVD Algorithm with Dynamic Ordering for Hermitian Matrices2018

    • Author(s)
      Gabriel Oksa・Yusaku Yamamoto・Martin Becka・Marian Vajtersic
    • Journal Title

      BIT Numerical Mathematics

      Volume: 58 Issue: 4 Pages: 1099-1123

    • DOI

      10.1007/s10543-018-0711-3

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Performance of the parallel block Jacobi method with dynamic ordering for the symmetric eigenvalue problem2018

    • Author(s)
      Shuhei Kudo, Kousuke Yasuda and Yusaku Yamamoto
    • Journal Title

      JSIAM Letters

      Volume: 10 Issue: 0 Pages: 41-44

    • DOI

      10.14495/jsiaml.10.41

    • NAID

      130007432998

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Solution of a Nonlinear Eigenvalue Problem Using Signed Singular Values2018

    • Author(s)
      Ooi Kouhei、Mizuno Yoshinori、Sogabe Tomohiro、Yamamoto Yusaku、Zhang Shao-Liang
    • Journal Title

      East Asian J. Appl. Math.

      Volume: 7 Issue: 4 Pages: 799-809

    • DOI

      10.4208/eajam.181016.300517c

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix2017

    • Author(s)
      Yusaku Yamamoto
    • Journal Title

      Appl. Math.

      Volume: 62 Issue: 4 Pages: 319-331

    • DOI

      10.21136/am.2017.0022-17

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] The danger of combining block red?black ordering with modified incomplete factorizations and its remedy by perturbation or relaxation2017

    • Author(s)
      Shioya Akemi、Yamamoto Yusaku
    • Journal Title

      Japan J. Indust. Appl. Math.

      Volume: 35 Issue: 1 Pages: 195-216

    • DOI

      10.1007/s13160-017-0277-5

    • NAID

      210000166851

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] On using the Cholesky QR method in the full-blocked one-sided Jacobi algorithm2017

    • Author(s)
      Shuhei Kudo and Yusaku Yamamoto
    • Journal Title

      Lecture Notes in Compu. Sci.

      Volume: 10777 Pages: 612-622

    • DOI

      10.1007/978-3-319-78024-5_53

    • ISBN
      9783319780238, 9783319780245
    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] テイラー展開に基づく行列指数関数計算の誤差解析2021

    • Author(s)
      山本有作, 工藤周平, 星健夫
    • Organizer
      日本応用数理学会「行列・固有値問題の解法とその応用」研究部会 第31回研究会
    • Related Report
      2021 Annual Research Report
  • [Presentation] ブロック赤黒順序付けされた摂動付き修正不完全分解前処理の収束性解析2021

    • Author(s)
      塩谷明美, 山本有作
    • Organizer
      日本応用数理学会2020年度年会
    • Related Report
      2021 Annual Research Report
  • [Presentation] 離散相対論的戸田方程式から導かれる交通流モデル2021

    • Author(s)
      関口真基, 岩崎雅史, 山本有作, 石渡恵美子
    • Organizer
      日本応用数理学会2020年度年会
    • Related Report
      2021 Annual Research Report
  • [Presentation] 原点シフト付きロトカ・ボルテラ系に対する超離散化から得られる箱玉系の性質について2021

    • Author(s)
      岡来美, 関口真基, 岩﨑雅史, 山本有作, 石渡恵美子
    • Organizer
      日本応用数理学会2020年度年会
    • Related Report
      2021 Annual Research Report
  • [Presentation] ベイズ推定による超並列計算の性能予測2021

    • Author(s)
      星健夫, 小橋恒士, 山本有作, 深谷猛
    • Organizer
      日本応用数理学会2020年度年会
    • Related Report
      2021 Annual Research Report
  • [Presentation] ブロック赤黒順序付けを用いた摂動付きMIC(0)分解のモデル問題による収束性解析2021

    • Author(s)
      塩谷明美,山本有作
    • Organizer
      日本応用数理学会「行列・固有値問題の解法とその応用」研究部会 第32回研究会
    • Related Report
      2021 Annual Research Report
  • [Presentation] Mobility and Eigenvalue/Singular Value Problems: Applications and Numerical Computation2021

    • Author(s)
      Yusaku Yamamoto
    • Organizer
      Mobility Workshop 2022 in Shizuoka: Mathematical Approaches to Mobility Problems
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] ヒルベルト行列と関連するある行列の固有値について2021

    • Author(s)
      山本有作
    • Organizer
      日本応用数理学会 第17回研究部会連合発表会
    • Related Report
      2020 Research-status Report
  • [Presentation] 実対称固有値問題向けのブロックヤコビ法に対する組み合わせ的前処理2020

    • Author(s)
      久賀谷匡貴, 山本有作
    • Organizer
      日本応用数理学会「行列・固有値問題の解法とその応用」研究部会 第29回研究会
    • Related Report
      2020 Research-status Report
  • [Presentation] 縦長行列の列ピボット付きQR分解に対するコレスキーQR型アルゴリズムの検討2020

    • Author(s)
      深谷猛, 中務祐治, 山本有作
    • Organizer
      日本応用数理学会2020年度年会
    • Related Report
      2020 Research-status Report
  • [Presentation] Efficient Implementations of the Modified Gram-Schmidt Orthogonalization in a Non-Standard Inner Product2019

    • Author(s)
      Yusaku Yamamoto, Akira Imakura
    • Organizer
      The 13th workshop in the series of the "Parallel Numerics" (ParNum 2019)(国際学会)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] ブロック赤黒順序付け緩和MILU(0)前処理法のGPU実装と性能評価2019

    • Author(s)
      塩谷明美,山本有作
    • Organizer
      2019年並列/分散/協調処理に関する『北見』サマー・ワークショップ (SWoPP2019)
    • Related Report
      2019 Research-status Report
  • [Presentation] 一般内積における直交化のためのMGS-HP 法の誤差解析2019

    • Author(s)
      山本有作,今倉暁
    • Organizer
      日本応用数理学会2019年度年会
    • Related Report
      2019 Research-status Report
  • [Presentation] エネルギー散逸型数値解法の設計におけるある逆固有値問題について2019

    • Author(s)
      山本有作
    • Organizer
      日本応用数理学会「行列・固有値問題の解法とその応用」研究部会 第28回研究会
    • Related Report
      2019 Research-status Report
  • [Presentation] エネルギー保存型並列解法MB4に基づく2次元量子ダイナミクス計算2018

    • Author(s)
      酒井翼・藤周平・井町宏人・宮武勇登・星健夫・山本有作
    • Organizer
      第47回数値解析シンポジウム
    • Related Report
      2018 Research-status Report
  • [Presentation] Application of an Energy-preserving Integrator to Quantum-mechanical Wavepacket Dynamics2018

    • Author(s)
      Yusaku Yamamoto・Tsubasa Sakai・Shuhei Kudo・Hiroto Imachi・Yuto Miyatake・Takeo Hoshi
    • Organizer
      13th SIAM East Asian Section Conference 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] 荻田・相島の固有ベクトル反復改良法における重複固有値の扱いについて2018

    • Author(s)
      白間久瑠美・工藤周平・山本有作
    • Organizer
      応用数理学会2018年度年会
    • Related Report
      2018 Research-status Report
  • [Presentation] Orthogonalization Algorithms and Dense Symmetric Eigenvalue Solvers Optimized for Strong Scaling2018

    • Author(s)
      Yusaku Yamamoto
    • Organizer
      International Symposium on Research and Education of Computational Science (RECS2018)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 荻田・相島の固有ベクトル反復改良法に基づく実対称行列の固有値分解追跡手法2018

    • Author(s)
      山本有作
    • Organizer
      RIMS共同研究(公開型)「次世代の科学技術を支える数値解析学の基盤整備と応用展開」
    • Related Report
      2018 Research-status Report
  • [Presentation] 一般化固有値問題に対する分割統治法におけるデフレーション手法について2018

    • Author(s)
      廣田悠輔・山本有作
    • Organizer
      日本応用数理学会「行列・固有値の解法とその応用」研究部会 第25回研究会
    • Related Report
      2018 Research-status Report
  • [Presentation] Block Red-Black Milu(0) Preconditioner with Relaxation on GPU2018

    • Author(s)
      Akemi Shioya and Yusaku Yamamoto
    • Organizer
      18th SIAM Conference on Parallel Processing for Scientific Computing
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Roundoff error analysis of the CholeskyQR2 and related algorithms2017

    • Author(s)
      Yusaku Yamamoto
    • Organizer
      PARNUM2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] ブロック赤-黒順序付けされた摂動/緩和MILU(0) 前処理法のGPUとマルチコアCPUにおける性能評価2017

    • Author(s)
      塩谷明美,山本有作
    • Organizer
      SWoPP 2017
    • Related Report
      2017 Research-status Report
  • [Presentation] On using the Cholesky QR method in the full-blocked one-sided Jacobi algorithm2017

    • Author(s)
      Shuhei Kudo and Yusaku Yamamoto
    • Organizer
      PPAM2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] 虚数シフトを行った実対称行列のための COCG法と一般化MINRES法の性能比較2017

    • Author(s)
      清藤 博暉, 山本 有作
    • Organizer
      日本応用数理学会「行列・固有値問題 の解法とその応用」研究部会 第24回研究会
    • Related Report
      2017 Research-status Report
  • [Book] 20世紀のトップ10アルゴリズム2022

    • Author(s)
      金田 行雄、笹井 理生、張 紹良
    • Total Pages
      360
    • Publisher
      共立出版
    • ISBN
      9784320122673
    • Related Report
      2021 Annual Research Report
  • [Book] 理工系の数理 ベクトル解析2020

    • Author(s)
      山本 有作、石原 卓
    • Total Pages
      182
    • Publisher
      裳華房
    • ISBN
      9784785315894
    • Related Report
      2020 Research-status Report
  • [Book] 計算科学のための基本数理アルゴリズム2019

    • Author(s)
      金田 行雄、笹井 理生、張 紹良
    • Total Pages
      256
    • Publisher
      共立出版
    • ISBN
      9784320122666
    • Related Report
      2019 Research-status Report
  • [Book] Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing2018

    • Author(s)
      Yusaku Yamamoto
    • Publisher
      Springer
    • ISBN
      9783319624266
    • Related Report
      2017 Research-status Report

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Published: 2017-07-21   Modified: 2023-01-30  

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