研究者代表のプロフィール

氏　名

中尾　充宏　(なかお　みつひろ)
Mitsuhiro T. NAKAO


所属部局　学科部門・職名

九州大学大学院数理学研究院 数理科学部門・教授

履　歴

生年月日	昭和22年 9月17日 （長崎県佐世保市）
昭和43年 3月	国立佐世保工業高等専門学校機械工学科卒業
昭和47年 3月	九州大学理学部数学科卒業
昭和49年 3月	同上大学院理学研究科修士課程数学専攻修了
昭和49年 4月	日本電信電話公社（現NTT）横須賀電気通信研究所研究員 コンピュータネットワークアーキテクチャの研究に従事
昭和51年10月	同上研究主任
昭和55年 1月	九州大学助手（理学部） 数値解析学（偏微分方程式の Galerkin 近似解の誤差評価） の研究に従事
昭和57年 4月	九州工業大学講師（工学部）, 59年10月同助教授
昭和59年 4月	理学博士（九州大学）
昭和63年10月	九州大学助教授（理学部）
平成 6年 6月	九州大学教授 （大学院数理学研究科, 平成12年4月より数理学研究院） 精度保証付き数値計算法の研究に従事. 特に偏微分方程式とそれから派生する無限次元問題の解に対する数値的検証法と応用解析学における計算機援用証明の研究.

主な学内外役職等

平成12年 4月	九州大学評議員（平成14年3月まで)
平成14年 7月	九州大学大学院数理学研究院長（平成18年7月15日まで）
平成19年 4月	九州大学産業技術数理研究センター長
平成11年 4月	文部科学省理学視学委員（平成15年3月まで）
平成15年 7月	２１世紀COEプログラム「機能数理学の構築と展開」 拠点リーダー(平成20年3月まで)
平成18年 9月	日本学術会議連携会員
平成19年10月	京都大学数理解析研究所運営委員
平成20年 5月	井上科学振興財団 選考委員会委員

研究上の興味

無限次元問題の解に対する精度保証付き数値計算法、特に偏微分方程式とそれに関連した問題の解に対する数値的検証法の研究を行っている。また併せて、有限要素法の信頼性、特に構成的誤差評価に関心を持ち、そのa prioriおよび a posterioriな評価手法の導出を目ざしている。これは非線形偏微分方程式の解の数値的検証の基礎を与える研究ともなっている。

偏微分方程式の解に対する精度保証については、筆者（中尾）が、楕円型境界値問題の有限要素近似とその構成的a priori誤差評価を、区間解析と有効に組み合わせることにより、厳密解が計算機上で捉えられことを、1988年世界に先駆けて立証した。この方法による検証結果は、解の存在検証とともに偏微分方程式の有限要素近似解に対するa posteriori誤差評価をも与えるものである。

その後、関連研究協力者との共同研究で、この原理にもとづく研究を進め、現在までに２次元有界領域上の２階半線形楕円型方程式のDirichlet問題については、多くの問題に対して、検証精度・効率の点で十分実用性のある方法として定着させるに至った。さらに適用対象も、楕円型作用素の固有値問題や逆固有値問題、定常Navier-Stokes方程式あるいは変分不等式にまで広げることに成功している。

最近では、応用解析学研究者との連携のもとに、理論解析が困難なNavier-Stokes方程式支配下の熱対流問題に対し、分岐解の性質に関する計算機支援証明を試みている。２次元問題については流れ関数表示による４階連立非線形楕円型問題に帰着させ、適切な関数空間の設定のもとで、スペクトルGalerkin法とその誤差評価を利用して数値的検証の定式化を行った。現在までに、分岐点から十分長いRayleigh数領域で分岐解が存在することの数値的証明や、２次分岐の存在検証など、理論的解明が困難ないくつかの課題について数値的検証を行い、内外の注目を集めている。現在、特にこの結果の３次元問題への拡張に取り組んでいる。３次元ではhexagonalやrectangularあるいはmixedタイプなど、興味ある解の存在が予想されており、数値検証の対象としても重要であるが検証コストをはじめ検証の困難性は飛躍的に増大する。現在までに、基本的定式化を終え、臨界Rayleigh数からの分岐後まもなくの解（例えば、相対Rayleigh数$\approx 1.1$程度）については、その存在検証を実現しており、今後の改良についても鋭意検討を進めている。 このほか、一層の検証効率化・高精度化と適用領域を広めるための新たな検証方式の開発や、精度保証の基礎となる各種a priori定数の計算機援用証明による精度保証付き算定を行うなど、将来の「計算機援用解析学」の創設に向けて意欲的に研究を進めている。

主な研究業績

[論文]
[1] 河岡、友永（中尾の旧姓）、高橋，通信制御プログラムに関するプロトコル試験手順の最適化，電子通信学会論文誌，J63-D，8 (1980)，618-625.

[2] Nakao, M., Some superconvergence estimates for a collocation-$H^{-1}$-Galerkin method for parabolic problems, Memoirs of Faculty of Science Kyushu University, Ser. A 35 (1981), 291-306.

[3] Tomonaga（中尾の旧姓）, M., Optimal error estimates for $H^{-1}-$Galerkin method for parabolic problems with time dependent coefficients, Memoirs of Numerical Mathematics No. 8/9 (1982), 65-85.

[4] Nakao, M., Collocation-$H^{-1}$-Galerkin method for some parabolic equations in two space variables, Memoirs of Faculty of Science Kyushu University, Ser. A 36 (1982), 129-143.

[5] Nakao, M., Interior estimates and superconvergence for $H^{-1}-$Galerkin method to elliptic equations, Bulletin of the Kyushu Institute of Technology (Math. \& Natur. Sci.) 30 (1983), 19-30.

[6] Nakao, M., Superconvergence estimates at Jacobi points of the collocation-Galerkin method for two point boundary value problems, Journal of Information Processing, 7 (1984), 31-34.

[7] Nakao, M., Some superconvergence estimates for a Galerkin method for elliptic problems, Bulletin of the Kyushu Institute of Technology (Math. \& Natur. Sci.) 31 (1984), 49-58.

[8] Nakao, M., A collocation$-H^{-1}-$Galerkin method for some elliptic equations, Mathematics of Computation 42 (1984), 417-426.

[9] Nakao, M.T., $L^{\infty}$ error estimates and superconvergence results for a collocation-$H^{-1}$-Galerkin method for elliptic equations, Memoirs of Faculty of Science Kyushu University, Ser. A 39 (1985), 1-25.

[10] Nakao, M., Some superconvergence of Galerkin approximations for parabolic and hyperbolic problems in one space dimension, Bulletin of the Kyushu Institute of Technology (Math. & Natur. Sci.) 32 (1985), 1-14.

[11] Nakao, M.T., Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension, Numerische Mathematik 47 (1985), 139-157.

[12] Nakao, M.T., Some superconvergence for a Galerkin method by averaging gradients in one dimensional problems, Journal of Information Processing, 9 (1986), 130-134.

[13] Nakao, M.T., Superconvergence of gradients of Galerkin approximations for elliptic problems, Mathematical Modelling and Numerical Analysis 21 (1987), 679-695.

[14] Nakao, M.T., A numerical approach to the proof of existence of solutions for elliptic problems, Japan Journal of Applied Mathematics 5 (1988), 313-332.

[15] Nakao, M.T., A computational verification method of existence of solutions for nonlinear elliptic equations, Lecture Notes in Num. Appl. Anal., 10, (1989) 101 - 120. In proc. Recent Topics in Nonlinear PDE {\tt4}, Kyoto, 1988, $North-Holland/Kinokuniya,$ 1989.

[16] Nakao, M.T., A numerical approach to the proof of existence of solutions for elliptic problems I\,I, Japan Journal of Applied Mathematics 7 (1990), 477-488.

[17] Nakao, M.T. & Yamamoto, N., Numerical verifications of solutions for elliptic equations with strong nonlinearity, Numerical Functional Analysis and Optimization 12 (1991), 535-543.

[18] Nakao, M.T., Solving nonlinear parabolic problems with result verification Part I: One space dimensional case, Journal of Computational and Applied Mathematics 38 (1991) 323-334.

[19] Nakao, M.T., A numerical verification method for the existence of weak solutions for nonlinear boundary value problems, Journal of Mathematical Analysis and Applications 164 (1992), 489-507.

[20] Nakao, M.T., Computable error estimates for FEM and numerical verification of solutions for nonlinear PDEs, Computational and Applied Mathematics, I (eds. C.Brezinski and U.Kulisch), North-Holland (1992), 357-366 .

[21] Watanabe, Y. & Nakao, M.T., Numerical verifications of solutions for nonlinear elliptic equations, Japan Journal of Industrial and Applied Mathematics 10 (1993), 165-178.

[22] Nakao, M.T., Solving nonlinear elliptic problems with result verification using an $H^{-1}$ residual iteration, Computing, Supplementum 9 (1993), 161-173.

[23] Nakao, M.T., Computable $L^{\infty}$ error estimates in the finite element method with application to nonlinear elliptic problems, Series in Applicable Analysis Vol.2, Contributions in Numerical Mathematics(ed. R.P. Agarwal), World Scientific (1993), 309-319.

[24] Yamamoto, N. & Nakao, M.T., Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains, Numerische Mathematik 65 (1993), 503-521.

[25] Nakao, M.T. & Watanabe, Y., On computational proofs of the existence of solutions to nonlinear parabolic problems, Journal of Computational and Applied Mathematics 50 (1994), 401-410.

[26] Nakao, M.T., Numerical verifications of solutions for nonlinear hyperbolic equations, Interval Computations 4 (1994), 64-77.

[27] Yamamoto, N. & Nakao, M.T., Numerical verifications for solutions to elliptic equations using residual iterations with higher order finite element, Journal of Computational and Applied Mathematics 60 (1995), 271-279.

[28] Nakao, M.T. & Yamamoto, N., A simplified method of numerical verification for nonlinear elliptic equations, in the Proceedings of International Symposium on Nonlinear Theory and its Applications (NOLTA'95), Las Vegas, USA, (1995), 263-266.

[29] Nakao, M.T., Yamamoto, N. & Watanabe, Y., Guaranteed error bounds for finite element solutions of the Stokes problem, in {\it Scientific Computing and Validated Numerics} (G. Alefeld et al. eds.), Akademie Verlag, Berlin (1996), 258-264.

[30] Watanabe, Y., Nakao, M.T. & Yamamoto, N., Verified computation of solutions for nondifferentiable elliptic equations related to MHD equilibria, {\it Nonlinear Analysis, Theory, Methods and Applications} 28, (1997), 577-587.

[31] Tsuchiya, T. & Nakao, M.T., Numerical verification of solutions of parametrized nonlinear boundary value problems with turning points, Japan Journal of Industrial and Applied Mathematics 14 (1997), 357-372.

[32] Minamoto, T. & Nakao, M.T., Numerical verifications of solutions for nonlinear parabolic equations in one-space dimensional case, {\it Reliable Computing} {\bf 3}, (1997), 137-147.

[33] Nakao, M.T. & Yamamoto, N., Numerical verification of solutions for nonlinear elliptic problems using $L^{\infty}$ residual method, Journal of Mathematical Analysis and Applications {\bf 217}, (1998), 246-262.

[34] Nakao, M.T., Yamamoto, N. & Kimura, S., On best constant in the optimal error stimates for the $H^1_0$-projection into piecewise polynomial spaces, Journal of Approximation Theory {\bf 93}, (1998), 491-500.

[35] Nakao, M.T., Yamamoto, N. & Watanabe, Y., Constructive $L^2$ error estimates for finite element solutions of the Stokes equations, Reliable Computing {\bf 4} (1998), 115-124.

[36] Nakao, M.T., Yamamoto, N. & Watanabe, Y., A posteriori and constructive a priori error bounds for finite element solutions of Stokes equations, Journal of Computational and Applied Mathematics {\bf 91}(1998), 137-158.

[37] Nakao, M.T., Yamamoto, N. & Nishimura, Y., Numerical verification of the solution curve for some parametrized nonlinear elliptic problem, in Proc. Third China-Japan Seminar on Numerical Matehmatics, Aug. 26-30, Dalian, China, 1996 (eds. Shi, Z.-C. \& Mori, M.), Science Press, Beijing , (1998), 238-245.

[38] Ryoo, C-S & Nakao, M.T., Numerical verification of solutions for variational inequalities, Numerische Mathematik {\bf 81 } (1998), 305-320.

[39] Nakao, M.T. & Ryoo, C-S, Numerical verifications of solutions for variational inequalities using Newton-like Method, INFORMATION, 2 (1999), 27-35.

[40] Nakao, M.T., Yamamoto, N. & Nagatou, K., Numerical verifications of eigenvalues of second-order elliptic operators, Japan Journal of Industrial and Applied Mathematics 16 (1999), 307-320.

[41] Nagatou, K., Yamamoto, N. & Nakao, M.T., An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness, Numerical Functional Analysis and Optimization {\bf 20} (1999), 543-565.\\{ } [42] Watanabe, Y., Yamamoto, N. & Nakao, M.T., A numerical verification method of solutions for the Navier-Stokes equations, Reliable Computing {\bf 5} (1999), 347-357.

[43]Yamamoto, N., Nakao, M.T. & Watanabe, Y., Validated computation for a linear elliptic problem with a parameter, GAKUTO International Series, Mathematical Sciences and Applications Vol. 12 ({\it eds. Kawarada et al.}), Advances in Numerical Mathematics ; Proc. Fourth Japan-China Joint Seminar on Numerical Mtahematics, Aug. 24-28, 1998, Chiba, Japan, (1999), 155-162.

[44] Watanabe, Y., Yamamoto, N. & Nakao, M.T., Verification method of generalized eigenvalue problems and its applications(in Japanese), Transaction of the Japan Society for Industrial and Applied Mathematics, Vol. 9 No. 3 (1999), 137-150.

[45] Nakao, M.T., Lee, S.H. & Ryoo, C.S., Numerical verification of solutions for elasto-plastic torsion problems, Computers & Mathematics with Applications 39 (2000), 195-204.

[46] Minamoto, T., Yamamoto, N. & Nakao, M.T., Numerical verification method for solutions of the perturbed Gelfand equation, Methods and Applications of Analysis 7 (2000), 251-262.

[47] Toyonaga, K. & Nakao, M.T., Numerical enclosure for the optimal threshold probability in discounted Markov decision processes, Bulletin of Informatics and Cybernetics, 32 (2000), 81-90.

[48] Nagatou, K. & Nakao, M.T., An enclosure method of eigenvalues for the elliptic operator linearlized at an exact solution of nonlinear problems, Linear Algebra and its Applications 324 (2001), 81-106.

[49] Nakao, M.T., Watanabe, Y. & Yamamoto, N., Verified numerical computations for an inverse elliptic eigenvalue problem with finite data, Japan Journal of Industrial and Applied Mathematics 18 (2001), 587-602.

[50] Nakao, M.T. & Yamamoto, N., A guaranteed bound of the optimal constant in the error estimates for linear triangular element, Computing Supplementum 15 (2001), 165-173.

[51] Nakao, M.T. & Yamamoto, N., A guaranteed bound of the optimal constant in the error estimates for linear triangular element Part �U: Details, Perspectives on Enclosure Methods (eds. U. Kulisch et al.), the Proceedings Volume for Invited Lectures of SCAN2000, Springer-Verlag, Vienna (2001), 265-276.

[52] M.T. Nakao & C-S Ryoo, Numerical verification methods for solutions of free boundary problems, the Proceedings of MSCOM(International Symposium on Mathematical Modeling and Numerical Simulation in Continuum Mechanics), Sept. 29-Oct. 3, 2000, Yamaguchi, Japan (Miyoshi et al. eds.), Lecture Notes in Computational Science and Engineering, Springer Verlag (2001), 195-208.

[53] Nakao, M.T. & Toyonaga, K., An improvement of the enclosure method for elliptic eigenvalue problems, in the Proceedings of Fifth China-Japan Seminar on Numerical Matehmatics, Aug. 21-25, 2000, Shanghai, China (eds. Shi, Z.-C. & Kawarada, H.), Science Press, Beijing (2002), 181-188.

[54] Ryoo, C-S & Nakao, M.T., Numerical verification of solutions for variational inequalities of the Second Kind, Computer and Mathematics with Applications 43 (2002), 1371-1380.

[55] Toyonaga, K., Nakao, M.T. & Watanabe, Y., Verified numerical computations for multiple or nearly multiple eigenvalues for elliptic operators, Journal of Computational and Applied Mathematics 147 (2002) 175-190.

[56] Nagatou, K., Nakao, M.T. & Wakayama, M., Verified numerical computations for eigenvalues of non-commutative harmonic oscillators, Numerical Functional Analysis and Optimization 23 (2002), 633-650.

[57] Nakao, M.T., Watanabe, Y., Yamamoto, N. & Nishida, T., Some computer assisted proofs for solutions of the heat convection problems, Reliable Computing 9 (2003), 359-372.

[58] Ryoo, C-S & Nakao, M.T., Numerical verification of solutions for obstacle problems, Journal of Computational and Applied Mathematics 161 (2003), 405-416.

[59] Y. Watanabe, N. Yamamoto, M. T. Nakao & T. Nishida, A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem, Journal of Mathematical Fluid Mechanics 6 (2004), 1-20.

[60] Nakao, M.T. & Watanabe, Y., An efficient approach to the numerical verification for solutions of elliptic differential equations, Numerical Algorithms 37, Special issue for Proceedings of SCAN2002 (2004), 311-323.

[61] Hashimoto, K., Abe, R., Nakao, M.T. & Watanabe, Y., A Numerical Verification Method for Solutions of Singularly Perturbed Problems with Nonlinearity, Japan Journal of Industrial and Applied Mathematics 22 (2005), 111-131

[62] Nakao, M.T., Hashimoto, K. & Watanabe, Y., A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing 75 (2005), 1-14.

[63] Hashimoto, K., Kobayashi, K. & Nakao, M.T., Numerical Verification Methods of Solutions for the Free Boundary Problems, Numerical Functional Analysis and Optimization 26 (2005), 523-542.

[64] Watanabe, Y., Yamamoto, N. & Nakao, M.T., An efficient approach to numerical verification for solutions of elliptic differential equations with local uniqueness(in Japanese), Transaction of the Japan Society for Industrial and Applied Mathematics, Vol.15, No.4 (2005), 509-520.

[65] M. T. Nakao, Y. Watanabe, N. Yamamoto & T. Nishida, A numerical verification of bifurcation points for nonlinear heat convection problems, in the proceedings of 2nd International Conference "From Scientific Computing to Computational Engineering", Athen, 5-8 July, 2006, 8 pages.

[66] M.-N. Kim, M.T. Nakao, Y. Watanabe & T. Nishida, Some computer assisted proofs on three dimensional heat convection problems, in Proceedings of Nonlinear Theory and its Applications NOLTA 2006, 11-14 September, Bologna, Italy (2006), 427-430.

[67] Nagatou, K., Hashimoto, K, Nakao, M.T., Numerical verification of stationary solutions for Navier-Stokes problems, Journal of Computational and Applied Mathematics 199 (2007), 424-431.

[68] Minamoto, T. and Nakao, M.T., Numerical method for verifying the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed Gelfand equation, Journal of Computational and Applied Mathematics 202 (2007), 177-185.

[69] Nakao, M.T., Hashimoto, K., Kobayashi, K., Verified numerical computation of solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains, Hokkaido Mathematical Journal, Vol. 36, Special Issue, Proceedings on "The First China-Japan-Korea Joint Conference on Numerical Mathematics" (2007), 777-799.

[70] Nakao, M.T., Hashimoto, K., Nagatou, K., A computational approach to constructive a priori and a posteriori error estimates for finite element approximations of bi-harmonic problems, GAKUTO International Series, Mathematical Sciences and Applications Vol. 28, Proceedings of the 4th JSIAM-SIMAI Seminar on Industrial and Applied Mathematics, May 26-28, 2005, Hayama, Japan (2008), 139-148.

[71] Nakao, M.T., Hashimoto, K., Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications, Journal of Computational and Applied Mathematics, 218 (2008), 106-115.

[72] Nakao, M.T., Kinoshita, T., Some remarks on the behaviour of the finite element solution in nonsmooth domains, Applied Mathematics Letters 21 (2008), 1310-1314.

[73] M.-N. Kim, M.T. Nakao, Y. Watanabe, T. Nishida, A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-B\'{e}nard problems, Numerische Mathematik 111 (2009), 389-406.

[74] Nakao, M.T., Kinoshita, T., On very accurate verification of solutions for boundary value problems by using spectral methods, JSIAM Letters 1(2009), 21-24.

[75] Kinoshita, T., Hashimoto, K. and Nakao, M.T., On the $L^2$ a priori error estimates to the finite element solution of elliptic problems with singular adjoint operator, Numerical Functional Analysis and Optimization 30 (2009), 289-305.

[76] Watanabe, Y., Plum, M, Nakao, M.T., A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow, Zeitschrift fuer Angewandte Mathematik und Mechanik(ZAMM) 89 (2009), 5-18

［解説論文］
[1] 河岡、友永（中尾の旧姓）、高橋，プロトコルの記述法と検証法、情報処理、２０巻７号(1979), 612-621.

[2] 中尾充宏、関数方程式の解の存在に対する数値的検証法、数学、{\bf 42} (1990),16-31. （英訳版：Numerical verification methods for the existence of solutions for functional equations, Sugaku Exposition {\bf 5} (1992), 71-91.）

[3] 中尾充宏、精度保証付き数値計算の現状と動向、情報処理、３１巻９号(1990), 1177-1190.\\{ } [4] Mitsuhiro T. Nakao, State of the art for numerical computations with guaranteed accuracy, Mathematica Japonica, Vol. 48, No.2 (1997), 323-338.

[5] 中尾充宏、精度保証による数値解析、数理科学、No. 417 (1998), 28-34.

[6] 渡部善隆、中尾充宏、偏微分方程式の精度保証、シミュレーション、１９巻３号(2000), 208-215.

[6] Nakao, M.T., Numerical verification methods for solutions of ordinary and partial differential equations, Numerical Functional Analysis and Optimization 22(3\&4) (2001), 321-356.

[7] 中尾充宏、偏微分方程式の解に対する数値的存在検証---研究の原点とその展開---、Fundamentals Review Vol.2 No.3、電子情報通信学会、基礎・境界ソサイエティ、(2008), 19-28.

［著書］
[1] 中尾充宏・山本野人,「精度保証付き数値計算」, 日本評論社, 1998.

[2] Alefeld, G., Nakao, M.T., and Rump, S. (Guest Editors), Special Issue: Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2004), Journal of Computational and Applied Mathematics 199/2 (2007), 453 pages.

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