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Computing with rational functions

Research Project

Project/Area Number 19K21027
Project/Area Number (Other) 18H05837 (2018)
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeMulti-year Fund (2019)
Single-year Grants (2018)
Review Section 0201:Algebra, geometry, analysis, applied mathematics,and related fields
Research InstitutionNational Institute of Informatics

Principal Investigator

中務 佑治  国立情報学研究所, 情報学プリンシプル研究系, 准教授 (10723554)

Project Period (FY) 2018-08-24 – 2020-03-31
Project Status Declined (Fiscal Year 2019)
Budget Amount *help
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords有理関数 / アルゴリズム / 行列分解 / 数値解析 / 応用数学
Outline of Annual Research Achievements

有理関数近似による数値計算を推し進める研究目的で、有理関数近似の理論解析や有理関数を使った行列分解アルゴリズム開発を行った.
例えば、Hawaii大学のEvan Gawlik,Randolph-Macon collegeのBrian Sutton氏と行列のCS分解を有理関数近似を用いて安定的に計算するアルゴリズムを開発し、SIAM Journal on Matrix Analysis and Applicationsに投稿、掲載された.
またOxfordのTrefethen教授と共同で、有理関数でx^nを近似する問題がnによらない指数関数的速度で近似できること、またこれが関数近似論の古典的な問題と深く関連することを示した.これはProceedings of the American Mathematical Societyに掲載された.
またKAUST(King Abdullah University of Science and Technology)のグループ(David Keyes教授を始め、Hatem Ltaief, Dalal E. Sukkar氏と共同)と有理関数を使って特異値分解を計算するアルゴリズムの並列計算実装の研究にも関わり、従来のScalapackなどのアルゴリズムよりも速く計算されることを実験で検証した.
またAlex Townsendの訪問で有理関数の機械学習への応用を検討し、共同研究を続けており、Evan Gawlik氏を訪問して有理関数近似が平方根や立方根の近似を効率的にできることを示す論文を執筆中である.現在も微分方程式への応用や、Barycentric representationの関数近似以外への応用なども検討中である.数学的理論だけでなく、実用問題への応用も広く考えられる、将来性の高いテーマだと考える.

Research Progress Status

翌年度、交付申請を辞退するため、記入しない。

Strategy for Future Research Activity

翌年度、交付申請を辞退するため、記入しない。

Report

(1 results)
  • 2018 Annual Research Report
  • Research Products

    (11 results)

All 2019 2018 Other

All Int'l Joint Research (3 results) Journal Article (3 results) Presentation (5 results) (of which Int'l Joint Research: 4 results,  Invited: 2 results)

  • [Int'l Joint Research] Univ. Hawaii at Manoa/Randolph-Macon college(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] KAUST(サウジアラビア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Oxford(英国)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Massively Parallel Polar Decomposition on Distributed-Memory Systems, to appear in ACM Transactions on Parallel Computing2019

    • Author(s)
      Hatem Ltaief, Dalal E. Sukkari, Aniello Esposito, Yuji Nakatsukasa and David E. Keyes
    • Journal Title

      ACM Transactions on Parallel Computing

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
  • [Journal Article] A Backward Stable Algorithm for Computing the CS Decomposition via the Polar Decomposition2018

    • Author(s)
      Gawlik Evan S.、Nakatsukasa Yuji、Sutton Brian D.
    • Journal Title

      SIAM Journal on Matrix Analysis and Applications

      Volume: 39 Issue: 3 Pages: 1448-1469

    • DOI

      10.1137/18m1182747

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Rational approximation of $\mathbf {x}^n$2018

    • Author(s)
      Nakatsukasa Yuji、Trefethen Lloyd N.
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 146 Issue: 12 Pages: 5219-5224

    • DOI

      10.1090/proc/14187

    • Related Report
      2018 Annual Research Report
  • [Presentation] Best and near-best rational approximation via adaptive barycentric representation2019

    • Author(s)
      Yuji Nakatsukasa
    • Organizer
      University of Hawaii at Manoa Applied Mathematics seminar
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems2018

    • Author(s)
      Yuji Nakatsukasa
    • Organizer
      CAM-ICCM 2018, Nanjing, China
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Approximate and integrate: Variance reduction in Monte Carlo integration via function approximation2018

    • Author(s)
      Yuji Nakatsukasa
    • Organizer
      MCQMC 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] The AAA algorithm for rational approximation2018

    • Author(s)
      Yuji Nakatsukasa
    • Organizer
      EASIAM 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Error bounds for Ritz vectors and approximate singular vectors2018

    • Author(s)
      Yuji Nakatsukasa
    • Organizer
      SIAM conference on Applied Linear Algebra 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research

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Published: 2018-08-27   Modified: 2024-03-26  

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