Enclosures of solution curve of differential equations by computer-assisted proof

• Project/Area Number: 21540134
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kyushu University
• Principal Investigator: WATANABE Yoshitaka 九州大学, 情報基盤研究開発センター, 准教授 (90243972)
• Co-Investigator(Renkei-kenkyūsha): NAGATOU Kaori 九州大学, マス・フォア・インダストリ研究所, 准教授 (40326426)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 計算機援用証明 / 精度保証付き数値計算 / 非線形偏微分方程式 / 非線形微分方程式 / 解曲線の追跡 / 線形化作用素の可逆性 / 無限次元Newton法 / 陰関数定理 / 固有値問題 / 解の存在検証 / 非自己共役固有値問題 / 固有値の除外 / 線形化作用素のノルム評価

Research Abstract

We developed a computer-assisted theory to verify the existence of the solutions for nonlinear elliptic boundary value problems with local uniqueness. We also proposed verified computations of eigenvalue exclosures for the linearization problems. Finally combining with developed method and the implicit function theorem, we introduced a procedure to enclose solution curves of nonlinear partial differential equations by computer-assisted proof and confirmed its effectiveness by some verified results.

Research Products

• [Journal Article] Tests of Inequivalence Among Absolutely Nonsingular Tensors Through Geometric Invariants (2012)
  • Author(s): Toshio Sakata, Kazumitsu Maehara, Takeshi Sasaki, Toshio Sumi, Mitsuhiro Miyazaki, Yoshitaka Watanabe, and Makoto Tagami
  • Journal Title: Universal Journal of Mathematics and Mathematical Sciences Volume: Vol.1, No.1 Pages: 1-28
• [Journal Article] Tests of inequivalence among absolutely nonsingular tensors through geometric invariants (2012)
  • Author(s): T.Sakata, et al
  • Journal Title: Universal Journal of Mathematics and Mathematical Sciences Volume: 1-1 Pages: 1-28
  • Peer Reviewed
• [Journal Article] 精度保証付き数値計算(4)-区間演算と有限次元問題の精度保証 (2011)
  • Author(s): 渡部善隆
  • Journal Title: 応用数理 Volume: Vol.21, No.4 Pages: 57-63
• [Journal Article] A Theorem for Numerical Verification on Local Uniqueness of Solutions to Fixed-Point Equations (2011)
  • Author(s): Nobito Yamamoto, Mitsuhiro T. Nakao, and Yoshitaka Watanabe
  • Journal Title: Numerical Functional Analysis and Optimization Volume: Vol.32, No.11 Pages: 1190-1204
• [Journal Article] 精度保証付き数値計算(3)-中尾理論入門 (2011)
  • Author(s): 渡部善隆
  • Journal Title: 応用数理 Volume: Vol.21, No.3 Pages: 47-53
  • NAID: 110009426225
• [Journal Article] 精度保証付き数値計算(2)-線形逆作用素のノルム評価 (2011)
  • Author(s): 渡部善隆
  • Journal Title: 応用数理 Volume: Vol.21, No.2 Pages: 54-60
  • NAID: 110008673554
• [Journal Article] 精度保証付き数値計算(1)-不動点定式化 (2011)
  • Author(s): 渡部善隆
  • Journal Title: 応用数理 Volume: Vol.21, No.1 Pages: 40-45
  • NAID: 110008593884
• [Journal Article] Numerical Verification Methods for Solutions of Semilinear Elliptic Boundary Value Problems (2011)
  • Author(s): Mitsuhiro T. Nakao, and Yoshitaka Watanabe
  • Journal Title: Nonlinear Theory and Its Applications, IEICE Volume: Vol.2, No.1 Pages: 2-31
  • NAID: 130000903439
• [Journal Article] A Computer-assisted Stability Proof for the Orr-Sommerfeld Problem with Poiseuille Flow (2011)
  • Author(s): Yoshitaka Watanabe, Kaori Nagatou, Michael Plum, and Mitsuhiro T. Nakao
  • Journal Title: Nonlinear Theory and Its Applications, IEICE Volume: Vol.2, No.1 Pages: 123-127
  • NAID: 130000903447
• [Journal Article] A theorem for numerical verification on local uniqueness of solutions to fixed-point equations (2011)
  • Author(s): N.Yamamoto, et al
  • Journal Title: Numerical Functional Analysis and Optimization Volume: 32-11 Issue: 11 Pages: 1190-1204
  • DOI: 10.1080/01630563.2011.594348
  • Peer Reviewed
• [Journal Article] 無限次元固有値問題に対する固有値の非存在証明,研究集会・数値解析と数値計算アルゴリズムの最近の展開 (2010)
  • Author(s): 渡部善隆, 長藤かおり, Michael Plum, 中尾充宏
  • Journal Title: 京都大学数理解析研究所講究録 Volume: Vol.1719 Pages: 118-129
• [Journal Article] Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems (2010)
  • Author(s): Mitsuhiro T. Nakao, Yoshitaka Watanabe, Nobito Yamamoto, Takaaki Nishida and Myoung-Nyoun Kim
  • Journal Title: Journal of Scientific Computing Volume: Vol.43, No.3 Pages: 388-401
• [Journal Article] Numerical Verification Methods for Solutions of Semilinear Elliptic Boundary Value Problems (2010)
  • Author(s): M.T.Nakao, Y.Watanabe
  • Journal Title: Nonlinear Theory and Its Applications, IEICE Volume: Vol.2, No.1 Pages: 2-31
  • NAID: 130000903439
  • Peer Reviewed
• [Journal Article] A Computer-assisted Stability Proof for the Orr-Sommerfeld Problem with Poiseuille Flow (2010)
  • Author(s): Y.Watanabe, et al.
  • Journal Title: Nonlinear Theory and Its Applications, IEICE Volume: Vol.2, No.1 Pages: 123-127
  • NAID: 130000903447
  • Peer Reviewed
• [Journal Article] Numerical Verification Method of Solutions for Elliptic Equations and Its Application to the Rayleigh-Benard Problem (2009)
  • Author(s): Yoshitaka Watanabe and Mitsuhiro T. Nakao
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Volume: Vol.26, No.2-3 Pages: 443-463
  • NAID: 10028169295
• [Journal Article] A Numerical Verification Method for Two-Coupled Elliptic Partial Differential Equations (2009)
  • Author(s): Yoshitaka Watanabe
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Volume: Vol.26, No.2-3 Pages: 233-247
  • NAID: 10028168961
• [Journal Article] Numerical Verification Method of Solutions for Elliptic Equations and Its Application to the Rayleigh-Benard Problem (2009)
  • Author(s): Y.Watanabe, et al.
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Vol.26, No.2-3 Pages: 443-463
  • NAID: 10028169295
  • Peer Reviewed
• [Journal Article] A Numerical Verification Method for Two-Coupled Elliptic Partial Differential Equations (2009)
  • Author(s): Y.Watanabe
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Vol.26, No.2-3 Pages: 233-247
  • NAID: 10028168961
  • Peer Reviewed
• [Journal Article] Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems (2009)
  • Author(s): M.T.Nakao, et al.
  • Journal Title: Journal of Scientific Computing online
  • Peer Reviewed
• [Journal Article] A Numerical Verification Method for a System of FitzHugh-Nagumo Type
  • Author(s): Chuting Cai, Kaori Nagatou, and Yoshitaka Watanabe
  • Journal Title: A Numerical Verification Method for a System of FitzHugh-Nagumo Type
• [Journal Article] 微分方程式の精度保証付き数値計算-逐次反復に基づく計算機援用証明
  • Author(s): 渡部善隆
  • Journal Title: 京都大学数理解析研究所別冊,掲載決定
• [Journal Article] A Posteriori Estimates of Inverse Operators for Boundary Value Problems in Linear Elliptic Partial Differential Equations
  • Author(s): Yoshitaka Watanabe, Takehiko Kinoshita, and Mitsuhiro T. Nakao
  • Journal Title: Mathematics of Computation Volume: (accepted)
• [Presentation] 無限次元作用素に対する固有値の除外理論と精度保証付き数値計算,九州大学数値解析学セミナー (2012)
  • Author(s): 渡部善隆,長藤かおり, Michael Plum,中尾充宏
  • Organizer: 九州大学大学院数理学研究院
  • Year and Date: 2012-04-21
• [Presentation] exflibにおける多倍長区間演算Fortranインターフェースとその応用 (2012)
  • Author(s): 渡部善隆,藤原宏志
  • Organizer: 日本応用数理学会2012年研究部会連合発表会
  • Place of Presentation: 九州大学
• [Presentation] Computer-assisted stability and instability proofs for the Orr-Sommerfeld problem with Poiseuille flow, Workshop on" Avoided? Crossing of Eigenvalue Curves (2012)
  • Author(s): Yoshitaka Watanabe
  • Organizer: Non-commutative Harmonic Oscillator, Special Functions and Number Theory-, Kyushu University
• [Presentation] 無限次元作用素に対する固有値の計算機援用非存在証明 (2011)
  • Author(s): 渡部善隆,長藤かおり, Michael Plum,中尾充宏
  • Organizer: 応用数学合同研究集会報告集
  • Place of Presentation: 龍谷大学
  • Year and Date: 2011-12-16
• [Presentation] Computer-assisted proof for functional equations based on infinite dimensional sequential iteration (2011)
  • Author(s): Y.Watanabe
  • Organizer: Japanese-German Workshop on Computer-Assisted Proofs and Verification Methods
  • Place of Presentation: Karlsruhe Institute of Technology, Germany
  • Year and Date: 2011-09-22
• [Presentation] rr-Sommerfeld方程式に対する局所一意性付き計算機援用証明 (2011)
  • Author(s): 渡部善隆,木下武彦,中尾充宏
  • Organizer: 日本数学会2011年度秋季総合分科会応用数学分科会講演アブストラクト
  • Place of Presentation: 信州大学
• [Presentation] 線形楕円型偏微分作用素の逆作用素に対する事後誤差評価について (2011)
  • Author(s): 木下武彦,渡部善隆,中尾充宏
  • Organizer: 日本数学会2011年度秋季総合分科会応用数学分科会講演アブストラクト
  • Place of Presentation: 信州大学
• [Presentation] Computer-assisted proof for functional equations based on infinite dimensional sequential iteration (2011)
  • Author(s): Yoshitaka Watanabe
  • Organizer: Japanese-German Workshop on Computer-Assisted Proofs and Verification Methods, Karlsruhe Institute of Technology, Germany
• [Presentation] 線形楕円型偏微分作用素の逆作用素に対する高精度な事後評価について (2011)
  • Author(s): 木下武彦,渡部善隆,中尾充宏
  • Organizer: 日本応用数理学会2011年度年会講演予稿集
  • Place of Presentation: 同志社大学
• [Presentation] 逐次反復に基づく関数方程式の計算機援用証明 (2011)
  • Author(s): 渡部善隆
  • Organizer: 日本応用数理学会2011年度年会講演予稿集
  • Place of Presentation: 同志社大学
• [Presentation] 微分方程式の精度保証付き数値計算,研究集会 (2011)
  • Author(s): 渡部善隆
  • Organizer: 可積分系数理の進化,京都大学数理解析研究所
• [Presentation] An enclosure method for solutions of FitzHugh-Nagumo equation (2010)
  • Author(s): Shuting Cai,長藤かおり,渡部善隆
  • Organizer: 応用数学合同研究集会報告集
  • Place of Presentation: 龍谷大学
  • Year and Date: 2010-12-16
• [Presentation] A self-validating norm computation of inverse for infinite dimensional linear operators and its applications (2010)
  • Author(s): Y.Watanabe
  • Organizer: SCAN 2010
  • Place of Presentation: ENS de Lyon, France
  • Year and Date: 2010-09-30
• [Presentation] 非自己共役作用素に対する固有値の除外法 (2010)
  • Author(s): 渡部善隆,長藤かおり,中尾充宏
  • Organizer: 日本応用数理学会2010年度年会
  • Place of Presentation: 明治大学
  • Year and Date: 2010-09-08
• [Presentation] MATLABで偏微分方程式の精度保証プログラムを作ってみました (2010)
  • Author(s): 渡部善隆
  • Organizer: 九州大学数値解析学セミナー,九州大学大学院数理学研究院
  • Year and Date: 2010-07-27
• [Presentation] Computer assisted proofs for partial differential equations (2010)
  • Author(s): Kaori Nagatou and Yoshitaka Watanabe
  • Organizer: Seminar at University of Kent, UK
  • Year and Date: 2010-03-31
• [Presentation] Computer-assisted existence proofs with local uniqueness for the Orr-Sommerfeld problem (2010)
  • Author(s): Y, Watanabe
  • Organizer: International Workshop on Numerical Verification and its Applications 2010
  • Place of Presentation: Hotel Lido Azzurro, Tokyo, Japan
  • Year and Date: 2010-03-11
• [Presentation] Legendre多項式を用いた高精度品質保証-およびその課題 (2010)
  • Author(s): 渡部善隆,木下武彦
  • Organizer: 日本応用数理学会2010年研究部会連合発表会
  • Place of Presentation: 筑波大学
  • Year and Date: 2010-03-08
• [Presentation] 逐次反復に基づく関数方程式の精度保証付き数値計算 (2010)
  • Author(s): 渡部善隆
  • Organizer: 研究集会:「科学計算の信頼性とその周辺に関するワークショップ」,西海国立公園九十九島ビジターセンター
  • Place of Presentation: 佐世保市
• [Presentation] 無限次元固有値問題に対する固有値の除外法 (2010)
  • Author(s): 渡部善隆
  • Organizer: 研究集会:「数値解析と計算の信頼性評価」,ハウステンボス,ユトレヒト
  • Place of Presentation: 佐世保市
• [Presentation] A self-validating norm computation of inverse for infinite dimensional linear operators and its applications (2010)
  • Author(s): Yoshitaka Watanabe
  • Organizer: 14th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics(SCAN 2010)
  • Place of Presentation: ENS de Lyon, France
• [Presentation] Computer-assisted existence proofs with local uniqueness for the Orr-Sommerfeld problem (2010)
  • Author(s): Yoshitaka Watanabe
  • Organizer: International Workshop on Numerical Verification and its Applications 2010(INVA2010)
  • Place of Presentation: Hotel Lido Azzurro, Hachijyo Island, Tokyo, Japan
• [Presentation] ある非自己共役作用素の複素固有値の除外に対する数値的検証法II (2009)
  • Author(s): 渡部善隆,中尾充宏, Michael Plum,長藤かおり
  • Organizer: 応用数学合同研究集会報告集
  • Place of Presentation: 龍谷大学
  • Year and Date: 2009-12-17
• [Presentation] Orr-Sommerfeld問題の解に対する局所一意性付き数値的検証 (2009)
  • Author(s): 渡部善隆
  • Organizer: 研究集会:「数値解析の現状と展望」-九州大学数値解析セミナーとその協力者からの発信
  • Place of Presentation: 公立はこだて未来大学
  • Year and Date: 2009-11-22
• [Presentation] Orr-Sommerfeld方程式の複素固有値の除外に対する計算機援用証明 (2009)
  • Author(s): 渡部善隆,中尾充宏, Michael Plum,長藤かおり
  • Organizer: 日本応用数理学会2009年度年会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-30
• [Presentation] 楕円型作用素の逆作用素の作用素ノルムの評価の改良について (2009)
  • Author(s): 木下武彦,渡部善隆,中尾充宏
  • Organizer: 日本数学会2009年度秋季総合分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-26
• [Presentation] Spectral problem for 3-D photonic crystals, Proceedings of the international Conference on Numerical Analysis and Applied Mathematics 2009(ICNAAM-2009) (2009)
  • Author(s): Kaori Nagatou, Malcolm Brown, Myoungnyoun Kim, Yoshitaka Watanabe and Ian Wood
  • Organizer: September 18-22, Rethymno, Crete, Greece. AIP Conference Proceedings
  • Year and Date: 2009-09-21
• [Presentation] 熱対流問題の精度保証-計算の品質評価 (2009)
  • Author(s): 渡部善隆
  • Organizer: 九州大学産業数理研究センター第7回ワークショップ
  • Year and Date: 2009-09-03
• [Presentation] 計算機援用証明による並行Poiseuille流れの安定性・不安性解析,九州大学数値解析学セミナー (2009)
  • Author(s): 渡部善隆
  • Organizer: 九州大学大学院数理学研究院
  • Year and Date: 2009-04-28
• [Presentation] An eigenvalue excluding method for the Orr-Sommerfeld problem (2009)
  • Author(s): Yoshitaka Watanabe
  • Organizer: International Conference on Engineering and Computational Mathematics(ECM2009)
  • Place of Presentation: The Hong Kong Polytechnic University, Hong Kong, China
• [Book] シミュレーション辞典,微分方程式と数値計算(日本シミュレーション学会編) (2012)
  • Author(s): 渡部善隆
  • Total Pages: 454
  • Publisher: コロナ社
• [Book] 実例で学ぶ精度保証付き数値計算～理論と実装～,臨時別冊・数理科学 (2011)
  • Author(s): 中尾充宏, 渡部善隆
  • Total Pages: 216
  • Publisher: サイエンス社
• [Book] 実例で学ぶ精度保証付き数値計算～理論と実装～ (2011)
  • Author(s): 中尾充宏, 渡部善隆
  • Total Pages: 216
  • Publisher: サイエンス社
• [Remarks]
  • URL: http://www.cc.kyushu-u.ac.jp/RD/watanabe/
• [Remarks]
  • URL: http://www.cc.kyushu-u.ac.jp/RD/watanabe/
• [Remarks]
  • URL: http://www.cc.kyushu-u.ac.jp/RD/watanabe/
• [Remarks]
  • URL: http://www.cc.kyushu-u.ac.jp/RD/watanabe/