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Optimal estimation for inverse of infinite-dimensional operator by self-validating numerical computations and its applications

Research Project

Project/Area Number 24340018
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyushu University

Principal Investigator

Yoshitaka Watanabe  九州大学, 情報基盤研究開発センター, 准教授 (90243972)

Co-Investigator(Kenkyū-buntansha) Nagatou Kaori  九州大学, マス・フォア・インダストリ研究所, 准教授 (40326426)
Co-Investigator(Renkei-kenkyūsha) Nakao Mitsuhiro T.  佐世保工業高等専門学校, 校長 (10136418)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥5,590,000 (Direct Cost: ¥4,300,000、Indirect Cost: ¥1,290,000)
Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2013: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2012: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Keywords精度保証付き数値計算 / 偏微分方程式 / 計算機援用証明 / 関数解析 / 無限次元固有値問題 / 不動点定理 / 応用数学 / 関数方程式論 / 微分方程式 / 精度保証 / 非線形解析 / 固有値解析
Outline of Final Research Achievements

"Self-validating numerical computations" stands for a computer-assisted approach to prove the existence of the solutions and its guaranteed error bounds for a given problem. By using self-validating numerical computations, this reserch proposed a computer-assisted procedure to assure the invertibility of a linear operator which is the sum of an unbounded bijective and a bounded operator in a Hilbert space, and to compute a bound for the norm of its inverse. We also showed that our bounds are expected to converge to the exact operator norm and to provide accurate and efficient enclosure results for the solution of nonlinear problem by infinite-dimensional Newton-type methods.

Report

(4 results)
  • 2015 Final Research Report ( PDF )
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • Research Products

    (26 results)

All 2015 2014 2013 2012 Other

All Journal Article (6 results) (of which Peer Reviewed: 6 results,  Acknowledgement Compliant: 2 results) Presentation (16 results) (of which Invited: 2 results) Book (1 results) Remarks (3 results)

  • [Journal Article] Some Considerations of the Invertibility Verifications for Linear Elliptic Operators2015

    • Author(s)
      Mitsuhiro T. Nakao, Yoshitaka Watanabe, Takehiko Kinoshita, Takuma Kimura, and Nobito Yamamoto
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 32 Issue: 2 Pages: 19-32

    • DOI

      10.1137/120894683

    • NAID

      210000173114

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Verified Computations of Eigenvalue Exclosures for Eigenvalue Problems in Hilbert Spaces2014

    • Author(s)
      Yoshitaka Watanabe, Kaori Nagatou, Michael Plum, and Mitsuhiro T. Nakao
    • Journal Title

      SIAM Journal on Numerical Analysis

      Volume: 52 Issue: 1 Pages: 975-992

    • DOI

      10.1007/s13160-014-0160-6

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] An improvement of the theorem of a posteriori estimates for inverse elliptic operators2014

    • Author(s)
      Takehiko Kinoshita, Yoshitaka Watanabe, and Mitsuhiro T. Nakao
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: 5 Issue: 1 Pages: 47-52

    • DOI

      10.1587/nolta.5.47

    • NAID

      130003386652

    • ISSN
      2185-4106
    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations2013

    • Author(s)
      Yoshitaka Watanabe, Takehiko Kinoshita and Mitsuhiro T. Nakao
    • Journal Title

      Mathematics of Computation

      Volume: 82 Issue: 283 Pages: 1543-1557

    • DOI

      10.1090/s0025-5718-2013-02676-2

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Simple Numerical Verification Method for Differential Equations Based on Infinite Dimensional Sequential Iteration2013

    • Author(s)
      Yoshitaka Watanabe
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: 4 Pages: 23-33

    • NAID

      130003375410

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Numerical Verification Method for a System of Fitz Hugh-Nagumo Type2012

    • Author(s)
      Shuting Cai, Kaori Nagatou, and Yoshitaka Watanabe
    • Journal Title

      Numerical Functional Analysis and Optimization

      Volume: 33 Pages: 1195-1220

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Presentation] 楕円型線形逆作用素評価の収束性について2015

    • Author(s)
      渡部 善隆, 木下 武彦, 中尾 充宏
    • Organizer
      日本数学会2015年度年会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Annual Research Report
  • [Presentation] 楕円型線形逆作用素評価の収束性について2015

    • Author(s)
      渡部 善隆, 木下 武彦, 中尾 充宏
    • Organizer
      日本応用数理学会2015年研究部会連合発表会
    • Place of Presentation
      明治大学中野キャンパス
    • Year and Date
      2015-03-06 – 2015-03-07
    • Related Report
      2014 Annual Research Report
  • [Presentation] Orr-Sommerfeld方程式の臨界Reynolds数に対する精度保証付き数値計算(上2014

    • Author(s)
      渡部 善隆
    • Organizer
      応用数学合同研究集会
    • Place of Presentation
      龍谷大学
    • Year and Date
      2014-12-18 – 2014-12-20
    • Related Report
      2014 Annual Research Report
  • [Presentation] A numerical verification method for the Orr-Sommerfeld problem2014

    • Author(s)
      渡部 善隆
    • Organizer
      研究集会: 流体方程式の構造と特異性に迫る数値解析・数値計算
    • Place of Presentation
      名古屋大学
    • Year and Date
      2014-12-08 – 2014-12-09
    • Related Report
      2014 Annual Research Report
  • [Presentation] 精度保証付き数値計算による楕円型作用素の逆作用素ノルム評価2014

    • Author(s)
      渡部 善隆
    • Organizer
      RIMS研究集会・新時代の科学技術を牽引する数値解析学, 京都大学数理解析研究所
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-10-08 – 2014-10-10
    • Related Report
      2014 Annual Research Report
  • [Presentation] Some remarks on the rigorous estimation of inverse linear elliptic operators2014

    • Author(s)
      Takehiko Kinoshita, Yoshitaka Watanabe, and Mitsuhiro T. Nakao
    • Organizer
      16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2014)
    • Place of Presentation
      University of Wuerzburg, Germany
    • Year and Date
      2014-09-21 – 2014-09-26
    • Related Report
      2014 Annual Research Report
  • [Presentation] 2階楕円型作用素における構成的Laplacianノルム評価2014

    • Author(s)
      渡部 善隆, 木下 武彦, 中尾 充宏
    • 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
    • Year and Date
      2014-09-03 – 2014-09-05
    • Related Report
      2014 Annual Research Report
  • [Presentation] An alternative approach of invertibility verifications and norm estimations for linear elliptic operators2014

    • Author(s)
      Takehiko Kinoshita, Yoshitaka Watanabe, Mitsuhiro T. Nakao
    • Organizer
      日本応用数理学会2014年研究部会連合発表会
    • Place of Presentation
      京都大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] 計算機援用証明によるOrr-Sommerfeld問題の安定性・不安定性解析2014

    • Author(s)
      渡部 善隆
    • Organizer
      日本数学会2014年度年会
    • Place of Presentation
      学習院大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] A comparison of computer-assisted proofs for the Kolmogorov problem2014

    • Author(s)
      Yoshitaka Watanabe
    • Organizer
      International Workshop on Numerical Verification and its Applications 2014
    • Place of Presentation
      早稲田大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 精度保証とHPC2013

    • Author(s)
      渡部善隆
    • Organizer
      HPCS2013
    • Place of Presentation
      東京工業大学(招待講演)
    • Year and Date
      2013-01-16
    • Related Report
      2012 Annual Research Report
  • [Presentation] 2階楕円型線形作用素の可逆性検証に関するいくつかの考察2013

    • Author(s)
      渡部善隆, 木下 武彦, 木村 拓馬, 山本 野人, 中尾 充宏
    • Organizer
      応用数学合同研究集会
    • Place of Presentation
      龍谷大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] A computer-assisted proof of the Kolmogorov problem of incompressible viscous fluid2013

    • Author(s)
      渡部 善隆
    • Organizer
      RIMS研究集会・非圧縮性粘性流体の数理解析
    • Place of Presentation
      京都大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] 無限次元非線形関数方程式に対するNewton反復型計算機援用証明2013

    • Author(s)
      渡部 善隆, 中尾 充宏
    • Organizer
      日本数学会2013年度秋季総合分科会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Annual Research Report
  • [Presentation] 楕円型偏微分作用素に対する逆作用素評価の効率化2013

    • Author(s)
      木下 武彦, 渡部 善隆, 中尾 充宏
    • Organizer
      日本応用数理学会2013年度年会
    • Place of Presentation
      アクロス福岡
    • Related Report
      2013 Annual Research Report
  • [Presentation] exflibによる平行Poiseuille流れの不安定性解析の高精度化2013

    • Author(s)
      渡部 善隆, 藤原 宏志, 中尾 充宏
    • Organizer
      日本応用数理学会2013年度年会
    • Place of Presentation
      アクロス福岡
    • Related Report
      2013 Annual Research Report
  • [Book] Special Section on Recent Progress in Verified Numerical Computations2012

    • Author(s)
      Z. Arai, et al. (eds)
    • Total Pages
      110
    • Publisher
      電子情報通信学会
    • Related Report
      2012 Annual Research Report
  • [Remarks] Watanabe, Yoshitaka's Home Page

    • URL

      http://yebisu.cc.kyushu-u.ac.jp/~watanabe/

    • Related Report
      2014 Annual Research Report
  • [Remarks] Watabane Yoshitaka's page

    • URL

      http://www.cc.kyushu-u.ac.jp/RD/watanabe/

    • Related Report
      2013 Annual Research Report
  • [Remarks]

    • URL

      http://www.cc.kyusyu-u.ac.jp/RD/watanabe/

    • Related Report
      2012 Annual Research Report

URL: 

Published: 2012-04-24   Modified: 2019-07-29  

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