Researcher Profile and Settings

URL

http://se.doshisha.ac.jp/doc/labo/keisansuuri/keisansuuri.html

researchmap

https://researchmap.jp/IMAI_hitoshi

Research Interests

• 金融工学
• 逆問題
• カオス
• 数値解析手法の開発
• 自由境界問題

Research Areas

• Informatics / Computational science
• Natural sciences / Applied mathematics and statistics
• Natural sciences / Basic mathematics

Research Experience

• Doshisha University, 教授, 2014/10 - Today
• The University of Tokushima, 教授, 1996/04 - 2014/09
• The University of Tokushima, 助教授, 1993/04 - 1996/03
• University of Tsukuba, 講師, 1990/10 - 1993/03
• University of Tsukuba, Lecturer, 1990 - 1993
• University of Tsukuba, 助手, 1988/04 - 1990/10

Education

• The University of Tokyo, The Graduate School of Engineering, Department of Applied Physics, 1985/04 - 1988/03
• The University of Tokyo, The Graduate School of Engineering, Department of Applied Physics, 1983/04 - 1985/03
• The University of Tokyo, The Faculty of Engineering, Department of Applied Physics, 1981/04 - 1983/03
• The University of Tokyo, College of Arts and Sciences, 理科I 類, 1979/04 - 1981/03

Degree

• Doctor of Engineering, The University of Tokyo

Association Memberships

• 日本応用数理学会
• 日本数学会
• 情報処理学会

Committee Memberships

• 応用数学分科会委員会委員、評議員、代議員、理論応用力学講演会運営委員, 2001 - 2005, 日本数学会, Society, 日本数学会
• 論文誌編集委員会委員、学会誌編集委員会委員、評議員, 1991 - 1997, 日本応用数理学会, Society, 日本応用数理学会

Published Papers

• Spatial Global Numerical Computation of the Integral Equations for the Ruin Probability
  Hiroko Soutome; Takuya Ooura; Naoyuki Ishimura; Hitoshi Imai
  Springer Proceedings in Mathematics and Statistics, Vol. 496. Mathematics for Sustainable Industry: ISMI 2024, Kuala Lumpur, Malaysia, September 9–11, 496, May 2025
• Efficient numerical computation of the integral equation for the ruin probability
  Hiroko Soutome; Naoyuki Ishimura; Hitoshi Imai
  2023 27th International Computer Science and Engineering Conference (ICSEC), 37 - 41, 2023
• Efficient numerical computation of the ruin probability
  Hiroko Soutome; Naoyuki Ishimura; Hitoshi Imai
  Proceedings of the 54th ISCIE International Symposium on Stochastic Systems Theory and Its Applications Nara, Oct. 14-15, 2022, 35 - 40, 2023
• Global in space numerical computation of the ruin probability
  H. Soutome; N. Ishimura; H. Imai
  Advances in Mathematical Sciences and Applications, 31(2) 397 - 406, 2022
• NUMERICAL REGULARITY MAP FOR FUNDAMENTAL ONE-DIMENSIONAL FRACTIONAL DIFFERENTIAL EQUATIONS WITH H\"{O}LDER CONTINUOUS SOLUTIONS
  Mana Kato; Hiroshi Fujiwara; Hitoshi Imai
  Advances in Mathematical Sciences and Applications, 30(2) 499 - 506, 2021
• Numerical Experiments on Regularity of a One-Variable Function by Using Chebyshev Collocation Derivative Matrices
  Hitoshi IMAI; Hideo SAKAGUCHI
  THE HARRIS SCIENCE REVIEW OF DOSHISHA UNIVERSITY, 62(2) 87 - 95, 2021
• Development of a high-precision numerical method for integration over one period of periodic functions with a sharp peak
  H. Ito; H. Imai; T. Ooura
  Advances in Mathematical Sciences and Applications, 30(1) 175 - 189, 2021
• Numerical Regularity Map for Blow-Up Solutions of Nonlinear Ordinary Differential Equations
  H. Soutome; H. Imai
  Advances in Mathematical Sciences and Applications, 29(2) 393 - 402, 2020, Scientific journal
• A simple numerical experiment on singularity of one-variable functions by using the spectral collocation method
  Hitoshi IMAI; Hideo SAKAGUCHI
  THE HARRIS SCIENCE REVIEW OF DOSHISHA UNIVERSITY, 59(4) 217 - 226, 2019
• Numerical experiments on analyticity of solutions to fractional differential equations
  H. Fujiwara; N. Higashimori; H. Imai
  Advances in Mathematical Sciences and Applications, 27(1) 169 - 180, 2018, Scientific journal
• Numerical limit and its application to a blow-up problem related to default risk
  Y. Sasaki; H. Soutome; H. Imai; N. Ishimura
  Advances in Mathematical Sciences and Applications, 26(1) 29 - 38, 2017, Scientific journal
• On numerical challenge for the distinction between smooth functions and analytic functions
  Krishna Chandra Datta; Hitoshi Imai; Hideo Sakaguchi
  Theoretical and Applied Mechanics Japan, Science Council of Japan, 62 107 - 117, 2014, Scientific journal
  

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• On Numerical Computation of Rank Deficit of Matrices in Multiple Precision
  Enkhbayar Azjargal; Hitoshi Imai; Hideo Sakaguchi
  Advances in Mathematical Sciences and Applications, 23(2) 477 - 486, 2013, Scientific journal
• Computing Attractors for a Stefan Problem with Heat Source Term Using the Spectral Collocation Method
  Shewli Shamim Shanta; Hitoshi Imai; Mohamad Faisal Abdul Karim; Indrojit Kumar Pramanik
  Far East Journal of Mathematical Sciences, 71(1) 87 - 104, 2012, Scientific journal
• Numerical Simulation on Local Solutions of Partial Differential Equations
  Hitoshi Imai; Hideo Sakaguchi
  Theoretical and Applied Mechanics Japan, 61 185 - 193, 2012, Scientific journal
• Simple Numerical Judgement on the Singularity of the Matrix by Using the Multiple-Precision Arithmetic
  Hitoshi Imai; Hideo Sakaguchi; Yuusuke Iso
  Theoretical and Applied Mechanics Japan, 60 343 - 351, 2012, Scientific journal
• Numerical Simulation on Non-Existence and Non-Uniqueness of Solutions for the Tricomi Equation
  Hitoshi Imai; Hideo Sakaguchi; Yuusuke Iso
  Gakuto International Series, Mathematical Sciences and Applications, 34 39 - 58, 2011, Scientific journal
• Numerical integration of delay differential equations by the spectral collocation method
  安部 公輔; 今井 仁司; 中村 正彰
  日本大学理工学部一般教育教室彙報, 日本大学理工学部, 89(89) 1 - 10, 2011, Research institution
• On Numerical Computation of the Toricomi Equation
  Hitoshi Imai; Hideo Sakaguchi; Yuusuke Iso
  Theoretical and Applied Mechanics Japan, 59 359 - 372, 2011, Scientific journal
• Numerical Continuation for the Laplace Equation with Higher Order Regularization
  Hitoshi Imai; Hideo Sakaguchi
  Gakuto International Series, Mathematical Sciences and Applications, 32 131 - 144, 2010, Scientific journal
• Numerical Computation of Continuation Problems in the Annular Domain
  Hitoshi Imai; Hideo Sakaguchi; Toshiki Takeuchi
  Theoretical and Applied Mechanics Japan, 58 153 - 164, 2010, Scientific journal
• Space-Precise Computation of a Singular Nonlinear Evolution Equation for the Risk Preference
  Hitoshi Imai; Naoyuki Ishimura; Masahiro Kushida
  IMECS 2009: INTERNATIONAL MULTI-CONFERENCE OF ENGINEERS AND COMPUTER SCIENTISTS, VOLS I AND II, 2135 - +, 2009, International conference proceedings
• Numerical Treatment of Nonlinear Partial Differential Equations for the Risk Preference
  Masahiro Kushida; Naoyuki Ishimura; Hitoshi Imai
  Theoretical and Applied Mechanics Japan, 57 487 - 492, 2009, Scientific journal
• Global in space numerical computation for the nonlinear Black-Scholes equation
  Naoyuki Ishimura; Hitoshi Imai
  Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, Nova Science Publishers, Inc., 219 - 242, Apr. 2008, In book
• リスク選好の偏微分方程式に対する数値解析
  櫛田 雅弘; 今井 仁司
  日本応用数理学会論文誌, 18(4) 681 - 686, 2008, Scientific journal
• Parallel Computing of Interval Arithmetic in Multiple Precision for Simultaneous Linear Equations
  Hideo Sakaguchi; Hitoshi Imai; Toshiki Takeuchi
  Gakuto International Series, Mathematical Sciences and Applications, 28 165 - 172, 2008, Scientific journal
• Global in space simulation for the black-scholes equation incorporating transaction costs
  Zhenyu Jin; Hideo Sakaguchi; Naoyuki Ishimura; Hitoshi Imai
  Theoretical and Applied Mechanics Japan, Science Council of Japan, 56 445 - 450, 2008, Scientific journal
  

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• Numerical computation on analyticity of the solution of a cauchy problem for the backward heat equation
  Hitoshi Imai; Hideo Sakaguchi
  Theoretical and Applied Mechanics Japan, Science Council of Japan, 56 291 - 300, 2008, Scientific journal
  

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• Numerical Treatment of a Singular Nonlinear Partial Differential Equation Arising in the Optimal Investment
  Hitoshi Imai; Naoyuki Ishimura; Masahiro Kushida
  Thai Journal of Mathematics, 5 321 - 326, 2007, Scientific journal
• Computational technique for treating the nonlinear Black-Scholes equation with the effect of transaction costs
  Hitoshi Imai; Naoyuki Ishimura; Hideo Sakaguchi
  KYBERNETIKA, 43(6) 807 - 815, 2007, Scientific journal
• 空間1次元熱伝導方程式の解の接続の数値計算
  今井 仁司; 坂口 秀雄
  日本応用数理学会論文誌, 17(4) 481 - 493, 2007, Scientific journal
• Numerical treatment of analytic continuation with multiple-precision arithmetic
  Hiroshi Fujiwara; Hitoshi Imai; Toshiki Takeuchi; Yuusuke Iso
  Hokkaido Mathematical Journal, 36(4) 837 - 847, 2007, Scientific journal
  

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• On the Hoggard-Whalley-Wilmott equation for the pricing of options with transaction costs
  Hitoshi Imai; Naoyuki Ishimura; Ikumi Mottate; Masaaki Nakamura
  Asia-Pacific Financial Markets, 13(4) 315 - 326, Dec. 2006, International conference proceedings
  

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• Some Numerical Experiments on Global Simulation of the Backward Heat Conduction Problem
  Toshiki Takeuchi; Hitoshi Imai; Yinglian Zhu
  Theoretical and Applied Mechanics Japan, 55 175 - 184, 2006, Scientific journal
  

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• 有界化による熱伝導逆問題の大域的数値計算
  祝 穎蓮; 竹内 敏己; 今井 仁司
  日本応用数理学会論文誌, 16(1) 27 - 36, 2006, Scientific journal
• Some Methods for Removing Singularity and Infinity in Numerical Simulation
  Hitoshi Imai
  Gakuto International Series, Mathematical Sciences and Applications, 23 103 - 118, 2005, Scientific journal
• 第一種積分方程式の高精度数値計算について
  藤原 宏志; 今井 仁司; 竹内 敏己; 磯 祐介
  日本応用数理学会論文誌, 15(3) 419 - 434, 2005
• Global Simulation of a Backward Heat Conduction Problem with a Variable Transform on Time
  Toshiki Takeuchi; Hitoshi Imai; Hideo Sakaguchi
  Theoretical and Applied Mechanics Japan, 54 319 - 326, 2005, Scientific journal
  

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• A Numerical Method for Tracking the Level Set in One-Dimensional Problems
  Hideo Sakaguchi; Hitoshi Imai
  Gakuto International Series, Mathematical Sciences and Applications, 20 277 - 288, 2004, Scientific journal
• One-Phase Stefan Problems for Sublinear Heat Equations: Asymptotic Behavior of Solutions
  Toyohiko Aiki; Hitoshi Imai; Naoyuki Ishimura; Yoshio Yamada
  Communications in Applied Analysis, 8(1) 1 - 15, 2004, Scientific journal
• 応用解析における多倍長計算
  今井 仁司
  数学, 55(3) 316 - 325, 2003, Scientific journal
• Direct Numerical Simulations of Cauchy Problems for the Laplace Operators
  Toshiki Takeuchi; Hitoshi Imai
  Advances in Mathematical Sciences and Applications, 13(2) 587 - 609, 2003, Scientific journal
• Well-posedness of one-phase Stefan problems for sublinear heat equations
  T Aiki; H Imai; N Ishimura; Y Yamada
  NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 51(4) 587 - 606, Nov. 2002, Scientific journal
  

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• Numerical computation of Lyapunov exponents related to attractors in a free boundary problem
  H Imai; T Takeuchi; SS Shanta; N Ishimura; T Aiki
  NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 47(6) 3823 - 3833, Aug. 2001, Scientific journal
  

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• A numerical approach to the asymptotic behavior of solutions of a one-dimensional free boundary problem of hyperbolic type
  H Imai; K Kikuchi; K Nakane; S Omata; T Tachikawa
  JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS, 18(1) 43 - 58, Feb. 2001, Scientific journal
• 自由境界問題の数値解法 - 数理ファイナンスへの応用 -
  石村 直之; 今井 仁司; 竹内 敏己
  一橋論叢, 126(4) 419 - 428, 2001, Scientific journal
• Some Advanced Applications of the Spectral Collocation Method
  Hitoshi Imai; Toshiki Takeuchi
  Gakuto International Series, Mathematical Sciences and Applications, 17 323 - 335, 2001, Scientific journal
• Numerical Computation of Attractors in Free Boundary Problems
  Shewli Shamim Shanta; Toshiki Takeuchi; Hitoshi Imai; Masahiro Kushida
  Advances in Mathematical Sciences and Applications, 11(2) 531 - 548, 2001, Scientific journal
• Stability of global solutions to one-phase Stefan problem for a semilinear parabolic equation
  T Aiki; H Imai
  CZECHOSLOVAK MATHEMATICAL JOURNAL, 50(1) 135 - 153, 2000, Scientific journal
  

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• Numerical Simulation of One-Dimensional Free Boundary Problems in Infinite Precision
  Tarmizi,Toshiki Takeuchi; Hitoshi Imai; Masahiro Kushida
  Advances in Mathematical Sciences and Applications, 10(2) 661 - 672, 2000, Scientific journal
• Numerical Simulation of Some One-Dimensional Free Boundary Problems in Arbitrary Precision
  Tarmizi,Toshiki Takeuchi; Hitoshi Imai; Masahiro Kushida
  Gakuto International Series, Mathematical Sciences and Applications, 14 440 - 452, 2000, Scientific journal
• Analysis on One-Phase Stefan Problems for Semilinear Parabolic Equations with the Dirichlet Boundary Condition
  Hitoshi Imai; Naoyuki Ishimura; Toyohiko Aiki
  Gakuto International Series, Mathematical Sciences and Applications, 14 176 - 183, 2000, Scientific journal
• A crystalline motion of spiral-shaped curves with symmetry
  H Imai; N Ishimura; T Ushijima
  JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 240(1) 115 - 127, Dec. 1999, Scientific journal
  

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• Motion of spirals by crystalline curvature
  H Imai; N Ishimura; T Ushijima
  RAIRO-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 33(4) 797 - 806, Jul. 1999, Scientific journal
• A Direct Approach to an Inverse Problem
  Hitoshi Imai; Toshiki Takeuchi; Masaaki Nakamura; Naoyuki Ishimura
  Gakuto International Series, Mathematical Sciences and Applications, 12 223 - 232, 1999, Scientific journal
• On Numerical Simulation of Partial Differential Equations in Infinite Precision
  Hitoshi Imai; Toshiki Takeuchi; Masahiro Kushida
  Advances in Mathematical Sciences and Applications, 9(2) 1007 - 1016, 1999, Scientific journal
• Global Existence of Solutions to One-Phase Stefan Problems for Semilinear Parabolic Equations.
  Toyohiko Aiki; Hitoshi Imai
  ANNALI DI MATEMATICA PURA ED APPLICATA, 175(1) 327 - 337, Dec. 1998, Scientific journal
• Numerical Computations of Free Boundary Problems in Quadruple Precision Arithmetic Using an Explicit Method
  Hitoshi Imai; Yoshitane Shinohara; Takeshi Konno; Makoto Natori; Weidong Zhou; Isamu Ohnishi; Yasumasa Nishiura
  Gakuto International Series, Mathematical Sciences and Applications, 11 193 - 207, 1998, Scientific journal
• A Model of a Contact Angle Problem and its Numerical Simulation
  Takao Hanada; Hitoshi Imai
  Gakuto International Series, Mathematical Sciences and Applications, 11 181 - 192, 1998, Scientific journal
• Convergence of Attractors for Simplified Magnetic Benard System
  Hitoshi Imai; Naoyuki Ishimura; Masaaki Nakamura
  Gakuto International Series, Mathematical Sciences and Applications, 11 136 - 144, 1998, Scientific journal
• On the Finite Determination of the Solutions for the MHD Equations
  Hitoshi Imai; Naoyuki Ishimura; Masaaki Nakamura
  Gakuto International Series, Mathematical Sciences and Applications, 11 126 - 135, 1998, Scientific journal
• Modified Hele-Shaw Moving Boundary Problem Related to Some Phase Transition Phenomena
  Isamu Ohnishi; Hitoshi Imai
  Bulletin of the University of Electro-Communications, The University of Electro-Communications, 11(1) 17 - 28, 1998, Scientific journal
• Existence and uniqueness of quasiperiodic solutions to perturbed nonlinear oscillators
  Zulfikar Ali; Yoshitane Shinohara; Hitoshi Imai; Hideo Sakaguchi; Kuniya Okamoto
  Japan Journal of Industrial and Applied Mathematics, Kinokuniya Co. Ltd, 15(2) 279 - 293, 1998, Scientific journal
  

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• Analytical approach to estimating the dimension of attractors
  T Hakamada; Imai, H; N Ishimura
  APPLIED MATHEMATICS AND OPTIMIZATION, 34(1) 29 - 36, Jul. 1996, Scientific journal
• The Optimal Parameter Preconditioning for Spectral Collocation Methods
  Zhou Weidong; Imai Hitoshi; Natori Makoto; Jin Chenghai
  Transactions of the Japan Society for Industrial and Applied Mathematics, The Japan Society for Industrial and Applied Mathematics, 6(3) 191 - 204, 1996, Scientific journal
  

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• Wavelet変換による海岸線データの間引き
  桧山 澄子; 花田 孝郎; 今井 仁司
  日本応用数理学会論文誌, 6(1) 83 - 99, 1996, Scientific journal
• Convergence of Attractors for the Simplified Magnetic Benard Equations
  Hitoshi Imai; Naoyuki Ishimura; Masaaki Nakamura
  European Journal of Applied Mathematics, 7 53 - 62, 1996, Scientific journal
• Chaos in the magnetic Benard problem(共著)
  Naoyuki Ishimura; Hitoshi Imai; Masaaki Nakamura
  BUTSURI, The Physical Society of Japan, 50(9) 697 - 703, 1995, Scientific journal
  

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• Numerical Simulation of the Plug Flow Appearing in the Bingham Fluid
  Masahiro Fukuda; Hitoshi Imai; Hideo Kawarada
  Advances in Mathematical Sciences and Applications, 4(1) 227 - 239, 1994, Scientific journal
• 自由表面を有する熱対流の数値シミュレーションと線形安定性解析
  周 偉東; 今井 仁司; 名取 亮
  日本応用数理学会論文誌, 4(1) 27 - 40, 1994, Scientific journal
• Numerical Analysis of the Simplified Magnetic Benard Problem
  Hitoshi Imai; Masaaki Nakamura
  Gakuto International Series, Mathematical Sciences and Applications, 2 405 - 419, 1993, Scientific journal
• Numerical Study of Natural Convection with a Free Surface by a Spectral Method
  Weidong Zhou; Hitoshi Imai; Makoto Natori
  Gakuto International Series, Mathematical Sciences and Applications, 1 49 - 60, 1993, Scientific journal
• Numerical Computations of Free Boundary Problems Using the Spectral Method
  Hitoshi Imai; Weidong Zhou; Makoto Natori; Hideo Kawarada
  Gakuto International Series, Mathematical Sciences and Applications, Gakuto, 1 39 - 47, 1993, Scientific journal
• Numerical Computation for Solidification Problems with Moving Surface
  Takao Hanada; Hitoshi Imai; Hideo Kawarada; Makoto Natori
  Gakuto International Series, Mathematical Sciences and Applications, 1 17 - 38, 1993, Scientific journal
• 磁気ベナール対流の簡約化モデルとそのシミュレーション
  今井 仁司; 中村 正彰
  シミュレーション, 12(2) 99 - 106, 1993, Scientific journal
• 実用的な曲線データ点の間引き法
  桧山 澄子; 花田 孝郎; 今井 仁司
  日本応用数理学会論文誌, 3(2) 85 - 104, 1993, Scientific journal
• Reorthogonalization in the Block Lanczos Algorithm
  Hiroko Iguchi; Makoto Natori; Hitoshi Imai
  Bulletin Greek Mathematical Society, 33 25 - 39, 1992, Scientific journal
• A New Reorthogonalization in the Lanczos Algorithm
  Hitoshi Imai; Makoto Natori; Eiji Kawamura
  Journal of Information Processing, 14(1) 56 - 59, 1991, Scientific journal
• A Method for Finding Bifurcation Points
  Hitoshi Imai; Hideo Kawarada
  Control and Cybernetics, 20(1) 7 - 19, 1991, Scientific journal
• Numerical Computations for Solidification Problems with Change of Volume
  Takao Hanada; Hitoshi Imai; Hideo Kawarada; Makoto Natori
  Bulletin Greek Mathematical Society, 31 29 - 49, 1990, Scientific journal
• An approximate resolution of a free boundary problem appearing in the equilibrium plasma by means of conformal mapping
  Hideo Kawarada; Toshio Sawaguri; Hitoshi Imai
  Japan Journal of Applied Mathematics, 6(3) 331 - 340, Oct. 1989, Scientific journal
  

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• One-component asymmetric plasmas in a symmetric vessel
  Hitoshi Imai; Hideo Kawarada
  Japan Journal of Applied Mathematics, 5(2) 173 - 186, Jun. 1988, Scientific journal
  

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• Numerical Analysis of a Free Boundary Problem Related to Two-Dimensional Plasma Equilibrium
  Hitoshi Imai
  The University of Tokyo, 1988, Doctoral thesis
• 表面電流を持つ２次元平衡プラズマの分岐現象の数値解析的研究
  今井 仁司
  東京大学, 1985, Master thesis

MISC

• 野球の打撃フォームにおける角度解析への初歩的アプローチ - テイクバック時の肩の方位角 -
  児玉祐軌; 松井孝真; 馬場誠太郎; 今井仁司
  同志社大学ハリス理化学研究報告, 57(2) 75 - 80, 2016
• Numerical Computation for Smoothness of the Solution of a One-Dimensional Hyperbolic Equation
  Enkhbayar Azjargal; Naoki Wada; Hitoshi Imai; Hideo Sakaguchi
  The Sixth International Conference on Science and Mathematics Education in Developing Countries, 78 - 87, 2014
• Tricomi方程式の数値計算について
  今井 仁司; 坂口 秀雄; 磯 祐介
  第59回理論応用力学講演会講演論文集, 357 - 358, 2010
• スペクトル選点法による遅延微分方程式の高精度数値計算
  安部 公輔; 今井 仁司; 中村 正彰
  第59回理論応用力学講演会講演論文集, 355 - 356, 2009
• 円環領域における解の接続問題の数値計算について
  今井 仁司; 坂口 秀雄; 竹内 敏己
  第58回理論応用力学講演会講演論文集, 481 - 482, 2009
• チコノフの正則化を用いた逆問題の解の設計について
  今井 仁司; 坂口 秀雄
  第58回理論応用力学講演会講演論文集, 479 - 480, 2009
• Some numerical experiments for continuation problems of the two-dimensional Laplace operator
  Takeuchi Toshiki; Imai Hitoshi
  NCTAM papers, National Congress of Theoretical and Applied Mechanics, Japan, 日本学術会議 「機械工学委員会・土木工学・建築学委員会合同IUTAM分科会」, 57(0) 239 - 239, 2008
• A Numerical Method for Distinction between Blow-up and Global Solutions of the Nonlinear Heat Equation
  Hideo Sakaguchi; Hitoshi Imai
  Journal of Mathematics, The University of Tokushima, The University of Tokushima, 42 27 - 44, 2008
• リスク選好に対する非線形偏微分方程式の数値解法 -最適投資問題の場合-
  櫛田 雅弘; 石村 直之; 今井 仁司
  第57回理論応用力学講演会講演論文集, 531 - 532, 2008
• 複素ニュートン法の熱伝導逆問題への応用
  今井 仁司; 坂口 秀雄
  第57回理論応用力学講演会講演論文集, 523 - 524, 2008
• ラプラス作用素のCauchy問題の解の接続に関するいくつかの数値計算
  竹内 敏己; 今井 仁司
  第57回理論応用力学講演会講演論文集, 515 - 516, 2008
• IPNS of Cauchy problems in the annulus
  Jin Zhenyu; Takeuchi Toshiki; Imai Hitoshi; Sakaguchi Hideo
  NCTAM papers, National Congress of Theoretical and Applied Mechanics, Japan, 日本学術会議 「機械工学委員会・土木工学・建築学委員会合同IUTAM分科会」, 56(0) 270 - 270, 2007
• 熱伝導逆問題の初期値問題に関連する数値計算
  今井 仁司; 坂口 秀雄
  数理解析研究所講究録, 京都大学, 1566 170 - 180, 2007
• 無限精度計算が切り開く応用解析・数値解析の未来
  今井 仁司
  数理解析研究所講究録, 京都大学, 1566 96 - 118, 2007
• 取引費用を考慮したBlack-Scholes方程式の空間大域的数値シミュレーション
  金 珍玉; 坂口 秀雄; 石村 直之; 今井 仁司
  第56回理論応用力学講演会講演論文集, 569 - 570, 2007
• 円環領域におけるCauchy問題の無限精度数値計算
  金 珍玉; 竹内 敏己; 今井 仁司; 坂口 秀雄
  第56回理論応用力学講演会講演論文集, 541 - 542, 2007
• 熱伝導逆問題の初期値問題の解の解析性に関する数値計算
  今井 仁司; 坂口 秀雄
  第56回理論応用力学講演会講演論文集, 539 - 540, 2007
• 643 Numerical computation of the heat equation with insufficient boundary conditions by the spectral collocation method
  IMAI Hitoshi; SAKAGUCHI Hideo
  The Computational Mechanics Conference, The Japan Society of Mechanical Engineers, 2006(19) 741 - 742, 02 Nov. 2006
• Direct Numerical Simulations of Some Inverse Problems in Multiple Precision
  Hitoshi Imai; Toshiki Takeuchi
  Proceeding of the 5th Asian Symposium on Applied Electromagnetics and Mechanics, 413 - 417, 2006
• 熱伝導逆問題の大域的数値計算に関するいくつかの数値実験
  今井 仁司; 竹内 敏己; 祝 穎蓮
  第55回理論応用力学講演会講演論文集, 473 - 474, 2006
• Numerical computation of an inverse problem governed by the heat equation with a variable transformation on time
  Takeuchi Toshiki; Imai Hitoshi; Sakaguchi Hideo
  NCTAM papers, National Congress of Theoretical and Applied Mechanics, Japan, 日本学術会議 「機械工学委員会・土木工学・建築学委員会合同IUTAM分科会」, 54(0) 260 - 260, 2005
• 有界化法による微分方程式の数値計算
  今井 仁司
  数理解析研究所講究録, 1441 173 - 186, 2005
• 無限精度数値シミュレーションの拓く計算力学の新たな可能性
  今井 仁司
  日本計算数理工学会誌計算数理工学レビュー, 日本計算数理工学会, 2005-1(1) 21 - 32, 2005
  

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• 多倍長計算を適用した精度保証数値計算
  坂口 秀雄; 渡部 善隆; 今井 仁司
  数理解析研究所講究録, 1441 165 - 172, 2005
• 時間に関する変数変換を用いた熱伝導逆問題に対する数値計算
  竹内 敏己; 今井 仁司; 坂口 秀雄
  第54回理論応用力学講演会講演論文集, 543 - 544, 2005
• 極座標変換に伴う微分方程式の特異性の回避公式について
  今井 仁司
  数理解析研究所講究録, 京都大学, 1362 161 - 168, 2004
• 等高点追跡の無限精度数値計算法の提案とその並列計算
  坂口 秀雄; 今井 仁司
  第53回理論応用力学講演会講演論文集, 405 - 406, 2004
• Laplace作用素のCauchy問題における数値誤差の影響について
  竹内 敏己; 今井 仁司; 坂口 秀雄
  第53回理論応用力学講演会講演論文集, 311 - 312, 2004
• 熱伝導方程式の逆問題に対するいくつかの数値実験
  今井 仁司; 竹内 敏己
  第53回理論応用力学講演会講演論文集, 309 - 310, 2004
• Parallel IPNS of an inverse problem for the heat equation
  Takeuchi Toshiki; Imai Hitoshi; Iso Yuusuke
  NCTAM papers, National Congress of Theoretical and Applied Mechanics, Japan, 日本学術会議 「機械工学委員会・土木工学・建築学委員会合同IUTAM分科会」, 52(0) 212 - 212, 2003
• IPNS of Cauchy problems for the elliptic operator
  Imai Hitoshi; Takeuchi Toshiki; Sakaguchi Hideo
  NCTAM papers, National Congress of Theoretical and Applied Mechanics, Japan, 日本学術会議 「機械工学委員会・土木工学・建築学委員会合同IUTAM分科会」, 52(0) 211 - 211, 2003
• Eguchi-Oki-Matsumura equation for phase separation: Numerically guided approach
  T Hanada; H Imai; N Ishimura; MA Nakamura
  RECENT DEVELOPMENT IN THEORIES & NUMERICS, 433 - 442, 2003
• Direct simulation of an integral equation of the first kind
  H Imai; T Takeuchi
  RECENT DEVELOPMENT IN THEORIES & NUMERICS, 247 - 254, 2003
• Exact American Option Pricing and the Free Boundary Problem
  今井 仁司; 竹内 敏己; 坂口 秀雄
  第52回理論応用力学講演会講演論文集, 501 - 502, 2003
• 熱伝導方程式に関する逆問題の無限精度並列数値シミュレーション
  竹内 敏己; 今井 仁司; 磯 祐介
  第52回理論応用力学講演会講演論文集, 197 - 198, 2003
• 楕円型作用素のコーシー問題に対する無限精度数値シミュレーション
  今井 仁司; 竹内 敏己; 坂口 秀雄
  第52回理論応用力学講演会講演論文集, 195 - 196, 2003
• On a method in analysis of the American put option
  石村 直之; 竹内 敏己; 今井 仁司
  Proceedings of the Annual Conference of the Japan Society for Industrial and Applied Mathematics, 一般社団法人 日本応用数理学会, 2002(0) 31 - 31, 2002
• Visualization with arbitrary manification for solutions of perturbed systems
  菱沼 哲也; 坂口 秀雄; 竹内 敏己; 今井 仁司
  Proceedings of the Annual Conference of the Japan Society for Industrial and Applied Mathematics, 一般社団法人 日本応用数理学会, 2002(0) 239 - 239, 2002
• A certain notes about the parallel computing in an infinite precision numerical simulation
  坂口 秀雄; 竹内 敏己; 今井 仁司
  Proceedings of the Annual Conference of the Japan Society for Industrial and Applied Mathematics, 一般社団法人 日本応用数理学会, 2002(0) 204 - 204, 2002
• On a linear system appearing in the applications of DDM to IPNS and iterative methods
  竹内 敏己; 坂口 秀雄; 金 成海; 今井 仁司
  Proceedings of the Annual Conference of the Japan Society for Industrial and Applied Mathematics, 一般社団法人 日本応用数理学会, 2002(0) 127 - 127, 2002
• Domain Decomposition Method and Infinite-Precision Numerical Simulation
  Toshiki Takeuchi; Hitoshi Imai
  RIMS Kokyuroku, Kyoto University, 1288 102 - 107, 2002
• A CGS-Like Method for Solving Nonsymmetric Linear Systems
  Cheng Hai Jin; Shao Liang Zhang; Hitoshi Imai
  Proceedings of Fifth China-Japan Seminar on Numerical Mathematics, 147 - 153, 2002
• Parallel Computing in Infinite Precision Numerical Simulation for PDE Systems
  Hitoshi Imai; Hideo Sakaguchi; Toshiki Takeuchi
  Proceedings of Fifth China-Japan Seminar on Numerical Mathematics, 141 - 146, 2002
• On One-Phase Stefan Problems for Sublinear Heat Equations
  Toyohiko Aiki; Hitoshi Imai; Naoyuki Ishimura; Yoshio Yamada
  Proceeding of the Third Asian Mathematical Conference 2000, 6 - 11, 2002
• On Super Numerical Simulation
  Hitoshi Imai; Toshiki Takeuchi; Hideo Sakaguchi; Tetsuya Hishinuma
  RIMS Kokyuroku, Kyoto University, 1265 9 - 17, 2002
• Methods for an Integral Equation of the First Kind
  今井 仁司; 竹内 敏己; 磯 祐介; 藤原 宏志
  第51回理論応用力学講演会講演論文集, 589 - 590, 2002
• アメリカン・オプションの数値解析 (〔第1回〕金融工学合同研究集会報告)
  今井 仁司; 竹内 敏己
  四国大学経営情報研究所年報, 四国大学経営情報研究所, (7) 181 - 187, Dec. 2001
• A New Numerical Technique for the Option Pricing Problem of American Type
  Toshiki Takeuchi; Naoyuki Ishimura; Hitoshi Imai
  World Multiconference on Systemics, Cybernetics and Informatics Proceedings, 2 233 - 236, 2001
• On Numerical Methods for Analysis of Chaotic Phenomena in Free Boundary Problems
  Hitoshi Imai; Toshiki Takeuchi
  RIMS Kokyuroku, Kyoto University, 1210 115 - 128, 2001
• On Numerical Methods for Solving Linear Systems Appearing in Infinite Precision Numerical Simulation
  Toshiki Takeuchi; Hideo Sakaguchi; Cheng Hai Jin; Hitoshi Imai
  RIMS Kokyuroku, Kyoto University, 1198 154 - 160, 2001
• 逐次的数値解法による連続場中の形状同定
  杉野 隆三郎; 今井 仁司; 登坂 宣好
  第50回理論応用力学講演会講演論文集, 455 - 456, 2001
• いくつかの逆問題の無限精度計算
  今井 仁司; 竹内 敏己; 磯 祐介
  第50回理論応用力学講演会講演論文集, 449 - 450, 2001
• Asymptotic Behavior of Solutions to One-phase Stefan Problems for Sublinear Heat Equations (Nonlinear Diffusive Systems : Dynamics and Asymptotics)
  Aiki Toyohiko; Imai Hitoshi; Ishimura Naoyuki; Yamada Yoshio
  RIMS Kokyuroku, Kyoto University, 1178 36 - 47, Dec. 2000
• Numerical Computation of Lyapunov Exponents Related to Attractors in a Free Boundary Problem
  Hitoshi Imai; Toshiki Takeuchi; Shewli Shamim Shanta; Naoyuki Ishimura
  NIFS-PROC, 46 21 - 29, 2000
• Infinite Precision Numerical Simulation for PDE Systems and its Applications
  Hitoshi Imai; Toshiki Takeuchi; Hideo Sakaguchi
  RIMS Kokyuroku, Kyoto University, 1147 42 - 50, 2000
• 準周期的 Duffing 方程式の解の存在と一意性および近似解の精度保証について
  篠原 能材; 今井 仁司; 竹内 敏己; 蔭西 義輝
  数理解析研究所講究録, 京都大学, 1147 28 - 31, 2000
• Numerical Computation of Attractors in Two-Phase Stefan Problems
  Toshiki Takeuchi; Hitoshi Imai; Shewli Shamim Shanta; Naoyuki Ishimura
  RIMS Kokyuroku, Kyoto University, 1145 220 - 228, 2000
• Numerical Simulation of One-Phase Stefan Problems in Arbitrary Precision
  Tarmizi,Hitoshi Imai; Toshiki Takeuchi; Masahiro Kushida
  RIMS Kokyuroku, Kyoto University, 1129 129 - 138, 2000
• ポテンシャル場における形状決定問題のBEM解析
  杉野 隆三郎; 今井 仁司; 登坂 宣好
  第49回理論応用力学講演会講演論文集, 359 - 360, 2000
• １次元１相ステファン問題の無限精度数値シミュレーション
  Tarmizi,今井 仁司,竹内 敏己,櫛田 雅弘
  信学技報, 99(20) 45 - 52, 1999
• 振動的な界面運動を伴う移動境界問題のBEM解析
  三原 登志男; 杉野 隆三郎; 今井 仁司; 登坂 宣好
  計算工学講演会論文集, 4(2) 997 - 998, 1999
• On Multiple Precision Calculation of Eigenvalues and Eigenvectors of Matrices
  Masahiro Kushida; Hitoshi Imai; Toshiki Takeuchi
  NIFS-PROC, 40 48 - 57, 1999
• Application of the Infinite-Precision Numerical Simulation to an Inverse Problem
  Hitoshi Imai; Toshiki Takeuchi
  NIFS-PROC, 40 38 - 47, 1999
• 液滴が示す非線形運動の数値計算
  三原 登志男; 杉野 隆三郎; 今井 仁司; 登坂 宣好
  第48回理論応用力学講演会講演論文集, 101 - 102, 1999
• 滑らかな解を持つ偏微分方程式の任意精度数値シミュレーション
  今井 仁司; 竹内 敏己; 坂口 秀雄; 篠原 能材; Tarmizi
  統計数理研究所共同リポート, 110 158 - 167, 1998
• Existence and Uniqueness of Quasiperiodic Solutions to Van der Pol Type Equations
  Zulfikar Ali; Yoshitane Shinohara; Hitoshi Imai; Atsuhito Kohda; Kuniya Okamoto; Hideo Sakaguchi; Haruo Miyamoto
  Journal of Mathematics, Tokushima University, The University of Tokushima, 31 69 - 80, 1998
• Numerical Analysis of a Free Boundary Problem Governed by a Hyperbolic Equation
  Hitoshi Imai; Seiro Omata; Kazuaki Nakane; Koji Kikuchi
  Proceedings of Third China-Japan Seminar on Numerical Mathematics, 214 - 221, 1998
• Convergence of Fractal Dimensions of Attractors for the Simplified Magnetic Benard System
  Hitoshi Imai; Naoyuki Ishimura; Masaaki Nakamura
  Proceedings of Third China-Japan Seminar on Numerical Mathematics, 73 - 81, 1998
• 偏微分方程式の任意精度数値シミュレーションについて
  今井 仁司; 竹内 敏己; 坂口 秀雄; 篠原 能材; Tarmizi
  数理解析研究所講究録, 京都大学, 1(1040) 92 - 99, 1998
• 摂動項を伴う非線形常微分方程式の準周期解の存在と一意性について(科学技術における数値計算の理論と応用II)
  ALI Zulfikar; 篠原 能材; 今井 仁司; 坂口 秀雄
  数理解析研究所講究録, 京都大学, 990 179 - 187, Apr. 1997
• Some Results on Behavior of Solutions to One-Phase Stefan Problems for a Semilinear Parabolic Equation
  Toyohiko Aiki; Hitoshi Imai
  RIMS Kokyuroku, Kyoto University, 1009 60 - 78, 1997
• 摂動項を伴う非線形振動の準周期解の存在と一意性について
  Zulfikar Ali; 篠原 能材; 今井 仁司; 坂口 秀雄; 岡本 邦也
  信学技報, 97(53) 9 - 16, 1997
• Numerical computation of free boundary problems in quadruple precision arithmetic using an explicit scheme
  IMAI Hitoshi; SHINOHARA Yoshitane; TAKEUCHI Toshiki; TARMIZI; ALI Zulfikar; NATORI Makoto; ZHOU Weidong
  IEICE technical report. Nonlinear problems, The Institute of Electronics, Information and Communication Engineers, 97(53) 1 - 8, 1997
• Convergence of Attractors for Simplified Magnetic Benard System
  Hitoshi Imai; Naoyuki Ishimura; Masaaki Nakamura
  RIMS Kokyuroku, Kyoto University, 989 56 - 64, 1997
• Numerical Simulation of Free Boundary Problems in Quadruple Precision Arithmetic Using Explicit Schemes
  Hitoshi Imai; Yoshitane Shinohara; Makoto Natori; Weidong Zhou; Isamu Ohnishi; Yasumasa Nishiura
  RIMS Kokyuroku, Kyoto University, 989 18 - 30, 1997
• On High Precision Numerical Conputations to Free Boundary Problems
  Hitoshi Imai; Yoshitane Shinohara; Makoto Natori; Weidong Zhou; Isamu Ohnishi; Yasumasa Nishiura
  統計数理研究所共同リポート, 105 129 - 142, 1997
• Behaviour of Blow-Up Solutions to One-Phase Stefan Problems with Dirichlet Boundary Conditions
  Toyohiko Aiki; Hitoshi Imai
  Free Boundary Problems, Theory and Applications，Proceedings of the Zakopane Congress '95, 3 - 15, 1996
• 摂動項を伴う非線形常微分方程式系の準周期解の存在と一意性の定理
  篠原能材; 今井仁司; Zul kar ALI
  信学技報, 96(72) 13 - 17, 1996
• スペクトル選点法を用いた簡約化磁気ベナール系のアトラクターの数値解析
  今井 仁司; 中村 正彰; 石村 直之; 篠原 能材
  信学技報, 96(72) 5 - 12, 1996
• On a Global Solution and a Blow-Up Solution to One-Phase Stefan Problem
  Toyohiko Aiki; Hitoshi Imai
  RIMS Kokyuroku, Kyoto University, 951 54 - 61, 1996
• 簡約化磁気ベナール系のアトラクターの解析 -スペクトル選点法を用いた計算-
  中村 正彰; 今井 仁司; 石村 直之
  計算工学講演会論文集, 1(1) 159 - 162, 1996
• いくつかの自由境界問題に対するスペクトル選点法の応用
  今井 仁司; 周 偉東; 名取 亮; 都田 艶子; 篠原 能材
  数理解析研究所講究録, 京都大学, 944 247 - 255, 1996
• On Numerical Methods for a Free Boundary Problem Governed by a Hyperbolic Equation
  Hitoshi Imai; Koji Kikuchi; Kazuaki Nakane; Seiro Omata; Tomomi Tachikawa
  統計数理研究所共同リポート, 85 49 - 54, 1996
• Blow-Up Points to One Phase Stefan Problems with Dirichlet Boundary Conditions
  Toyohiko Aiki; Hitoshi Imai
  Proceedings of Modelling and Optimization of Distributed Parameter Systems Applications to Engineering, Chapman & Hall, 83 - 89, 1996
• Global Existence of Solutions to One-Phase Stefan Problems for Semilinear Parabolic Equations
  Toyohiko Aiki; Hitoshi Imai
  Technical Reports of Mathematical Sciences, Chiba University, 11(11) 1 - 9, 1995
• Existence and Uniqueness of Quasiperiodic Solutions to Quasiperiodic Nonlinear Differential Equations
  Yoshitane Shinohara; Atsuhito Kohda; Hitoshi Imai
  Numerical Analysis of Ordinary Differential Equations and its Applications，Proceedings of Kyoto Workshop on Numerical Analysis of ODEs, 147 - 163, 1995
• On Spectral Collocation Methods in Space and Time for Free Boundary Problems
  Hitoshi Imai; Yoshitane Shinohara; Tsuyako Miyakoda
  Computational Mechanics '95, Springer, 1 798 - 803, 1995
• On Validity of Application of the Fuzzy Theory and Spectral Collocation Methods to an Ill-Posed Shape Desigh Problem with a Free Boundary
  Hitoshi Imai
  統計数理研究所共同リポート, 72 216 - 223, 1995
• 凝固現象における数値解析 (接触角の変化と表面張力)
  花田 孝郎; 今井 仁司
  数理解析研究所講究録, 京都大学, 891 64 - 69, 1995
• Magnetic Benard System and its Simplified Model
  Hitoshi Imai; Naoyuki Ishimura; Masaaki Nakamura
  Lecture Notes in Numerical and Applied Analysis（Vol.14），Proceedings of the Second Japan-China Seminar on Numerical Mathematics, Kinokuniya, 14 79 - 92, 1995
• Application of Spectral Collocation Methods in Space and Time to Free Boundary Problems
  Hitoshi Imai; Yoshitane Shinohara; Tsuyako Miyakoda
  Hellenic European Research on Mathematics and Informatics '94, 2 781 - 786, 1994
• 時間積分におけるスペクトル選点法の有効性について
  今井 仁司
  統計数理研究所共同リポート, 55 185 - 190, 1994
• Application of the Fuzzy Theory and Spectral Collocation Methods to an Ill-Posed Shape Desigh Problem with a Free Boundary
  Hitoshi Imai
  Inverse Problems in Mechanics, ASME AMD-Vol.186 103 - 107, 1994
• Adaptive Fuzzy Control for an Ill-Posed Problem
  Hitoshi Imai; Hideo Kawarada; Makoto Natori
  Inverse Problems，Proceedings of the International Conference on Computational Engineering Science, 51 - 60, 1993
• 不適切な問題に対する適用可能な数値解法の検討
  今井 仁司; 周 偉東; 河原田 秀夫; 沢栗 利男; 名取 亮
  数理解析研究所講究録, 京都大学, 836 102 - 112, 1993
• An Application of the Fuzzy Theory for an Ill-Posed Problem
  Hitoshi Imai; Akira Sasamoto; Hideo Kawarada; Makoto Natori
  Inverse Problems in Engineering Mechanics: Proceedings of the IUTAM Symposium, 31 - 38, 1993
• On the Bifurcation Diagram of the Simplified Magnetic Benard Problem
  Hitoshi Imai; Masaaki Nakamura
  Unstable and Turbulent Motion of Fluid, World Scientific, 71 - 78, 1993
• 自由表面の運動を陽に取り入れた熱対流モデルとその数値解法について
  今井 仁司; 周 偉東; 名取 亮
  数理解析研究所講究録, 京都大学, 812 94 - 98, 1992
• 体積変化と接触点の運動を考慮したステファン問題の数値解析
  今井 仁司; 花田 孝郎; 河原田 秀夫; 名取 亮
  数理解析研究所講究録, 京都大学, 812 56 - 66, 1992
• 性質の悪い形状設計問題へのファジィ理論の応用
  今井 仁司; 笹本 明; 河原田 秀夫; 名取 亮
  情処研報(92-NA-41), 92(46) 9 - 12, 1992
• A Practical Method for an Ill-Conditioned Optimal Shape Desigh of a Vessel in Which Plasma Is Confined
  Akira Sasamoto; Hitoshi Imai; Hideo Kawarada
  Inverse Problems in Engineering Sciences，ICM-90 Satellite Conference Proceedings, Springer-Verlag, 120 - 125, 1991
• 体積変化を伴う凝固現象の数値解析
  花田 孝郎; 今井 仁司; 名取 亮; 河原田 秀夫
  数理解析研究所講究録, 京都大学, 746 116 - 129, 1991
• 凝固現象における自由境界問題の数値解析
  花田 孝郎; 今井 仁司; 名取 亮; 河原田 秀夫
  数理解析研究所講究録, 京都大学, 744 100 - 111, 1991
• 動的計算格子制御について
  今井 仁司; 畑上 到; 河原田 秀夫; 名取 亮
  数理解析研究所講究録, 京都大学, 744 78 - 84, 1991
• An Optimum Data Reduction Algorithm for General Plane Curves
  Sumiko Hiyama; Takao Hanada; Hitoshi Imai
  Technical Report, Institute of Information Sciences and Electronics, University of Tsukuba, ISE-TR-90-87 1 - 23, Nov. 1990
• The Parallel Processing in the Fuzzy Control System Governed by Partial Differential Equations
  Hitoshi Imai; Makoto Natori; Lun; Shan Gao; Hideo Kawarada
  Technical Report, Institute of Information Sciences and Electronics, University of Tsukuba, ISE-TR-90-88 1 - 10, 1990
• いくつかの自由境界問題とその数値解析
  今井 仁司; ソー ウィン マウン; 河原田 秀夫
  数理解析研究所講究録, 京都大学, 724 1 - 24, 1990
• 偏微分方程式系へのファジィ制御の応用とその並列処理
  今井 仁司; 名取 亮; 高 崙山; 河原田 秀夫
  Advances in Numerical Methods for Large Sparse Sets of Linear Equations, 6 1 - 6, 1990
• Global Structure of Bifurcation Appearing in the Equilibrium Plasma
  Hitoshi Imai; Hideo Kawarada
  Proceedings of the Fifth International Symposium on Numerical Methods in Engineering, 2 519 - 524, 1989, Introduction international proceedings
• A Method to Solve Drop Formation from a Capillary Jet
  Hitoshi Imai; Hideo Kawarada; Daisuke Takahashi
  Proceedings of the Fifth International Symposium on Numerical Methods in Engineering, 2 515 - 518, 1989, Introduction international proceedings
• Fuzzy Control of Systems Governed by Elliptic Partial Differential Equation
  Lun Shan Gao; Hitoshi Imai; Hideo Kawarada
  Technical Reports of Mathematical Sciences, Chiba University, 5(8) 1 - 12, 1989
• A Shape Design Problem of a Vessel in Which Plasma Is Confined
  Hitoshi Imai; Akira Sasamoto; Hideo Kawarada
  Technical Reports of Mathematical Sciences, Chiba University, 5(7) 1 - 10, 1989
• Numerical Simulation of Free Boundaries Appearing in Bingham Fluid
  Masahiro Fukuda; Hitoshi Imai; Hideo Kawarada
  Technical Reports of Mathematical Sciences, Chiba University, 5(4) 1 - 13, 1989
• A Method to Find Bifurcation Points
  Hitoshi Imai; Hideo Kawarada
  Technical Reports of Mathematical Sciences, Chiba University, 5(1) 1 - 18, 1989
• A New Reorthogonalization in the Lanczos Algorithm
  Makoto Natori; Hitoshi Imai; Eiji Kawamura
  Advances in Numerical Methods for Large Sparse Sets of Linear Equations, 5 38 - 43, 1989
• NUMERICAL ANALYSIS OF BIFURCATION PHENOMENA CONCERNING PLASMA SHAPES
  今井 仁司; 河原田 秀夫
  情報処理学会研究報告ハイパフォーマンスコンピューティング（HPC）, 1988(46) 1 - 8, 08 Jul. 1988
• 平衡プラズマの形状に関する分岐現象の数値解析
  今井 仁司; 河原田 秀夫
  数理解析研究所講究録, 京都大学, 676 105 - 125, 1988
• プラズマ形状の分岐現象の数値解析
  今井 仁司; 河原田 秀夫
  情処研報(NA-25), 88(46) 25-3-1 - 25-3-8, 1988
• On the Asymmetric Configuration of the 2-Dimensional Equilibrium Plasma in the Vessel Which Has a Symmetric Cross Section
  Hitoshi Imai
  Technical Reports of Mathematical Sciences, Chiba University, 2(6) 1 - 21, 1986
• Approximation of the Shape of the 2-Dimensional Equilibrium Plasma in the Vessel Which Has an Arbitrary Cross Section
  Hideo Kawarada; Toshio Sawaguri; Hitoshi Imai
  Technical Reports of Mathematical Sciences, Chiba University, 2(5) 1 - 13, 1986
• 平衡プラズマにおける自由境界値問題の数値解法
  今井 仁司
  数理解析研究所講究録, 京都大学, 553 134 - 144, 1985
• ２次元平衡状態におけるプラズマの境界決定の逐次近似法
  河原田 秀夫; 花田 孝郎; 今井 仁司
  数理解析研究所講究録, 京都大学, 532 146 - 162, 1984

Research Projects

• Effective use of multi-precision arithmetic on floating number system of digital computers aiming at numerical computations of differential equations with singulari or ill-posedness
  磯 祐介; 藤原 宏志; 川越 大輔; 木村 正人; 今井 仁司; 田口 智清
  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2021/04 -2025/03, Grant-in-Aid for Scientific Research (B), Kyoto University
• Study of Mathematical Modeling and Analysis for Antidune in Rivers
  磯 祐介; 藤原 宏志; 川越 大輔; 今井 仁司
  反砂堆(antidune)現象は主として砂を組成とする河床の現象で、河の流れの反対方向に砂が遡上して堆積する「移動境界」ものであり、河川のほか大陸棚等で観測される。水路実験では比較的短時間で発生して消滅する現象として自然現象としても興味深いものであるが、近年では河川氾濫と関連する現象として注目を受けている。現状ではこの現象の定義自体が確立されているとはいい難く、したがってその数理モデルも現象を特徴付ける仮定に依存して幾つかの提案がなされている。本課題研究では、1963年に J. F. Kennedy が提唱した古典的な数理モデルを採用し、反砂堆現象の信頼できる数値シミュレーションを行い、また数理モデルの解の安定性を数学解析によって明らかにすることを目的としている。 本課題研究の現状は Kenneddy の仮定による渦無し完全流体の流れを前提に、反砂堆を河床（数理モデルにおいては流体現象を記述する偏微分方程式の境界）の動的挙動として捉えて実験式を踏まえた数理モデルを前提としたうえで、反砂堆が発生している場合の数値シミュレーションと安定性を論じている。研究代表者およびその研究組織による先行研究によって Kennedy の提案する非線型の境界条件の役割についてのモード解析が行なわれているが、その成果を前提に、初年度は Kennedy が導入したパラメータの役割について論じて成果をあげた。なお、近年の防災研究においては河床に生じる剪断力を論じることの重要性が指摘され、Kennedy モデルとは異なる数理モデルが用いられることも多い。このため、近年の別仮定下での数理モデルにおけるモード解析との比較が「数理モデルによる現象の理解」では重要であることを初年度の研究遂行を通して認識をした。, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2021/07 -2024/03, Grant-in-Aid for Challenging Research (Exploratory), Kyoto University
• Mathematical Sciences aiming at medical application of light propagation in biomedical tissues and related topics
  Iso Yuusuke
  The present research program focuses on Diffused Optical Tomography (DOT), and the key idea of the research is dealing with the radiative transport equation (RTE) as the mathematical model of the aimed phenomena. DOT is considered an inverse problem of RTE, and we set the inverse problem to determine the scattering coefficient in the equation by observation of discontinuity of the solution of RTE. We show some mathematical results based on the method of characteristic lines and also numerical ones based on the discontinuous Galerkin method. We also show some new results the concerning photo-acoustic technology and, furthermore, obtain new knowledge on NIRS. We show some interesting numerical results aiming at numerical computation for inverse problems., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2016/04 -2021/03, Grant-in-Aid for Scientific Research (A), Kyoto University
• Basic research on quality assurance of numerical simulations by visualizing the regularity of solutions of differential equations
  Imai Hitoshi
  We developed numerical methods to investigate the location of singular points of one-variable functions and their smoothness at singular points. We found that the interpolated function oscillates violently near the boundary, and that the oscillation is localized by local averaging. For nonlinear ordinary differential equations with blow-up solutions, we developed highly accurate numerical methods for the blow-up time using the numerical limit, and succeeded in creating numerical regularity maps of the solution. For fractional differential equations with Helder continuous solutions, we succeeded in creating numerical regularity maps of the solution. They show that the property of the Caputo derivative changes whether the order is greater than or less than 1. We also found that the accuracy spike phenomenon occurs. We developed highly accurate numerical integration methods when the integrand has a sharp peak, and also developed methods for fast numerical computations., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C), 2018/04 -2021/03, Principal investigator, Grant-in-Aid for Scientific Research (C), Doshisha University
• Mathematical modeling for glucose concentration in blood based on inverse problem analysis of fractional differential equations
  ISO Yuusuke; SHEEN Dongwoo; HIGASHIMORI Nobuyuki
  We have shown some new results both in mathematical and numerical analysis for fractional order ordinary differential equations (F-ODE), which are equivalent to weakly singular integral equation of the Volterra type. We focus on regularity of unknown functions up to the the initial points, and we have clarified assumptions in the fundamental theorem for existence and uniqueness of solutions. We showed new results on numerical instability for a well-known scheme and give a new reliable numerical one. We also focus on limitation of F-ODE as mathematical models for phenomena including glucose concentration., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2016/04 -2019/03, Grant-in-Aid for Challenging Exploratory Research, Kyoto University
• Fundamental study for high-resolution diffused optical tomography based on numerical analysis and applied analysis
  Iso Yuusuke; FUKUYAMA Hidenao; NISHIDA Takaaki; YAMAMOTO Masahiro; LIU Tai-Ping; Dmitri Anikonov
  This project is aimed at innovation of the diffused optical tomography(DOT) from the view point of mathematical science, and DOT is consideredas an inverse problem to detect unknown coefficients in the radiative transport equation (RTE). There are two approaches to the inverse problem;the one is based on time resolution data as observation, and the other is based on stationary data. We obtained prominent result for the former case,and we focus on discontinuity of the coefficients as to show nice images numerically through reconstruction of the coefficients having discontinuity.We also prove convergence of the CIP (Cubic Interpolation Pseudoparticle) method to solve hyperbolic partial differential equations, which is used in our numerical simulation, to give an affirmative answer to an open problem about the CIP scheme. We propose new idea on iteration approach to the latter case, and we show new effective algorithm for numerical simulation of the stationary RTE., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2013/04 -2016/03, Grant-in-Aid for Scientific Research (B), Kyoto University
• Numerical analysis to support splitting and merging phenomena in interfacial dynamics
  TOMOEDA Kenji; IMAI Hitoshi; KURAMA Hiroyuki
  Numerical experiments suggest interesting properties in the several fields. Among such properties, there are support splitting and merging phenomena in the behaviour of non-stationary seepage. The model equation in one dimensional space is written in the form of the initial-boundary value problem with the effect of a non-linear filtration. In this study, such phenomena are realized by use of finite difference schemes, and are justified from numerical and analytical points of view. Thus we obtained following results: (1) Our difference scheme has the property of the stability and the convergence; (2) The solution of the initial-boundary value problem converges to the stationary solution when the boundary conditions are constant; (3) Repeated support splitting and merging phenomena appears when the period of the boundary conditions is sufficiently long; (4) We are able to show some example such that the support splitting phenomena never appears when the period is sufficiently short., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2011/04 -2015/03, Grant-in-Aid for Scientific Research (C)
• Estimation of modeling errors and their regularization in applied inverse problems
  ISO Yuusuke; KUBO Masayoshi; FUJIWARA Hiroshi; OOISHI Naoya; IMAI Hitoshi; HIGASHIMORI Nobuyuki
  We have obtained some new results as fundamental study for the diffused optical tomography (DOT), which is considered as a coming technology. The results are deduced through deep consideration, from both mathematics and computing, about mathematical models corresponding to DOT.In the project study, we focus on modeling errors which arise in mathematical modeling; we restrict ourselves to modeling errors in the transport equation model which is regarded as the fundamental mathematical one for propagation of photons in biomedical tissues. Through theoretical arguments and computings, we have pointed out some fatalmisunderstanding in the past studies for DOT based on mathematical analysis of the inverse problems., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2011 -2012, Grant-in-Aid for Challenging Exploratory Research, Kyoto University
• Breakthrough in numerical analysis and numerical computation related with infinitely-precision arithmetic
  ISO Yuusuke; FUJIWARA Hiroshi; KUBO Masayoshi; NISHIDA Kotoba; SAKAJYOU Takashi; OONISHI Kazuei; NAKAMURA Yoshimasa; IMAI Hitoshi
  We have shown some new results aimed to accurate scientific computationsbased on effective use of infinitely-precision arithmetic. The project study has been carried out both by theoretical approaches and computationalones. In the theory of numerical analysis, convergence of an abstract difference scheme is proved with a technique of the Banach scale, and numerical instability of difference schema are discussed from a viewpoint of proper meaning of the CFL condition. In computation, we have obtained accurate numerical results for the problems appeared in the particle physics and the eigen-value problems. The GPGPU programmingsare also discussed in order to realize fast computational environment., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2010 -2012, Grant-in-Aid for Scientific Research (B), Kyoto University
• Mathematical analysis of the nonlinear systems arising in the industry.
  NAKAMURA Masaaki; ISHIMURA Naoyuki; IMAI Hitoshi; MIZUTANI Akira; CHEN Yungang
  We have obtained the following results.1. We showed the effectiveness of the multiple precision arithmetic to approximate the solutions of PDE‘s which have transition phase and the blow up in finite time.2. We derived the default risk model equation and analyze theoretically and numerically.3. We showed the existence and the finite dimensional property of the attractors in magnetic Benard problem4. We applied the multi-precision computation to the equation with delay term,which is hard to compute, and showed that in the case the solution is analytic, it is effective., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2009 -2011, Grant-in-Aid for Scientific Research (C), Nihon University
• 高解像光トモグラフィの実現に向けての数学的基礎研究
  磯 祐介; 久保 雅義; 藤原 宏志; 今井 仁司; 東森 信就
  次世代の医用トモグラフィと期待される光トモグラフィの基礎研究として、輸送方程式の数学解析および数値解析の研究を展開している。平成22年度には方程式中の係数等の摂動における弱解の安定性、ならびに定常問題の場合の新たな数値計算アルゴリズムの提案を行うと共に、多くの数値計算事例の蓄積を行った。先行研究においては基礎方程式である輸送方程式の解を拡散方程式の解で近似したことが開発研究上の閉塞を招いたと考え、輸送方程式を直接扱う研究の展開を行っている。しかし平成21年度に集取した文献の精査を行った結果、光トモグラフィの高精度な実現に対しては、輸送方程式モデルにおいても検討すべき課題が残っていることがわかった。これは減衰項と散乱項の正当性に関するもので、このような根本的な問題意識に至ったことは、本課題研究による大きな進歩と考えられる。同時にこれらの項のL^∞摂動に対してL^P-弱解が安定であることも証明され、光トモグラフィを輸送方程式の逆問題として実現することの困難さも改めて確認した。 定常モデルの場合には効率の良い反復型数値計算法の新たな提案を行ったが、計算機の並列性とこのアルゴリズムの効率の問題は、数値計算例による事例研究の蓄積に留まっている。2年間の研究を通し、輸送方程式モデルを利用して光トモグラフィの開発研究を行うことは、拡散方程式を用いて行われた先行研究よりも優位性があることが幾つかの点で指摘された。しかし一方で過去に看過されてきた幾つかの事項の重要性がわかり、輸送方程式を基礎方程式とする光トモグラフィの基礎研究の一層の深化が必要であることが確認された。また、多倍長数値計算に+係る幾つかの成果も研究課程で得られた。, 日本学術振興会, 科学研究費助成事業, 2009 -2010, 挑戦的萌芽研究, 京都大学
• Foundation of high accuracy computational methods on the multiple-precision computer environment and its applications to analysi of inverse problems
  ISO Yuusuke; NISHIMURA Naoshi; YAMAMOTO Yutaka; SUGIMOTO Naozou; FUJIWARA Hiroshi; HIGASHIMORI Nobuyuki; IMAI Hitoshi; NISHIDA Takaaki; OONISHI Kazuei; YAMAMOTO Masahiro; NISHIDA Kotoba; MATSUDA Tetsuya; OKAMOTO Hisashi; SAKAJYO Takashi; NAKAMURA Gen; TANUMA Kazumi; SHIROTA Kenji; NAKAO Mitsuhiro; TOMOEDA Kenji
  The multiple-precision computer environment has been considered as a new research tool which can bear progress of sciences and technologies in top. In this project, we have been succeeded in not only development of multiple-precision arithmetic but its applications to hard problems in inverse and ill-posed analysis. We have established so excellent results that we lead research trends in computational mechanics., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2007 -2009, Grant-in-Aid for Scientific Research (B), Kyoto University
• Studies on dynamical systems of nonlinear phenomena with energy dissipation and the theory of stability
  KENMOCHI Nobuyuki; ITO Akio; OHARU Shinnosuke; OTANI Mitsuharu; IMAI Hotoshi; KADOYA Atsushi; AIKI Toyohiko; SHIRAKAWA Ken; FUKAO Takeshi; YAMAZAKI Noriaki
  We tried to construct a new mathematical theory for clarifying the mechanisms of nonlinear and complex phenomena in the real world and to make use of it to handle some concrete problems arising in the material and life science. In fact, our new mathematical theory provided an effective approach for some open questions. This is one of the most important achievements in this research, and the so-called "Life Mathematics" has been evolved in the same framework., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2006 -2009, Grant-in-Aid for Scientific Research (B), Chiba University
• 爆発・成長現象のダイナミクスを明らかにする数値計算手法の開発と応用
  今井 仁司; 坂口 秀雄
  非線形偏微分方程式のなかには、有限の時刻で解の値が無限大となるような爆発解や時刻無限大まで存在する大域解を持つものがある。このような方程式の理論解析は極めて困難で、数値計算であっても解の値もしくは解の存在時刻が無限大であるためその適用は極めて困難となる。 本研究では、このような解析に有効な数値的判定法を開発した。数値的判定は次のように行う。時間に関して以前我々が開発した有界化を適用して、[0、∞)の無限区間を[0、1)の有限区間に変換する。この変換された問題を、空間変数を差分法で時間変数を陽的オイラー法で離散化して、粗い数値計算を行う。この粗い計算時に、1未満の明らかなる時刻で解の数値がオーバーフローすれば爆発解がとらえられたことになる。時刻1まで数値計算できた場合は、時刻1の非常にそばで(元時間では長大な時間が経過した後に)爆発する解や成長もしくは有界な大域解を区別する精査を行う。精査は時空間スペクトル選点法で行う。スペクトル選点法は陰解法であるために、得られる離散化方程式は非線形になる。この非線形方程式をニュートン法で解く。このニュートン法が収束して数値解が収束すれば、有界あるいは減少する大域解が存在する。ニュートン法が収束しなければ、計算領域内に特異性が存在しているので、爆発解あるいは成長する大域解が存在することになる。これらの分類は、爆発時刻を精査するか、解に関する有界化をさらに施すことで行う。ニュートン法が収束しない場合には解の情報が全く得られないので、この致命的欠点を克服するために変数を複素数に拡張した複素ニュートン法の適用する。 本研究で開発した数値的判定法を最も有名な空間1次元の非線形熱伝導方程式に適用して、数値実験によってその有効性を検証した。開発した手法は一般の偏微分方程式に適用可能である。, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Exploratory Research, 2006 -2007, Principal investigator, Grant-in-Aid for Exploratory Research, The University of Tokushima
• The Frontier of Numerical Analysis for Dynamics of Interfaces and Developments in Sciences and Engineering
  TOMOEDA Kenji; MIMURA Masayasu; KAWAGUCHI Masami; TABATA Masahisa; GIGA Yoshikazu; NAKAKI Tatsuyuki
  We were concerned with the following numerical methods to the phenomena appearing in the repre-sentative dynamical interfaces : i) Pattern dynamics in the reaction-diffusion system, ii) Viscous fingering phenomena in Hele-Shaw Cell, iii) Dynamical behavior of the region occupied by the water in the process of evaporation. We obtained several results : 1) The TCD (Threshold Competition Dynamics) method is developed for the numerical computation in reaction-diffusion system, and enables us to realize the dynamical behavior of free boundary in R^n (n=1, 2, 3.). The idea of this method is based on the theory of "Singular limit method". 2) The mathematical model for the crystal growth is considered in the, form of the reaction-diffusion equation with the effect of a convection, and gives us interesting mathematical results. 3) In viscous fingering phenomena, the buoyancy-driven path instabilities of bubble rising in Hele-Shaw Cell are examined. As an interesting phenomenon there appears a wake which is similar to a comet. However, such a wake is not realized in numerical method yet. 4) Multi-scale FEM based on crystallographic homogenization method is developed to predict the dynamics of interfaces in the formability of sheet metal. 5) The repeated support splitting and connecting property in the process of evaporation is investigated, where the the support means the region occupied by the water. The numerical methods for this process are established and the shape of the initial distribution for which such a property appear is explicitly obtained., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2004 -2007, Grant-in-Aid for Scientific Research (B), Osaka Institute of Technology
• Establishment of New Numerical Methods for Applied Inverse and Ill-Posed Problems
  ISO Yuusuke; NISHIMURA Naoshi; FUJIWARA Hiroshi; ONISHI Kazuei; IMAI Hitoshi; YAMAMOTO Masahiro
  The aim of this research project is mathematical analysis and numerical analysis of ill-posed problems written in partial differential equations connecting with applied inverse problems which are important in physics, medical science, and engineering. Especially, considering the future requirement in practice, it is one of our originalities that we have developed a new fast multiple-precision arithmetic environment for the sake of large scale numerical computation of the ill-posed problems with high accuracy, in addition to mathematical theory and algorithms. In the scientific computations including numerical simulations of inverse problems, approximation by floating-point arithmetic are usually used in representation and arithmetic of real numbers on digital computers. Nowadays the double precision arithmetic defined in the IEEE754 standard is the common way. This means that scientific numerical computations are carried out on the assumption that real numbers have 15 decimal digits accuracy in the usual end-user environments. In the floating-point arithmetic we cannot omit rounding errors and cannot treat real numbers exactly on the digital computers. Of course we must also take discretization errors into account which appear in discretization of functional equations and partial differential equations in numerical computations. In ill-posed problems which typically appear in inverse problems, the error is fatal defect for reliable numerical computations. This is the most different point between well-posed problems which induce stable numerical schemes. Conventional numerical analysis for ill-posed problems treated only discretization errors or measurement errors, and consideration of rounding errors is not enough. The most significant points of our research is development of a new multiple-precision arithmetic in discussion on rounding errors besides the conventional numerical analysis for discretization errors and measurement errors. In the multiple-precision arithmetic environment, the new aspects have been found in high accurate discretization of functional equations, and new computational schemes have been developed and established in the project. One of the concrete results is the fast multiple-precision arithmetic environment "exflib", which was designed and implemented in the predecessor research, has been improved by co-researcher Prof. Hiroshi Fujiwara, who has succeed in implementation of special functions and in porting to supercomputers to treat scientific numerical simulations. We also apply the spectral methods, which achieve quite high accurate numerical solutions than the conventional discretization methods. Combining the multiple-precision arithmetic and the spectral methods, we have proved the proposed approach is quite effective for numerical analysis of ill-posed problems. And we give a remark on the regularization method under high accurate numerical methods, especially the relation between measurement errors, regularization parameters, and computation precisions. The remark is important in practical applied inverse problems in which we must take measurement error into account. Each problem has its own ill-posedness. Because the matter is different in each setting in inverse problems, we place mathematical analysis for inverse problems as fundamental subjects in the project and we discuss uniqueness and conditional stability of solutions. Co-researcher Professor Masahiro Yamamoto obtain sharp results in inverse scattering problems. In application of the results in mathematical and numerical analysis to practical problems, we need the fundamental research from the computational mechanics viewpoints. All co-researchers have discussed applied inverse problems in their fields. We also discuss computer aided proof and succeed in numerical verification techniques which is one of the applications of the fast multiple-precision arithmetic., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2004 -2006, Grant-in-Aid for Scientific Research (B), KYOTO UNIVERSITY
• Synthetic approach for the development of computer assisted analysis from the numerical verification methods
  NAKAO Mitsuhiro; TABATA Masahisa; IMAI Hitoshi; TSUCHIYA Takuya; NISHIDA Takaaki; CHIN Shokun
  In this research, we newly developed the numerical verification methods which can be applied to wide mathematical and analytical problems, as well as extended or improved the existing techniques. And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems etc. The important research results obtained by investigators and co-investigators are as follows : 1.Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the constructive error estimates for the finite finite element projections of the Poisson and the bi-harmonic equations on various kinds of domains, particularly, on nonconvex polygonal domains. These results played important and essential roles for the numerical verification of solutions of nonlinear elliptic equations and the two dimensional stationary Navier-Stokes problems. 2.Nagatou numerically proved the stability of the flow on the torus called Kolmogorov problem. 3.Minamoto presented a formulation of the verification condition for the double turning point and applied it to the perturbed Gelfand equation. 4.Oishi established some refinements on the fast algorithm for the solutions of linear equations. 5.Nishida et al. presented the computed results with guaranteed error bounds for the symmetry breaking bifurcation point of the solution of two dimensional heat convection problems, as well as they formulated the numerical verification algorithm for the three dimensional problems with some prototypical verified examples. 6.Chin obtained some numerical verification results on the existence of solutions and a posteriori error estimates for the linear complementarity problems., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2003 -2006, Grant-in-Aid for Scientific Research (A), KYUSHU UNIVERCITY
• 多倍長数値計算環境下での逆問題・非適切問題の数値解析手法の確立
  磯 祐介; 若野 功; 藤原 宏志; 今井 仁司
  逆問題は、医用CTなど、各種非破壊検査や断層撮影などの先端技術と関係する実用的かつ重要な問題である。この問題の多くは微分方程式や積分方程式で記述されるが、数学的には「Hadamardの意味で非適切(ill-posed)」な問題に分類されるものが殆どであり、逆問題の数値シミュレーションにおいては、その計算結果の信頼性の保証が重要課題となる。医学・工学などの応用分野では、個々の逆問題解析において、新たな解析手法が次々に提案されているが、その多くは数学的に見ればアドホックな手法と言わざるを得ない。本課題研究では、これらの手法の中から有効な普遍的的アイデアを精査し、その新しいアプローチに基づく逆問題あるいは非適切問題に対する数値解析手法を数学の視点から確立すると共に、それを多倍長数値計算環境で実現することを目標として研究が行なわれ、成果を得ている。 具体的には、平成16年度には、スペクトル法、特にスペクトル選点法の高精度性と利便性に目をつけ、データ誤差を含まない第1種積分方程式の直接離散化による数値解法を提案し、その多倍長数値計算環境上での実現を図った。前年度の研究により、Tikhonov正則化法の離散化を利用した第1種積分方程式の数値計算では、観測誤差・打ち切り誤差・正則化誤差・丸め誤差のバランスの重要性がしてされていた。今年度は、この前年度の問題点への取り組みを観測誤差と正則化誤差を切り放した純粋数学的条件下で議論を行なったうことを試み、スペクトル法の活用を行なった。この結果、これらの誤差のない環境では、今年度の成果として提案する「誤差を任意に制限できる離散化手法」の有効性が数値実験によって確認され、将来の観測誤差・正則化誤差を含めた解析に対する一つの判断基準が与えられた。さらに、この数値計算において重要な役割を果たす、多倍長数値計算環境上での高精度数値積分公式も得ている。, 日本学術振興会, 科学研究費助成事業, 2003 -2004, 萌芽研究, 京都大学
• Precision Improvement of Numerical Simulation Including Iteration Processes for Nonlinear Evolution Equations
  HATAUE Itaru; IMAI Hitoshi; ZHANG Shao-Liang; IWASA Manabu; SAISHO Yasumasa
  For the purpose of construction of noble schemes in which influence of errors are extremely removed, we did consideration about setting of the reasonable condition for convergence of numerical solutions in iterative steps in solving the linear systems and about speed-up of convergence. In addition, we applied them to the issues of real fluid calculation and the problem of traffic jam in order to analyze several factors which govern nonlinear phenomena. First, we analyzed the structure of dynamical systems which are produced by discretizing the nonlinear differential equations from several viewpoints such as numerical experiments, visualization of nonlinear structure of attractors and statistical consideration. Concretely, we clarified essential structure of numerical solutions in discrete dynamical system and proposed the criterion to judge reliability of solutions. Next, we tried to discuss how structure of numerical solutions of a stochastic differential equation changes by the insertion of errors with the random style from the viewpoints of probabilistic approaches. Concretely, we tried to investigate the dependence of the structure of numerical solutions on insertion of random errors. As a fundamental study, the stochastic differential equation based on the deterministic logistic differential equation was considered and the relation between the size of noise and characteristics of obtained numerical solutions was discussed. Furthermore, we tried to discuss the dependence of the structure of numerical solutions of incompressible fluid equations on insertion of random errors in solving simultaneous equations. Next, direct numerical simulation to an integral equation of the first kind was carried out by using IPNS(Infinite-Precision Numerical Simulation). Numerical results are very satisfactory in accuracy. Moreover, they also show some interesting facts. These numerical results show IPNS facilitates numerical analysis for such inverse problems., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2002 -2004, Grant-in-Aid for Scientific Research (C)
• 応用数学の日本・イタリア研究集会のための企画調査研究
  中村 正彰; 大森 克史; 今井 仁司; 石村 直之; 岡田 正巳
  本年度の研究計画に基づき研究を実施し,次のような成果を得た。 ・イタリア応用数理学会会長のフィレンツェ大学M.Primicerio教授と研究連絡を実施し、次のような方向で開催をする事に合意した。 1.2004年5月の3日間、日本で開催する。候補地として葉山をあげた。 2.参加者は1講演50分、講演者は両国15-20人とする。 3.テーマは金融工学、ナノテク・ナノサイ、非線形現象、PDEとその近似を基本項目とする。 ・上のテーマのための下記の分野の調査と研究を実施し成果を挙げた。 Navier-Stokes方程式を基礎とする摩擦境界条件などの非線形問題の数学的および数値的解析に成功した。 二種の金属合金の相分離を記述する江口・沖松村方程式の1次元定常解の構造を数値的かつ理諭的に解析した。 多倍長計算手法と選点法を用いた離散化手法を組み合わせた高精度計算手法の開発と応用に成功した。 適切でない可能性のある逆問題へのスペクトル選点法を用いた多倍長高精度計算法の応用。 数理金融工学における種々のオプションに現れる自由境界問題の数値的理論的な解析。 有限要素法,差分法、モルタル法、領域分割法を用いた手法による各種流体問題の解析に成果をあげた。 燃焼合成反応におけるヘリカル波の出現は平面進行波からの安定分岐であることの解析。 2流体問題に対する質量保存的有限要素数値シミュレーションスキーム、と界面の収束性の誤差評価。 二重指数関数型変換とsinc近似を用いた不定積分の数値積分法の効率の高さの解析。 競合拡散系、Stefan様問題、非線形拡散系など数理生物学の諸問題の応用解析。 仮想領域法と差分法を用いた、オイル汚染など環境Eco systemに現れる流体問題の数値解析。 仮想領域法と混合有限要素法を用いたradiation problemの数値解析, 日本学術振興会, 科学研究費助成事業, 2003 -2003, 基盤研究(C), 日本大学
• 特定領域「応用逆問題解析」の申請へ向けての調査と国内調整
  磯 祐介; 今井 仁司; 西村 直志; 山本 裕; 久保 司郎; 山本 昌宏
  本課題研究は、逆問題の分野で特定領域申請を将来行なうことの可否・妥当性に対する調査を行なうことが目的であった。結論は、極めて近未来に逆問題解析に関して特定領域申請を行ない、理学・工学・医学の分野横断的な研究発展を目指すことが重要と考えられる。 応用逆問題は、非破壊検査・医用CT・各種断層撮影と極めて多岐にわたるものである。そしてその研究進展には、これまでの他の計算工学など以上に、最近の数学・数値解析の諸結果の効率的な反映が不可欠と考えられる。殆どの逆問題は数学的にはHadamardの意味での非適切問題(ill-posed problem)であり、通常の意味での近似の適用が不可能である。この事実は経験的に多くの分野で知られており、逆問題の解析に対して物理・工学・医学の各分野では、それぞれの分野で固有の先験情報を考慮することによって非適切性の排除を試みている。しかしながら、これまでの研究の多くは、個別論に埋没したアドホックなケースも多く、さらに数学的には誤った理論を誤用している場合も見られる。このような問題を解決して研究の一層の進展を図るには、分野横断的なブレークスルーを図り、さらに数学の視点から先験情報の構造を解析することが有効であると考えられる。 我が国においては、学問においても縦割り的な様相が強く、複数の分野を横断的に纏めた研究を行なうためには、現在の枠組みでは特定領域の制度を活用することが合理的であると考えられる。さらに、分担者による各学会の逆問題解析研究者の意見収集では、応用逆問題に関しては分野横断的に研究を行なえるだけの個別の実績が有ると判断される。 さらに、我が国の科学技術戦略として、組織規模まで同定できる医用断層撮影技術や地雷探査等の戦後処理技術の確立が議論されているが、これらの技術革新は応用逆問題の適用例であり、逆問題解析の研究レベルの向上は我が国の近未来の方針とも合致するものと判断される。, 日本学術振興会, 科学研究費助成事業, 2003 -2003, 基盤研究(C), 京都大学
• 解が存在しない微分方程式の数値シミュレーション
  今井 仁司; 竹内 敏己; 坂口 秀雄
  解が存在しない微分方程式として,解析的な係数を持つ楕円型偏微分方程式の初期値問題をとりあげた.昨年度は,初期条件のパラメータの値によって解の存在非存在がコントロールできるテスト問題をとりあげて,差分法と無限精度数値シミュレーション法を比較した.その結果,無限精度数値シミュレーションのみが存在非存在に正確に対応する計算結果を示すことを実証した.ただし,初期条件としては解析関数か低階微分のみ可能な関数というはっきりと区別できるものを採用したので,無限精度数値シミュレーションの有効性を確認しやすい状況であった.本年度は,初期条件として解析関数か無限回微分可能であるが解析的でない関数という究極の状況をとりあげた.メモリ使用量が20GBに迫るような超大規模の並列計算を行ったところ,選点数(近似の次数)を空間方向によって変えないとうまくいかないことが判明した.逆にいえば,選点数(近似の次数)を空間方向によって変えればうまくいくということが判明したのである.即ち,解の存在非存在に関するこの究極の状況下でさえ,数値シミュレーションがうまくできる環境が無限精度数値シミュレーションにはあるという驚くべき事実が判明した.このような直接的な研究以外にも関連する研究として,極座標変換による特異性を回避する公式の発見や進行波解や平衡解の存在や安定性を数値的に示す手法の開発など,様々な微分方程式の数値解析を可能にする手法の開発を行った., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Exploratory Research, 2002 -2003, Principal investigator, Grant-in-Aid for Exploratory Research, The University of Tokushima
• Domain Decomposition Method for fee Boundary Problems and Its Applications
  TAKEUCHI Toshiki; IMAI Hitoshi; NAKAMURA Masaaki; IKEDA Tsutomu; SAKAGUCHI Hideo
  It is known that there are travelling wave solutions in one-dimensional hyperbolic equations and reaction-diffusion equations. As a numerical method to capture the travelling wave solutions, we propose a numerical method for tracking the level set with the arbitrary precision. The feature of this method is that the level set is considered to be a free boundary and the original problem is transformed into a free boundary problem. Free boundary problems are boundary value problems defined on domains whose boundaries are unknown and must be determined as the solution. Many practical problems are formulated as free boundary problems. Recently, numerical methods for free boundary problems have been developed and improved. But investigation of the reliability of numerical results is not easy because of the unknown shape of the domain. So, we use the domain decomposition method and the fixed domain method together Otherwise, IPNS(Infinite-Precision Numerical Simulation) was developed. It consists of the spectral collocation method and multiple precision arithmetic. The spectral collocation method is used for the control of truncation errors. Multiple precision arithmetic is used for the control of rounding errors. The method is applicable to PDE systems with smooth solutions. It facilitates investigation of the reliability of numerical results It is Solved by IPNS for the free boundary problem. Numerical results are very satisfactory, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2001 -2003, Grant-in-Aid for Scientific Research (C), The University of Tokushima
• Development of Verified Methods for Numerical Simulations with Domain Decompsition Methods
  NAKAMURA Masaaki; TABATA Masahisa; IKEDA Tsutomu; IMAI Hitoshi; ZHANG Shao-liang; MORI Masatake
  In the term of the project (three years) many results are obtained. Important results are shown as follows. 1. Development of methods fot numerical simulation New method of a fast numerical computation by vectorization. Simulation by paralel computaions. Numerical simulation in Infinite precisions. New pre-condition methods. Analysis of numerical integrations. Mesoscopic simulation of concrete materials. 2. Applied and numerical analysis of nonlinear problems. Analysis of the system of phase separations. Error estimates for finite element methods. Code of finite element methods for mandtle convections. Analysis of reaction-diffusion systems. Navier-Stokes equations with friction type boundary conditions. Numerical methods for free boundary problems. Analysis microwave problems using multi grids methods. 3. Numerical simulations of several fluid dynamics. Two-phase flows. Navier-Stokes flows. Blood flow in brain, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2001 -2003, Grant-in-Aid for Scientific Research (B), Nihon University
• Numerical and Mathematical Analysis for the reconstruction for solutions of inverse and ill-posed problems by regularization methods
  ISO Yuusuke; IMAJ Hitoshi; YAMAMOTO Masahiro; NISHIMURA Noashi; OONISHI Nobuyoshi; OONISHI Kazuei
  We consider "Numerical Analysis by Regularization Methods" in wide sense, and we have aimed, in the present research, to propose and develop new methods to deal with inverse and ill-posed problems. The computer tomography and non-destructive tests are important technologies to support our present life, and they are typical inverse problems from the mathematical view points. Almost all the inverse problems are ill-posed in the sense of Hadamard, and it is too difficult to analyze them by the standard numerical methods ; ill-posedness of the problems implies numerical instability in computation and prevents reliable construction of numerical solutions. Regularization methods are proposed to reduce ill-posed problems to series of well-posed ones with the regularization terms, but we are obliged to satisfy with numerical treatments of the regularized solutions which are sometimes quite different from the exact ones. In order to seek accurate and reliable numerical solutions for the ill-posed problems, we have clarified demerits of regularization methods, and we have proposed some new techniques and methods for analysis of inverse and ill-posed problems in the present project. The most remarkable results in the present research is to design and to implement new and fast multiple-precision arithmetic on 64-bits computers as a software. The software enables us numerical treatments of ill-posed problems without rounding errors which cause numerical instability. And we propose a new algorithm based on the spectral collocation methods, and we give a keen remark for the choice of the suitable regularization parameter by many numerical experiments using our software. We propose new methods to reconstruct solutions of inverse and ill-posed problems in both mathematics and in computation: localized Dirichlet -Neumann map, regularization based on the filter theory, interval analysis approach etc. And we give some mathematical foundations for the analysis of inverse problems in the near future; analysis of propagation of waves on Fractals, a new mathematical model for brain, keen analysis for cracks in elasticity etc., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2001 -2003, Grant-in-Aid for Scientific Research (B), Kyoto University
• Construction of Numerical Analysis for High-performance Large-Scale Computation
  TABATA Masahisa; KANAYAMA Hiroshi; USHIJIMA Teruo; IMAI Hitoshi; NAKAO Mitsuhiro; KIKUCHI Fumio
  1.In devising numerical schemes for flow problems, how to approximate the convection torn is a crucial point. Characteristic finite element approximation is based on the approximation of the material derivative, which is the sum of the time derivative term and the convection term. So far finite element schemes of characteristic method of the first-order accuracy in time increment have been used. We have developed a finite element scheme of the second-order accuracy in time increment and obtained the best possible error estimate. This scheme is more robust than the first-order scheme with respect to numerical integration error and can solve flow problems more stably and accurately. 2.We have developed a finite element scheme and established an error estimate for heat convection problems with temperature-dependent viscosity. The viscosity of heat conduction problems such as mantle convection in the Earth and melting glass convection in the furnace is strongly dependent on the temperature. The dependence plays an important role in die formation of convection patterns. Our scheme is applicable for the general Rayleigh-Benard problems with temperature-dependent viscosity, thermal conductivity, and thermal expansion coefficient. Using this scheme we have carried out large-scale numerical simulation of Earth's mantle convection in three-dimensional spherical shell and succeeded in obtaining complex heat convection patterns. 3.In the infinite precision computation we have succeeded a large-scale parallel computation using a cluster of high-performance computers with 10CPU and 20GB memory. For one-dimensional boundary-value problems very precise results with precision 4995 digits have been obtained. We have used this system to perform direct numerical simulation of inverse problems and made possible a numerical analysis of inverse problems. 4.Formulating eddy current problems in magnetic vector potential and electric scalar potential, we have solved them using a hierarchical domain decomposition method. This solution has been shown to be effective under the environment of parallel computation. By this method we have carried out large-scale numerical simulation of nonlinear static magnetic problems in magnetic vector potential., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2001 -2003, Grant-in-Aid for Scientific Research (A), Kyushu University
• Synthetic approach for new developments of self-validating numerics
  NAKAO Mitsuhiro; OISHI Shin'ichi; IMAI Hitoshi; ISO Yusuke; YAMAMOTO Tetsurou; NISHIDA Takaaki
  In this research, we newly developed the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as extended or improved the existing techniques. And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems. The important research results obtained by investigators and co-investigators are as follows : 1. Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the numerical verification methods of solutions for elliptic problems. Namely, they succeeded the numerical computation with guaranteed error bounds for the inverse eigenvalue problems of second order elliptic operator. They also obtained some results for enclosing the solutions for elliptic variational inequlities. Moreover, they computed an optimal constant with guaranteed accuracy appearing in the a priori error estimates for the finite element projection of the Poisson problem, which is an important contribution for the numerical verification for nonlinear elliptic problems. 2. Nagatou and Minamoto obtained interesting computer assisted proofs for the Kolmogorov problem and for the perturbed Gelfand equation, respectively. 3. Oishi established some fast algorithms for the fundamental validated computations for the solutions of linear equations. 4. Nishida et al. computed with guaranteed error bounds for the non-trivial solution of heat convection problems, which is an important result for a computer assisted proof in the fluid mechanics. 5. T. Yamamoto obtained some convergence results of the finite difference scheme for the singular solutions of two point boundary value problems., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2001 -2002, Grant-in-Aid for Scientific Research (B), Kyushu University
• 逆問題の解の再構成手法の確立
  磯 祐介; 山本 昌宏; 西村 直志; 木上 淳; 今井 仁司; 中村 玄
  物理・工学・医学などの応用分野においてはCT(断層撮影法)など種々の逆問題が技術として利用されており、その成果は我々の生活を支えている。しかしこれらの逆問題は数学的にはHadamardの意味で非適切であり、通常の単純離散化による手法での解の再構成、特に数値的再構成は殆んど不可能である。このため、Tikhonov正則化法を始めとする幾つかの正則化法(緩和的手法)や、問題固有の情報(先見情報)を援用した解析手法が提案されている。しかしこれらの先見情報の利用は問題個別の議論が殆んどであり、その正当性などの数学解析は殆んどの場合は未だ示されていない。また同種の手法を分野固有の述語による記述をするため、分野を越えての共通理解が滞り、此れが研究推進を阻害している場合も見受けられる。この様な状況を鑑み、分野横断的な逆問題解析の研究を我が国において行う可能性を調査することが本研究の目的である。今回の調査研究では、逆問題の分野で医学・工学・物理・数学において我が国をリードする研究者を分担者として組織して調査を行い、主として分担者による個別調査によるかたちで調査を行い、メールを通しての相互意見交換によって今後の研究の進め方を議論した。 この結果、現在は複数の分野にわかれて独立して研究している研究者が「逆問題」「非適切問題」をキーワードに研究交流・共同研究を行うことが重要であるという共通認識が持たれ、さらには京都大学を中心に現在進行している多倍長数値計算環境の援用を医学・工学の応用逆問題の解の再構成に適用することの意義が指摘された。また分担者の何人かがロシア・香港での逆問題・非適切問題の国際研究集会に参加し、海外での応用逆問題の研究動向を調査し、海外においても応用逆問題の解の再構成-特に数値解析に対する関心が高いことを実感した。 この調査研究の結果をふまえ、平成14年度には特定領域の新設に向けての申請準備を行うことが妥当であるという結論に達し、平成13年度末には既に準備を開始している。, 日本学術振興会, 科学研究費助成事業, 2001 -2001, 基盤研究(C), 京都大学
• Studies on the treatment of Numerical Calculations Including Considerable Errors for Construction of Proper Mathematical Discrete models
  HATAUE Itaru; IMAI Hitoshi
  In this study we want to analyze the structure of dynamical systems which are produced by discretizing the nonlinear differential equations(DEs) from some viewpoints such as analytical and/or numerical approaches, and qualitative and/or quantitative ones. The numerical nonlinear dynamics approaches such as asymptotic numerical solutions. The structure of the asymptotic numerical solutions which were calculated by using implicit schemes were studied. Analytical discussions and numerical tests for fully implicit schemes of the Burgers' equation and several types of their linearized schemes were done. Furthermore, we tried to investigate the characteristics of ghost numerical solutions of incompressible fluid equations from viewpooints of the effect of popular fourth order artificial viscosity terms. Next, we tried to investigate the characteristics of ghost numerical solutions of incompressible fluid equations from the viewpoints of the effect of popular fourth order artificial viscosity terms and to discuss influence of the condition for convergence in iterative steps in solving the linear systems on the structure of numerical solutions. Second purpose of the present study is try to evaluate the dimensions of numerical solutions which are produced by discretizing the noonlinear differential equations by calculating the approximately generalized dimension, especially the correlation dimension. The dimensions of attractors constructed from the time series of one of the valiables in the numerical results which were given by solving Navier-Stokes equations directly are calculated. Futhermore, wavelet analysis is applied in order to clarify the difference of the complicated structure of numerical solutions in detail. Third purpose of the present study is that we apply IPNS(Infinite Precision Numerical Simulation) approach in order to remove numerical errors. A Cauchy problem of an elliptic operator and an integral equation of the first kind are solved. Another application is numerical computation of attractors in free boundary problems., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 2000 -2001, Grant-in-Aid for Scientific Research (C), Kumamoto University
• Computation Mechanics and Domain Decompsition Methods
  NAKAMURA Masaaki; IKEDA Tsutomu; TABATA Masahisa; IMAI Hitoshi; MORI Masatake; ZHANG Shao-liang
  In the term of the project (three years) many results are obtained. Important results are shown as follows. 1. Development of numerical simulation for computation mechanics. New method of a fast numerical computation. Gauss elimination package using PVM and paralel computaions. Numerical simulation in Infinite precisions. New pre-condition methods. Analysis of numerical integrations. Mesoscopic simulation of concrete materials. 2. Applied and numerical analysis of nonlinear problems. Analysis of the system of phase separations. Error estimates using Hausdorff metric. Code of finite element methods for mandtle convections. Analysis of reaction-diffusion systems. Navier-Stokes equations with friction type boundary conditions. Numerical methods for free boundary problems. Analysis of Ginzburg-Landau equation. 3. Numerical simulations of several fluid dynamics. Two-phase flows. Navier-Stokes flows., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1998 -2000, Grant-in-Aid for Scientific Research (B)., Nihon University
• Coordinative research on numerical analysis of complex systems and optimal control
  KAWARADA Hideo; KAKO Takashi; MORI Masatake; FUJITA Hiroshi; TSUTSUMI Masayoshi; NAKAO Mitsuhiro
  This project has continued for three years from 1998 to 2000. Fruiful and hopeful research results have been obtained by collaborations of excellent cooperators. We are sure that these results promise great success in the field of environments, industrial and applied mathematics. Main topics of research results are the followings. Developments of modeling for environment problems. Developments of numerical algorithm and programing for various complex phenomena. Developments of hybrid algorithm for global optimization problems. Developments of analysis and numerical simulation in the fluid dynamics., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1998 -2000, Grant-in-Aid for Scientific Research (B)., Chiba University
• New Numerical Methods for Flow Problems and their Numerical Simulation
  TABATA Masahisa; KAWARADA Hideo; KANAYAMA Hiroshi; USHIJIMA Teruo; NAKAO Mitsuhiro; TOMOEDA Kenji
  (1) We have developed a numerical method for computing accurately drag and lift exerted by the fluid to a body immersed in the flow field. Transforming those values to equivalent integrals in the domain, we have succeeded in establishing error estimates. We have obtained accurate drag coefficients of a sphere by this method. The restults have been extended to evolutional problems and best possible error estimates have been obtained. (2) The system of Rayleigh-Benard equations with infinite Prandtl number is a mathematical model describing heat convection phenomena in slow flows such as the Earth's mantle convection.We have developed a finite element scheme for this system, established error estimates, made an effective parallel code for three-dimensional computation, and performed numerical simulation for the Earth's mantle convection. (3) We have developed mathematical theory and computation algorithms to find exact solutions of partial differential equations from numerical computation results. Those methods have been applied to the stationary Stokes equations, the Navier-Stokes equations, stationary bifurcation solution of heat convection problems. (4) We have applied Dirichlet-Neumann mapping to exterior problems and developed combined numerical methods with a charge simulation method for the harmonic equation and with a domain decomposition method for the Helmholtz equation. (5) We have constructed a mathematical model to analyze effect to ecological system caused by coastal oil pollution and performed the numerical simulation and the visualization. The obtained results are in good agreement with physical experimental results. (6) The interface of porous media flow takes various behavior depending on the initial state. Using finite difference method we have given a sufficient condition for the separation of the support of the solution and upper and lower estimates of the waiting time for the initial interface to keep invariant., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1998 -2000, Grant-in-Aid for Scientific Research (A)., Kyushu University
• Research on infinite precision numerical simulation of partial differential equations
  IMAI Hitoshi; TOMOEDA Kenji; TABATA Masahisa; IKEDA Tsutomu; NISHIDA Takaaki; NAKAO Mitsuhiro
  In the term of the project (three years) many results were obtained. Important results are shown as follows. 1. Building of the parallel computing environment. High-performance workstations were purchased and connected by the high-speed network. PVM (Parallel Virtual Machine) was implemented on these workstations. Thus, the parallel computing environment was built at Imai's laboratory at Tokusima University. 2. Development of Infinite Precision Numerical Simulation. Errors in numerical simulations originate from truncation errors in discretization and rounding errors. Infinite Precision Numerical Simulation was developed by combining the spectral (collocation) method and multiple precision arithmetic. The spectral (collocation) method is used for the control of truncation errors. Multiple precision arithmetic is used for the control of rounding errors. The method was applied to PDE systems with smooth solutions. The feature of the method, i.e. arbitrary reduction of errors, was observed. In a one-dimensional boundary value problem errors were approximately 10^<-2300>. This is incredible compared with results by other numerical methods. 3. Development of the library and its release. The library for Infinite Precision Numerical Simulation was developed. The subroutine of Gauss elimination in multiple precision arithmetic was developed. Its parallelization was performed by using PVM.The library was released by up-loading the report of the research project on Imai's home page. 4. Related results. Many related results were obtained as for development of infinite magnifying in visualization, development of parallel computing by using domain decomposition, basic research and application of Infinite Precision Numerical Simulation to inverse problems, free boundary problems and fluid mechanics, research on verification., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)., 1998 -2000, Principal investigator, Grant-in-Aid for Scientific Research (A)., The University of Tokushima
• Numerical analysis of the chaotic bahavior of the free boundary
  IMAI Hitoshi; SAKAGUCHI Hideo; OKAMOTO Hisashi; IKEDA Tsutomu; NISHIURA Yasumasa; TAKEUCHI Toshiki
  In the term of the project (three years) many results are obtained. Important results are shown as follows. 1. Development of numerical methods for analysis of chaotic phenomena. The group of Tokushima Univ. developed a method for analysis of chaotic phenomena in free boundary problems.It is based on numerical methods, so it is general. Moreover, it has a surprising feature that attractors in the infinite-dimensional solution space can be approximated arbitrarily. This means the method connects theoretical analysis and numerical analysis, therefore it enhances development of theoretical and numerical researches. 2. A proposal of a free boundary problem with attractors. The group of Tokushima Univ. proposed a one-dimensional free boundary problem with some parameters. It has exact solutions for special values of the parameters, so it is covenient for the check of the method and programs. Various attractors are found numerically. 3. Analysis of the pattern. Nishiura solved self-replicating dynamics of the dissipative system. He also revealed the minimizer of the system which describes the pattern formation of the diblock copolymer has the fine structure with the meso-scale. Sakaguchi proposed a model to formation of colony patterns by a bacterial cell population. From numerical simulation various patterns are observed in spite of the simplicity of the model. 4. Development of fast numerical computation and its application. The group of Tokushima Univ., Ikeda and Okamoto developed methods for fast numerical computation in the environment of parallel computing offered by PVM. These methods enabled large-scaled numerical simulation of free boundary problems or fluid mechanics. Some new phenomena were found., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B), 1997 -1999, Principal investigator, Grant-in-Aid for Scientific Research (B), The University of Tokushima
• Mathematical Open Problems of the Navier-Stokes Equations
  OKAMOTO Hisashi; NAKAKI Tatsuyuki; SHOJI Mayumi; IMAI Hitoshi; IKEDA Hideo; OHKITANI Koji
  Progress is made in the study of the Navier-Stokes equation, the Burgers equations, and the reaction-diffusion equations. Okamoto and Shoji performed numerical experiments on the bifurcation of surface waves. New bifurcation diagrams are found and will be published in a form of textbook by World Scientific Inc. Okamoto and Sakajo compute numerically two-dimensional and three-dimensional vortex sheet motion. T.Ikeda and H.Ikeda consider a certain system of reaction-diffusion equations for three competing species. They clarify the structure of steady-states and traveling pulses. Their stability is also determined. K.Ohkitani and M.Yamada consider what is called the shell model of the turbulence. By numerical methods, they compute the Lyapunov numbers of the system and they study the scaling properties of the numbers. They derive an asymptotic formula as the viscosity tends to zero. T.Nakaki consider the motion of vortex patches as well as point vortices. He finds that the motion of the patches are quite similar to that of point vortices if the size of the patches are small enough and that the motion of vortex patches are substantially different if the sizes are large. Y.Kimura considers the motion of point vortices on two-dimensional hyperbolic surfaces. Its Hamiltonian formalism are derived and the algebraic properties of the invariants are studied. T.Nishida numerically computes the Boussinesq equations, which are the master equations for the thermal convection. In particular he obtains numerically the bifurcation from the trivial solution to the stationary convective flow. He applies the numerical verification technique and derives new criteria., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1997 -1998, Grant-in-Aid for Scientific Research (A), KYOTO UNIVERSITY
• 非線形複雑系の3次元数値計算
  池田 勉; 三村 昌泰; 友枝 謙二; 岡本 久; 今井 仁司; 高橋 大輔; 中村 正彰; 木田 重雄; 河原田 秀夫; 四ツ谷 晶二
  研究代表者池田は,反応拡散方程式系の並列計算スキームの有限要素法によるコーディングを完成した.これは領域分割法と前処理付き共役勾配法に基づくものである.領域を分割することにより生成される内部界面では整合しない三角形分割も許されるモルタル法のような手法もあるが,我々は分かり易さを前面に押し出す方針を採用した.その結果,三角形分割は内部界面において整合条件を満たさなければならないが,通常の有限要素法と同一の数値計算結果が得られるという特徴を備えている.大腸菌のある種の変異株が,複雑な過程を経て,形成するストライプ・スポット状のパターンの並列計算を行い,並列スキームがほぼ予定通りに機能することを確認した.研究分担者三村は,反応拡散系と界面ダイナミクスの数理解析を分担し,発熱反応拡散系に現れる時空間パターンの特異極限解析と計算機シミュレーション,競争拡散系に現れる棲み分けパターンを捉えるための界面ダイナミクス法の導出,活性抑制因子反応拡散系に現れるパルスダイナミクスの数理解析,バクテリアコロニーの時空間パターンのモデル化とその計算機シミュレーションを行った. 渦と乱流場に関しては,岡本が水の波の数値計算および渦層の巻き上げの数値計算において重要な貢献を行い,木田が渦構造の同定と抽出,管状渦と剪断流の3次元相互作用の2点を中心に詳細な研究を行った.高橋はすべての独立変数と従属変数を離散化する超離散化法の理論を交通流モデルおよびソリトン系に応用することに成功した.今井は自由境界問題の高精度数値解法の開発と応用,ポアソン方程式の差分離散化法およびSOR法の収束率などの研究を,友枝は多孔質媒体を流れる流体によって形成される浸透領域のダイナミクスなどの研究を推進した., 日本学術振興会, 科学研究費助成事業, 1995 -1997, 基盤研究(A), 龍谷大学
• 境界の運動方程式を陽に含まない自由境界問題の数値解析手法の開発
  今井 仁司
  本研究では以下のような成果が得られた. 1.自由境界の運動に関する時間発展方式を陽に持たない自由境界問題に対する数値解法の開発とその応用. 自由境界問題の中には自由境界の運動に関する時間発展方程式を陽に持たないものがある.このような問題は数値的にといえども解くのは容易ではないし,中には高精度計算を行わないと計算が途中で止まったり意味のない解が得られてしまうような微妙なものもある.そこで,本研究では,自由境界の時間発展方程式を導出して既存の高精度数値解法を適用し,このような自由境界問題を高精度に数値計算することを考えた.自由境界の運動方程式を陽に導くことは,写像関数を用いた固定領域法を適用することで成功した.また,この手法は同時に問題を固定領域の問題に変換するため,既存の高精度数値解法が適用できるという特徴も持っている.本手法を,自由境界の運動に関する時間発展方程式を陽に持たない,双曲型方程式に支配される自由境界問題に適用したところ,クリティカル時間など興味深い数理現象を発見した. 2.高精度数値手法の開発とその応用. 1.の自由境界問題の数値解法の開発に関連した研究も行った.いままでは自由境界問題に対して任意精度で数値計算することはできなかった.それをスペクトル選点法と写像関数を用いた固定領域法を併用することで可能にした.ただし,この手法は時間に関して陰解法となるために反復計算が必要となり計算コストがかさむ.(したがって,パラメータサーベイが必要とされる1.の自由境界問題に対して本手法は適用しなかった.)そこで問題の線形化手法の開発と効率的な反復法の開発を行った. この高精度数値手法であるスペクトル選点法を用いて,カオスの解析を実用的な計算精度限界である4倍精度で行った.今回もアトラクターのフラクタル次元の不連続性が確認された., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Encouragement of Young Scientists (A), 1996 -1996, Principal investigator, Grant-in-Aid for Encouragement of Young Scientists (A), The University of Tokushima
• Studies on mathematical analysis and numerical computation of the Nevier-Stokes equations
  OKAMOTO Hisashi; SHOJI Mayumi; NAKAKI Tatsuyuki; IKEDA Hideo; SUGIHARA Masaki; NISHIMASA Yasumasa
  The present study carried out mathematical and numerical analysis of the Navier-Stokes equations and the Euler equations, which are the master equations of incompressible fluid. In addition, some abstract analysis of numerical schemes which are necessary for the fluid computations. The study consists of three categories : (1) mathematical analysis of the Navier-Stokes equations, (2) numerical experiments on the bifurcation of water waves, and (3) numerical computation of the Euler equations by the vortex method. (1) mathematical analysis of the Navier-Stokes equations. New exact solutions of the Navier-Stokes equation outside a cylinder are discovered ; they are generalizations of Tamada's solution and Wang's solution. Kolmogorov's problem is studied and we find that some stationary solutions tends to C^1 but not C^2 vector field as the Reynolds number tends to infinity. This solution represents a kind of internal layr, which may well serve as a key to the understanding of the turbulent power spectra. Some self-similar solutions of the Navier-Stokes equations, which are represented by the congruent hypergeometric functions, are discovered. Some stationary solutions having inflows and outflows and their stability were considered. Some of them are found to be stable for all the Reynolds number. (2) numerical experiments on the bifurcation of water waves. Two-dimensional irrotational flows with free surface are considered. The free surface are assumed to be periodic in its profile and permanent in time. Varying the Weber number and the Froude number, we compute many now bifurcating solutions. We also compute water waves with negative surface tension. Some of them are, in its limiting form, found to be the same as Euler's elastica. (3) numerical computation of the Euler equations by the vortex method. Two-dimensional vortex sheets in shear flows are computed by the vortex method. Many studies on vortex sheet motion are known, but our research is new in that we study the relation between vortex sheet and background shear flow., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1995 -1996, Grant-in-Aid for Scientific Research (A), KYOTO UNIVERSITY
• 知識工学を用いた効率的な偏微分方程式ソルバの開発
  今井 仁司
  本年度の研究では,以下に示すように,ある偏微分方程式で記述された逆問題に対してファジィ理論を応用したソルバを開発した.また,知識工学を用いて効率化が必要とされる新たな応用分野を開拓した. 1.核融合炉の設計に関連した楕円型偏微分方程式の初期値問題として表される不適切な形状設計問題に対して,ファジィ理論を応用したソルバを開発した.これは問題の物理的な性質をファジィ理論を用いてインプリメントしたものであり,以下のような特徴を持つ. (1)柔軟性がある.これによって,振動現象をユーザーの思うように抑えることができる. (2)並列処理が実現されている.これによって,速く形状設計できるようになった. (3)入力データに対して出力データが非線形である.これは,このソルバが非線形であることを示している.他の偏微分方程式系の問題に対しても,問題の性質がわかっていたら,ファジィ理論を用いることで,同様のアプローチが可能であることがわかった. 2.偏微分方程式で記述される自由境界問題に対して,時間空間のどちらに対しても任意次数で数値計算することはできなかった.それをスペクトル選点法と写像関数を用いた固定領域法を併用することで原理的には解決した.この手法に基づく数値計算を決められた精度内で実行するためには,次数や時間刻み幅の動的な制御が必要になる.そこに知識工学の応用が有効であると思われる. 3.カオスの分野でアトラクターの次元の不連続性を数値的に発見した.これは,現在実用的な範囲で一番高次の4次の公式を用いて見つけた.この興味ある現象をさらに確定的にするためには,より高次の公式を用いてかつ次数や時間刻み幅の動的な制御を行って,より高精度に数値計算する必要がある.その制御に知識工学の応用が有効であると思われる., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Encouragement of Young Scientists (A), 1995 -1995, Principal investigator, Grant-in-Aid for Encouragement of Young Scientists (A), The University of Tokushima
• 連立非線形方程式系の大域における数値解法とその応用
  篠原 能材; 宮本 陽生; 香田 温人; 深貝 暢良; 今井 仁司; 長町 重昭
  非線形常微分方程式系dx/dt=X(x,t)の準周期解および概周期解の数値解析的研究を,京都大学数理解析研究所の短期共同研究「常微分方程式系の数値解析とその応用」(研究代表者:篠原能材,1994年11月14日〜11月16日)を通じて行った。研究課題の目的を達成するために,第1日は,Prof.Linda PETZOLD(ミネソタ大)の講演「Computational Challenge in the Solution of nonlinear oscillatorg multibody Systems」から始まり,大石進一(早稲田大・理工)は「非線形常微分方程式の2点境界値問題の精度保証つき数値解析について」,新谷尚義(広島大・学校教育)は「微分・代数方程式の準陽公式について」,柳原弘毅(九州産業大)は「予測子・修正子法の安定性について」の各研究発表があった。 第2日は柏木雅英(早稲田大・理工)は「テーラー級数法による精度保証つき数値解法」,香田温人(徳島大・工)は「非線形振動解析のためのガレルキン法について」,渡辺二太(核融合科学研)は「HIOMによる微分・代数方程式の数値計算法」,川上博(徳島大・工)は「電子回路に現れる常微分方程式のカオス現象について」,牛田明夫(徳島大・工)は「分布定数回路の解析手法について」,室谷義昭・石渡恵美子(早稲田大・理工)は「特異摂動問題から得られる非対称行列に対する順序付き改良SOR法について」,早川透(早稲田大・理工)は「抵抗回路の区間解析について」の各研究成果を報告した。 第3日は小藤俊幸(電気通信大)は「時間おくれをもつ微分方程式に対するルンゲ・クッタ法の安定性について」,小野令美(千葉大・工)は「ある8段陽的ルンゲ・クッタ法について」,小沢一文(東北大)は「微分方程式y″=f(x,y)に対する4次のP安定ブロツク法について」,山田進(東北大)は「BDF型ブロツク法の並列性と計算効率の解析」,鈴木千里(静岡工科大)は「2点エルミートーバーコフ型のA安定な数値積分公式のクラスについて」の各研究成果を報告した。, 日本学術振興会, 科学研究費助成事業, 1994 -1994, 一般研究(C), 徳島大学
• 選点法を用いた高精度時間積分公式の開発と非定常微分方程式系への応用
  今井 仁司
  スペクトル選点法は既に開発されている手法であるが,いままでは無限精度を持つことだけ強調されて応用されてきた.それをここでは,無限精度ではなくむしろ任意精度に設定できるという特徴に着目して,任意次数の時間積分公式として新たな位置づけを行った.このような観点から見ると,いままでの次数が代わると公式自体が大幅に変わってくる時間積分公式とは全く異なった,選点数によって次数が任意に自由に設定できるという全く新しいしかも使い易い時間積分公式が誕生したといえる.また,それを応用した新たな手法をいくつか開発してスペクトル選点法による数値シミュレーションの適用範囲を大幅に広げた.具体的な内容を以下に述べる. 1.スペクトル選点法による時間積分公式の高精度性との安定性解析 スペクトル選点法による時間積分法は,理論的には選点数と次数が比例し絶対安定であることが予想されたが,数値実験によりそれが確認された.このことから,時間刻みを大きくとり任意の高次精度の計算をすることによって,丸め誤差の影響を受けにくい長時間積分が可能になった. 2.スペクトル選点法の高速化手法の開発 差分法などでも(非線形)連立一次方程式が反復で解かれるのと同様,スペクトル選点法でも反復法が用いられる.スペクトル選点法は,精度が高い反面計算規模が大きくなると行列の性質が悪くなり,前処理なしでは反復計算が収束しない.ここでは,いままでの拡張となっている前処理を新たに開発した.この前処理法によっていままでより少ない計算時間でスペクトル選点法の大規模数値シミュレーションが可能になった. 3.非定常偏微分方程式系への応用 時間空間ともにスペクトル選点法で離散化し写像関数を用いた固定領域法を併用することで,自由境界問題に対する新たな高精度数値計算手法を開発し,数値計算でその高精度性を確認した., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Encouragement of Young Scientists (A), 1994 -1994, Principal investigator, Grant-in-Aid for Encouragement of Young Scientists (A), The University of Tokushima
• 常微分方程式系の初期値問題の数値解析的研究
  篠原 能材; 宮本 陽生; 香田 温人; 深貝 暢良; 長町 重昭; 今井 仁司
  常微分方程式系の初期値問題(dx)/(dt)=f(t,x),x(t_o)=x_o…(1)の解x(t)を,オイラー法,ルンゲ・クッタ法やアダムス法で代表される差分近似法x_=x_n+h_npsi(t_n,x_n,h_n)(n=0,1,2,…),h_n=t_-t_n(n=0,1,2,…)で数値計算によって、コンピュータで求めるとき、これ等の差分近似法の局所打切誤差と丸め誤差の影響で,各ステップt=t_nにおいて誤差γ_nが発生するため,実際の数値計算においては,元の初期値問題と異なった問題: (dy)/(dt)=f(t,y),(t≠t_n),y(t_n+O)-y(t_n-O)=γ_n,y(t_o)=y_o=x_o(n=0,1,2,…)…(2)の不連結解y(t)を求める問題となっている。したがって連結解x(t)と不連結解y(t)との差e(t)=y(t)-x(t)の挙動を調べることによって,差分近似法で解き易い初期値問題と解き難い初期値問題とを判別する数学的基準を確立することを本研究の目的とした。平成5年度の研究成果は次の定理である。 「定理.初期値問題(1)の解x(t)と初期値問題(2)の解y(t)は次の関係式を満す。 y(t)=x(t)+Σ^^m__〓^1_oΦ(t,t_k,y_k+(θ-1)γ(t_k)γ(t_k)dθ for t puch that t_0
• Algorithms for Large Scale Linear Systems in the Numerical Simulation
  NATORI Makoto; IMAI Hitoshi; SAKURAI Tetsuya; KITAGAWA Takashi; INAGAKI Toshiyuki; IKEBE Yasuhiko
  In this research, algorithms to solve large scale linear systems appeared in the numerical simulation were investigated. (1)We investigated preconditioned conjugate gradient methods to solve large scale linear systems of equations. Especially, We considered BiCGSTAB method which is known to be stable for asymmetric matrics. We developed a new preconditioner which enables parallel processing. (2)We investigated Lanczos algorithm to compute eigenvalus of large sparse matrices. Especially, we concidered the block Lanczos algorithm for matrices which have degenerate eigenvalues. We developed a new reorthoganalization mehod for the block Lanczos algorithm. (3)We investigated regularization methods for ill - posed problems. We developed new algorithms to estimate the value of the optimal regularization parameters. (4)We carryed out the numerical simulations of solidification problems with change of valume and natural convection problems with a free surface., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1992 -1993, Grant-in-Aid for General Scientific Research (C), University of Tsukuba
• 大型線形計算のアルゴリズム
  名取 亮; 小柳 義夫; 今井 仁司; 北川 高嗣; 池辺 八洲彦
  本研究では,大型線形計算のアルゴリズムのうち,大型線形方程式系の高速数値解法と大型疎行列の固有値問題に対するランチョス法について重点的に研究した。 1.大型線形方程式系の数値解法については,特に線形最小2乗問題を取り上げて,Orthomin(k)法の前処理および収束特性について研究した。 2.大型疎行列の固有値問題については,ランチョス法を取り上げて研究した。ランチョス法は大型疎行列に適したアルゴリズムであるが,計算を進めるにしたがってランチョスベクトルの間の直行性が丸め誤差のために崩れるという欠点を持っている。この直交性の崩れを防ぐために計算の途中で再直交化を行う必要がある。我々は再直交化のアルゴリズムとして,新しい方法(RIC)を考案し,従来の再直交化法(PROとSO)にくらべて優れていることを示した。また,単純なランチョス法では重複固有値を正確に求めることは不可能で,ブロックランチョス法を用いる必要がある。我々は,単純なランチョス法に対して考案した再直交化法(RIC)をブロックランチョス法に拡張することを試みた。そのために,ランチョスベクトル間の直交性を表す量が満す漸化式をブロックランチョス法に対しても導き,それを用いて直交性の崩れを検出するようにした。この方法も従来の方法にくらべて優れていることを示すことができた。特に,すべての固有値を正確に求めることができない場合には,どの固有値が信頼できる固有値であるかを示すことができる点が我々の方法の特徴である。, 日本学術振興会, 科学研究費助成事業, 1991 -1991, 一般研究(C), 筑波大学
• 位相的に変化する領域を伴う自由境界問題の数値解法の開発
  今井 仁司
  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Encouragement of Young Scientists (A), 1990 -1990, Principal investigator, Grant-in-Aid for Encouragement of Young Scientists (A), University of Tsukuba
• Numerical Analysis of Partial Differential Equations with Free Boundaries
  NATORI Makoto; MIYAMOTO Sadaaki; IMAI Hitoshi; INAGAKI Toshiyuki; OYANAGI Yoshio; IKEBE Yasuhiko
  In this research, numerical methods to solve partial differential equations with free boundaries were investigated. (1) Domidov developed a method using a hodograph transformation to obtain shapes of equilibrium plasma in the vessel. This method is valid only when the vessel has the polygonal boundary. We proposed an approximate resolution to obtain the shapes of plasma for the vessels which have not the polygonal boundary. The characteristics of our method is to use the conformal mapping. The validity of our method is numerically shown. (2) Bifurcation phenomena in the free boundary problem of equilibrium plasma have been studied by Domidov and Imai and Kawarada, for one-component plasma shapes. We considered two-component plasma and bifurcation of both one and two component plasma shapes is analyzed numerically. Numerical results show that in spite of the simplicity of the problem, many types of bifurcation occur. (3) We developed a method for finding bifurcation points along solution curves in free boundary problems. In this method, a point along a solution curve is determined as a bifurcation point where the smallest eigenvalue of a linearized problem is equal to zero. In order to verify the proposed method, numerical computations are carried out. (4) We proposed a numerical method which is useful for simulating flows in drop formation from a capillary. In our method, the boundary fit method is used to solve the Navier-Stokes equation with a free boundary. At the same time, a new method on the idea of the MAC method is proposed in the case of the splitting of free boundaries when deformation is large. (5) We developed a new method for the reorthogonalization in the Lanczos algorithm., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1989 -1990, Grant-in-Aid for General Scientific Research (C), University of Tsukuba
• 領域の位相的変化を伴う圧縮-非圧縮2相流の数値解析
  今井 仁司
  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research Grant-in-Aid for Encouragement of Young Scientists (A), 1989 -1989, Principal investigator, Grant-in-Aid for Encouragement of Young Scientists (A), University of Tsukuba
• Research in Numerical Analysis of Free Boundary Problems
  NATORI Makoto; IMAI Hitoshi; INAGAKI Toshiyuki; OYANAGI Yoshio; IKEBE Yasuhiko; MORI Masatake
  In this research, numerical methods to solve free boundary problems for partial differential equations were invastigated. The finite difference method combined with the integrated penalty method proposed by Kawarada and Natori, the boundary element method(BEM) and the finite element method(FEM) are used. A sharp interface problem arising in the flow of two immiscible fluids, slag and moltem metal, in a blast furnace was formulated usign a two and three dimensional model. For two dimensional case, the BEM and the FEM were compared. It turned out that the CPU time by the FEM was about half of that by the BEM. The ICCG method for the present sparse symmetric matrix equation plays a significant role in reducing the CPU time. A free boundary problem for the shape of the two dimensional equilibrium plasma in the vessel was studied. It was munerically shown that asymmetric one-conponent plasmas exist in the vessel which has a symmetric cross section. Numerical methods to solve large sparse system of linear equations were investigeted. Especially the preconditioned conjugate gradient method(PCG, ICCG) and the related methods were studied., Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, 1987 -1988, Grant-in-Aid for General Scientific Research (C), University of Tsukuba