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A study on the numerical verification method of solutions with high accuracy for the nonlinear mathematical models in infinite dimension

Research Project

Project/Area Number 15K05012
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionKyushu University (2016-2017)
Sasebo National College of Technology (2015)

Principal Investigator

NAKAO Mitsuhiro  九州大学, マス・フォア・インダストリ研究所, 学術研究者 (10136418)

Co-Investigator(Renkei-kenkyūsha) WATANABE Yoshitaka  九州大学, 情報基盤研究開発センター, 准教授 (90243972)
KIMURA Takuma  佐賀大学, 理工学部, 准教授 (60581618)
KINOSHITA Takehiko  京都大学, 学際融合教育研究推進センター, 特定講師 (30546429)
Research Collaborator Michael Plum  Karlsruhe大学, 教授
Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywords精度保証付き数値計算法 / 解の数値的検証 / 数値解析 / 解の事後誤差評価 / 計算機援用証明 / 非線形現象 / 非線形偏微分方程式 / 数値的検証法 / 構成的高精度事前誤差評価 / 非線形楕円型方程式 / 非線形発展方程式
Outline of Final Research Achievements

We studied the numerical verification method of solutions for infinite dimensional nonlinear mathematical models including elliptic and parabolic equations. We presented several computational techniques for the numerical estimations of the linearized inverse operators associated with nonlinear problems. From the viewpoint of efficiency and accuracy in the verified computations, we proposed several techniques which enebles us the actual effectiveness by showing numerical examples related to elliptic problems of second and fourth order. We also derived the constructive a priori error estimates with optimal order for a full discrete numerical scheme of the heat equation, which is based on the finite element Galerkin method with an interpolation in time using the fundamental matrix for ODEs. Furthermore, under the general setting in Hilbert space, we presented a principle of the verified computational method of solutions using Newton-type formulation.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (27 results)

All 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (3 results) Journal Article (9 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 9 results,  Open Access: 4 results,  Acknowledgement Compliant: 6 results) Presentation (15 results) (of which Int'l Joint Research: 7 results,  Invited: 5 results)

  • [Int'l Joint Research] Karlsruhe Institute of Technology(Germany)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Karlsruhe Institute of Technology(ドイツ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Karlsruhe Institute of Technology(ドイツ)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Validated Constructive Error Estimations for Biharmonic Problems,2017

    • Author(s)
      T. Kinoshita, Y. Watanabe, M. T. Nakao
    • Journal Title

      Reliable Computing

      Volume: 25 Pages: 168-177

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Optimal Order Constructive a Priori Error Estimates for a Full Discrete Approximation of the Heat Equation2017

    • Author(s)
      T. Kimura, T. Minamoto, M. T. Nakao
    • Journal Title

      Reliable Computing

      Volume: 25 Pages: 201-212

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On the compactness of a nonlinear operator related to stream function-vorticity formulation for the Navier-Stokes equations2017

    • Author(s)
      Watanabe Yoshitaka、Nakao Mitsuhiro T.、Nagatou Kaori
    • Journal Title

      JSIAM Letters

      Volume: 9 Issue: 0 Pages: 77-80

    • DOI

      10.14495/jsiaml.9.77

    • NAID

      130006252917

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Some remarks on the rigorous estimation of inverse linear elliptic operators2016

    • Author(s)
      T. Kinoshita, Y. Watanabe, M. T. Nakao
    • Journal Title

      Lecture Notes in Computer Science (LNCS) 9553, the proceedings of SCAN 2014

      Volume: 9553 Pages: 225-235

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Norm bound computation for inverses of linear operators in Hilbert spaces2016

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

      Journal of Differential Equations

      Volume: 260 Issue: 7 Pages: 6363-6374

    • DOI

      10.1016/j.jde.2015.12.041

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Some remarks on a priori estimates of highly regular solutions for the Poisson equation in polygonal domains2016

    • Author(s)
      T. Kinoshita, Y. Watanabe, N. Yamamoto, M. T. Nakao
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 33 Pages: 629-636

    • NAID

      210000183342

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A numerical verification method for nonlinear functional equations based on infinite-dimensional Newton-like iteration2016

    • Author(s)
      Y. Watanabe, M.T. Nakao
    • Journal Title

      Applied Mathematics and Computation

      Volume: 276 Pages: 239-251

    • DOI

      10.1016/j.amc.2015.12.021

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Recurrence relations of orthogonal polynomials in <I>H</I><sub>0</sub><sup>1</sup> and <I>H</I><sub>0</sub><sup>2</sup>2015

    • Author(s)
      T. Kinoshita, Y. Watanabe, M. T. Nakao
    • Journal Title

      Nonlinear Theory and Its Applications, IEICE

      Volume: 6 Issue: 3 Pages: 404-409

    • DOI

      10.1587/nolta.6.404

    • NAID

      130005085581

    • ISSN
      2185-4106
    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / 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
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Laplacianノルム評価を援用した楕円型線形作用素の可逆性検証2018

    • Author(s)
      渡部善隆、木下武彦、中尾充宏
    • Organizer
      日本応用数理学会2018年 研究部会連合発表会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Some remarks on the numerical norm estimation of the inverse operator in Hilbert spaces2017

    • Author(s)
      Mitsuhiro T. Nakao, Takehiko Kinoshita, Yoshitaka Watanabe
    • Organizer
      The International Workshop on Numerical Verification and its Applications 2017(INVA 2017)
    • Place of Presentation
      Miyako-jima, Okinawa, Japan
    • Year and Date
      2017-03-14
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the guaranteed error bounds of a new approximation scheme for parabolic problems2017

    • Author(s)
      Mitsuhiro T. Nakao
    • Organizer
      BIRS Workshop, Rigorous Numerics for Infinite Dimensional Nonlinear Dynamics
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the numerical norm estimation of the inverse operator in Hilbert space2017

    • Author(s)
      Mitsuhiro T. Nakao
    • Organizer
      Dagstuhl Seminar, Reliable Computation and Complexity on the Reals
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 値域が共役空間となる2 階楕円型作用素に対する可逆性検証法の改良2017

    • Author(s)
      木下武彦、渡部善隆、中尾充宏
    • Organizer
      日本応用数理学会2017年度年会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Hilbert空間における線形作用素に対する逆作用素ノルム評価の改良2017

    • Author(s)
      木下武彦、渡部善隆、中尾充宏
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2017 Annual Research Report
  • [Presentation] コンパクト作用素のレゾルベントに対する下側評価2016

    • Author(s)
      木下武彦, 渡部 善隆, 中尾 充宏
    • Organizer
      2016年度応用数学合同研究集会
    • Place of Presentation
      龍谷大学瀬田キャンパス
    • Year and Date
      2016-12-16
    • Related Report
      2016 Research-status Report
  • [Presentation] 有限要素法の構成的誤差評価とその計算機援用証明への応用2016

    • Author(s)
      中尾充宏
    • Organizer
      RIMS研究集会:現象解明に向けた数値解析学の新展開II
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-10-20
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Validated constructive error estimatations for bi-harmonic problems2016

    • Author(s)
      Yoshitaka Watanabe, Takehiko Kinoshita and Mitsuhiro T. Nakao
    • Organizer
      17th International Symposium on Scientific Computing, Computer Arithmetics and Verified Numerics(SCAN 2016)
    • Place of Presentation
      Uppsala University, Uppsala, Sweden
    • Year and Date
      2016-09-26
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the refinement of constructive a priori error estimates of the finite element methods with applications to verified computation for PDEs2016

    • Author(s)
      Mitsuhiro T. Nakao
    • Organizer
      The Sixth China-Japan-Korea Joint Conference on Numerical Mathematics
    • Place of Presentation
      NIMS, Daejeon, Korea
    • Year and Date
      2016-08-22
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 精度保証付き数値計算による平行Poiseuille 流れの高精度不安定性解析2016

    • Author(s)
      渡部善隆、木下武彦、中尾充宏
    • Organizer
      日本数学会年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-19
    • Related Report
      2015 Research-status Report
  • [Presentation] H3 and H4 regularities of the Poisson equation on polygonal domains2015

    • Author(s)
      Takehiko Kinoshita, Yoshitaka Watanabe, Mitsuhiro T. Nakao
    • Organizer
      6th International Conference on Mathematical Aspects of Computer and Information Sciences
    • Place of Presentation
      Zuse Institute Berlin, Germany
    • Year and Date
      2015-11-11
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] H3 正則性を持つ一般多角形領域上のPoisson 方程式の解について2015

    • Author(s)
      木下武彦、渡部善隆、中尾充宏
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-16
    • Related Report
      2015 Research-status Report
  • [Presentation] 線形化、Kolmogorov 問題に対する固有値の除外について2015

    • Author(s)
      渡部善隆、Nagatou Kaori、Plum Michael、中尾充宏
    • Organizer
      2015年度日本応用数理学会年会
    • Place of Presentation
      金沢大学
    • Year and Date
      2015-09-09
    • Related Report
      2015 Research-status Report
  • [Presentation] On constructive higher order a priori error estimates of the finite element methods with applications to validated computation for PDEs2015

    • Author(s)
      Mitsuhiro T. Nakao
    • Organizer
      Workshop, Nonlinear PDE days
    • Place of Presentation
      Karlsruhe Institute of Technology, Germany
    • Year and Date
      2015-07-14
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2015-04-16   Modified: 2022-02-16  

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