工学修士, 京都大学

博士(理学), 九州大学

Eigenvalue problems

PDEs

numerical analysis

Validated numerics

固有値問題

偏微分方程式

数値解析

精度保証付き計算

Natural sciences, Applied mathematics and statistics

Natural sciences, Basic mathematics

01 Apr. 2006
電気通信大学, 教授

Dec. 1987
Kyoto University, Graduate School, Division of Engineering, 数理専攻

Mar. 1986
Kyoto University, Graduate School, Division of Engineering, 数理専攻

Mar. 1984
Kyoto University, Faculty of Science

01 Apr. 1975 - 31 Mar. 1978
静岡県立藤枝東高等学校

01 Apr. 2021 - 31 Mar. 2022
代表会員, 日本応用数理学会, Others

01 Apr. 2020 - 31 Mar. 2021
代表会員, 日本応用数理学会, Others

01 Mar. 2015 - 28 Feb. 2017
理事, 日本応用数理学会

Mar. 2011 - Feb. 2013
a member of the council, Society

Mar. 2011 - Feb. 2013
評議員, 日本数学会, Society

Mar. 2010 - Feb. 2012
a member of the council, Society

Mar. 2010 - Feb. 2012
評議員, 日本応用数理学会, Society

Oct. 2005 - Sep. 2007
応用数学分科会委員, 日本数学会, Society

Numerical verification method on complex ODEs for existence of global solutions within finite domains
Koki Nitta; Nobito Yamamoto
Last, JSIAM Letters, The Japan Society for Industrial and Applied Mathematics, 15, 69-72, Aug. 2023, Peer-reviwed
Scientific journal

Inclusion method of optimal constant with quadratic convergence for H01-projection error estimates and its applications
Takehiko Kinoshita; Yoshitaka Watanabe; Nobito Yamamoto; Mitsuhiro T. Nakao
Journal of Computational and Applied Mathematics, Elsevier BV, 417, 114521-114521, Jan. 2023
Scientific journal

Simulation and Verified Numerics
山本野人
Lead, Journal of the Japan society for simulation technology, 41, 3, 176-182, Sep. 2022, Peer-reviwed, Invited
Japanese

A numerical verification method to specify homoclinic orbits as application of local Lyapunov functions
Koki Nitta; Nobito Yamamoto
Japan Journal of Industrial and Applied Mathematics volume, Springer, 39, 2, 467-513, May 2022, Peer-reviwed, We propose a verification method for specification of homoclinic orbits as application of our previous work for constructing local Lyapunov functions by verified numerics. Our goal is to specify parameters appeared in the given systems of ordinary differential equations (ODEs) which admit homoclinic orbits to equilibria. Here we restrict ourselves to cases that each equilibrium is independent of parameters. The feature of our methods consists of Lyapunov functions, integration of ODEs by verified numerics, and Brouwer’s coincidence theorem on continuous mappings. Several techniques for constructing continuous mappings from a domain of parameter vectors to a region of the phase space are shown. We present numerical examples for problems in 3 and 4-dimensional cases.
Scientific journal, English

On numerical verification methods to construct local Lyapunov functions around non-hyperbolic equilibria for two-dimensional cases
Koki Nitta; Toshiki Sasaki; Nobito Yamamoto
JSIAM Letters, 14, 33-36, 16 Mar. 2022, Peer-reviwed
Scientific journal, English

Erratum: Errata to “On the construction of Lyapunov functions with computer assistance” [J. Comp. Appl. Math. 319 (2017) 385-412] ( Journal of Computational and Applied Mathematics (2017) 319(385-412) (S0377042717300067), (10.1016/j.cam.2017.01.002))
Kaname Matsue; Tomohiro Hiwaki; Nobito Yamamoto
Journal of Computational and Applied Mathematics, 384, 01 Mar. 2021, This note states the correction of arguments in the proof of Theorem 3.2 in the original paper.
Scientific journal

Errata to ‘‘On the construction of Lyapunov functions with computer assistance’’
Kaname Matsue a; Tomohiro Hiwaki b; Nobito Yamamoto
Journal of Computational and Applied Mathematics, 384, 113175, 01 Mar. 2021, Peer-reviwed
Scientific journal, English

Construction of local Lyapunov functions around non-hyperbolic equilibria by verified numerics for two dimensional cases
Gen Terasaka; Masao Nakamura; Koki Nitta; Nobito Yamamoto
JSIAM Letters, 12, 37-40, 21 Jul. 2020, Peer-reviwed
Scientific journal, English

On the construction of Lyapunov functions with computer assistance
Kaname Matsue; Tomohiro Hiwaki; Nobito Yamamoto
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, ELSEVIER SCIENCE BV, 319, C, 385-412, Aug. 2017, Peer-reviwed, This paper aims at applications of Lyapunov functions as tools for analyzing concrete dynamical systems with computer assistance, even for non-gradient-like systems. We want to know concrete form of Lyapunov functions around invariant sets and their domains of definition for applying Lyapunov functions to various analysis of both continuous and discrete dynamical systems. Although there are several abstract results for the existence of Lyapunov functions, they cannot induce a systematic and concrete procedure of Lyapunov functions with explicit forms. In this paper, we present a numerical verification method which can validate Lyapunov functions with explicit forms and their explicit domains of definition, which can be applied to arbitrary dynamical systems with (hyperbolic) equilibria or fixed points. The proposed procedure provides us with a powerful validation tool for analyzing asymptotic behavior of dynamical systems. (C) 2017 Elsevier B.V. All rights reserved.
Scientific journal, English

Some remarks on a priori estimates of highly regular solutions for the Poisson equation in polygonal domains
Takehiko Kinoshita; Yoshitaka Watanabe; Nobito Yamamoto; Mitsuhiro T. Nakao
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS, SPRINGER JAPAN KK, 33, 3, 629-636, Dec. 2016, Peer-reviwed, This paper presents two expressions for and semi-norms for H-3 and H-4 the solutions of the Poisson equation in two-dimensional polygonal domains. These equalities enable us to obtain higher order constructive a priori error estimates for finite element approximation of the Poisson equation with validated computing.
Scientific journal, English

Some remarks on numerical verification of closed orbits in dynamical systems
T. Hiwaki; N. Yamamoto
Nonlinear Theory and its Applications, IEICE, The Institute of Electronics, Information and Communication Engineers, E-6N, 3, 397-403, 01 Jul. 2015, Peer-reviwed, We consider numerical verification methods for existence of a closed orbit in a dynamical system which is described by ODEs. Besides Zgliczynski's method using Poincaré map, the authors proposed a method of verification for closed orbits and their time period. In this paper, we derive a relationship between our method and one of bordering methods which gives some explanation of superiority of this bordering.
Scientific journal, English

Some considerations of the invertibility verifications for linear elliptic operators
Mitsuhiro T. Nakao; Yoshitaka Watanabe; Takehiko Kinoshita; Takuma Kimura; Nobito Yamamoto
Japan Journal of Industrial and Applied Mathematics, Springer-Verlag Tokyo, 32, 1, 19-31, 2015, Peer-reviwed, This paper presents three computer-assisted procedures for verifying the invertibility of second-order linear elliptic operators and for computing a bound on the norm of its inverse. One of these procedures is an improvement of a theorem by Nakao et al. (Computing 75:1–14, 2005) that uses projection and constructive a priori error estimates and was proposed by two of the authors of this paper. Results verifying these procedures are presented for several numerical examples.
Scientific journal, English

Validated Computation of Global Solutions to ODEs
M.Harikae; N.Yamamoto
Nonlinear Theory and its Applications, IEICE, The Institute of Electronics, Information and Communication Engineers, 4, 1, 88-96, Jan. 2013, Peer-reviwed
Scientific journal, English

力学系における閉軌道の存在領域の精度保証法による同定
樋脇知広; 山本野人
日本応 用数理学会論文誌, The Japan Society for Industrial and Applied Mathematics, 22, 4, 269-276, Dec. 2012, Peer-reviwed, We propose a numerical verification method for existence and bounds of closed orbits of solutions to ODEs, where the methods for validated computation are adopted. The closed orbit and its period are specified simultaneously using Poicare Map and the Newton method, which allows us simpler implementation than existing methods. A numerical example on the Rossler equation is shown to verify the existence of a closed orbit in practice.
Scientific journal, Japanese

常微分方程式の解の精度保証法
山本野人
シミュレーション, 31, 3, 149-153, Sep. 2012
Scientific journal, Japanese

On the basic operations of interval multiple-precision arithmetic with center-radius form
N.Matsuda; N.Yamamoto
Nonlinear Theory and Its Applications, IEICE, The Institute of Electronics, Information and Communication Engineers, 2, 1, 54-67, Jan. 2011, Peer-reviwed, Multiple-precision arithmetic with interval variables has been developed for computation with guaranteed high accuracy. There are several computer program packages which deal with interval variables of the inf-sup form, e.g. MPFI, etc. On the other hand, it is impressed by INTLAB that interval multiple-precision arithmetic using the center-radius form has advantages on memory size and computing time. However, arithmetic of the center-radius form sometimes makes the radius of an interval larger than the inf-sup form does, which would be one of the reasons why there is no practical program package for interval multiple-precision arithmetic with the center-radius form.
The authors intend to establish a computer program package for multiple-precision arithmetic using intervals of the center-radius form which is still under construction. The present paper treats the problem of the center-radius form about the expansion of radii caused by the fundamental rules and the operation of square root. We propose several methods for calculation of multiplication, division, and square root, among which one can choose an appropriate method according to one's situation. Theoretical consideration and numerical examples are given for these methods.
Scientific journal, English

A THEOREM FOR NUMERICAL VERIFICATION ON LOCAL UNIQUENESS OF SOLUTIONS TO FIXED-POINT EQUATIONS
Nobito Yamamoto; Mitsuhiro T. Nakao; Yoshitaka Watanabe
NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, TAYLOR & FRANCIS INC, 32, 11, 1190-1204, 2011, Peer-reviwed, We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Frechet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods.
We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.
Scientific journal, English

Erratum: Computer assisted proofs of bifurcating solutions for nonlinear heat convection problems (J Sci Comput (10.1007/s10915-009-9303-3))
Mitsuhiro T. Nakao; Yoshitaka Watanabe; Nobito Yamamoto; Takaaki Nishida; Myoungnyoun Kim
Journal of Scientific Computing, 44, 1, 107, Jul. 2010, Peer-reviwed
Scientific journal, English

Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems
Mitsuhiro T. Nakao; Yoshitaka Watanabe; Nobito Yamamoto; Takaaki Nishida; Myoung-Nyoung Kim
JOURNAL OF SCIENTIFIC COMPUTING, SPRINGER/PLENUM PUBLISHERS, 43, 3, 388-401, Jun. 2010, Peer-reviwed, In previous works (Nakao et al., Reliab. Comput., 9(5):359-372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1-20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known as Rayleigh-B,nard problem. In the present paper, to make the arguments self-contained, we first summarize these results including the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example. Finally, an extension to the three dimensional case will be described.
Scientific journal, English

精度保証付き多倍長演算の方法と構成
山本野人; 松田望
計測と制御, 49, 5, 297-302, 2010, Peer-reviwed
Scientific journal, Japanese

An Application of Taylor Models to the Nakao Method on ODEs
Nobito Yamamoto; Takashi Komori
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS, KINOKUNIYA CO LTD, 26, 2-3, 365-392, Oct. 2009, Peer-reviwed, The authors give short survey on validated computaion of initial value problems for ODEs especially Taylor model methods. Then they propose an application of Taylor models to the Nakao method which has been developed for numerical verification methods an PDEs and apply it to initial value problems for ODEs with some numerical experiments.
Scientific journal, English

連続した入力パタンのあいだの順序関係を認識する神経回路モデル -- 情報の予測・抽象化に向けて --
田中一穂; 矢野慎一郎; 山本野人
日本応用数理学会論文誌, The Japan Society for Industrial and Applied Mathematics, 18, 1, 87-105, 2008, Peer-reviwed, We propose a neural network model which commits the order relation among sequential input patterns to memory, using associative memory for sequential patterns. The network is constructed by association of Leaky Integrate and Fire neuron models. The model may give the basic concept to some functions of the brain, especially the functions of forecast and abstraction. We implement our model to computer and investigate the numerical results in order to consider what the model gives to the concept of forecast and abstraction.
Scientific journal, Japanese

常微分方程式境界値問題の精度保証法の初期値問題への適用
小森喬; 山本野人
日本応用数理学会論文誌, The Japan Society for Industrial and Applied Mathematics, 18, 3, 303-319, 2008, Peer-reviwed, We propose a numerical verification method for initial value problems(IVP) of ODEs in order to enclose the solutions at the end time by considerably small bounds. The method is based on Nakao's theory which is established for numerical verification methods for PDEs. We construct boundary value problems(BVP) of ODEs whose solutions can be used to define the bounds of the solutions to the IVP. Then Nakao's theory is applied to the BVPs. Numerical examples are shown to compare our method with existing validated computation methods for ODEs.
Scientific journal, Japanese

On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains
Nobito Yamamoto; Kenta Genma
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, ELSEVIER SCIENCE BV, 199, 2, 286-296, Feb. 2007, Peer-reviwed, Numerical verification methods, so-called Nakao's methods, on existence or uniqueness of solutions to PDEs have been developed by Nakao and his group including the authors. They are based on the error estimation of approximate solutions which are mainly computed by FEM.
It is a standard way of the error estimation of FEM to estimate the projection errors by elementwise interpolation errors. There are some constants in the error estimation, which depend on the mesh size parameters h. The explicit values of the constants are necessary in order to use Nakao's method. However, there were not so many researches for the computation of the explicit values of the constants. Then we had to develop the computation by ourselves, especially with guaranteed accuracy. Note that the methods of the computation depend on the dimension, the degree of bases, and the shape of the domain, etc.
The present paper shows how we have developed the methods to calculate the constants and describes new results for nonconvex domains. (c) 2006 Elsevier B.V. All rights reserved.
Scientific journal, English

On error estimation of finite element approximations to the elliptic equations in nonconvex domains
N.Yamamoto; K.Genma
Journal of Computational and Applied Mathematics, 1, 199, 347-359, Jan. 2007, Peer-reviwed
Scientific journal, English

A numerical verification of bifurcation points for nonlinear heat convection problems
Mitsuhiro T. Nakao; Yoshitaka Watanabe; Nobito Yamamoto; Takaaki Nishida
The proceedings of 2nd International conference "From Scientific Computing to Computational Engineering" (2nd IC-SCCE), 1-8, Jul. 2006, Peer-reviwed
International conference proceedings, English

多倍長演算を利用したBessel関数の精度保証付き数値計算
山本 野人; 松田 望
日本応用数理学会論文誌, The Japan Society for Industrial and Applied Mathematics, 15, 3, 347-359, 2005, Peer-reviwed, Both multiple-precision arithmetic and validated computation have close relationship with the quality of computation. They improve and insure the quality, respectively. We propose a method to compute Bessel functions with guaranteed accuracy, which works on MATLAB. Using multiple-precision arithmetic, the method gives as precise results as one wants together with information of how precise the results are. When it is sufficient to get the results in double-precision, one can use a fast version.
Scientific journal, Japanese

楕円型方程式の解に対する局所一意性付き数値\\的検証法の効率化
渡部 善隆; 山本 野人; 中尾 充宏
日本応用数理学会論文誌, The Japan Society for Industrial and Applied Mathematics, 15, 4, 509-520, 2005, Peer-reviwed, One of the authors have proposed a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computer. However, for second-order elliptic boundary value problems, in the case the equation includes the first-order term, it turned out that there is a possibility that the verification algorithm come to an end unsuccessfully. The purpose of this paper is to propose an alternative method to overcome this difficulity. Numerical examples compared with the previous algorithm confirm the effectiveness of the new method.
Scientific journal, Japanese

A numerical verification of nontrivial solutions for the heat convection problem
Yoshitaka Watanabe; Nobito Yamamoto; Mitsuhiro T. Nakao; Takaaki Nishida
Journal of Mathematical Fluid Mechanics, 6, 1, 1-20, 2004, Peer-reviwed, A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Bénard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set. including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.
Scientific journal, English

Error estimation with guaranteed accuracy of finite element method in nonconvex polygonal domains
N Yamamoto; K Hayakawa
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, ELSEVIER SCIENCE BV, 159, 1, 173-183, Oct. 2003, A new method to estimate errors of the finite element method (FEM) for nonconvex polygonal domain is proposed. It gives mathematically rigorous upper bounds for the errors using calculations with guaranteed accuracy. Numerical examples are shown and their orders concerning mesh sizes are compared with theoretical orders. (C) 2003 Elsevier B.V. All rights reserved.
Scientific journal, English

Some computer assisted proofs for solutions of the heat convection problems
Mitsuhiro T. Nakao; Yoshitaka Watanabe; Nobito Yamamoto; Takaaki Nishida
Reliable Computing, 9, 5, 359-372, Oct. 2003, This is a continuation of our previous results (Y. Watanabe, N. Yamamoto, T. Nakao, and T. Nishida, "A Numerical Verification of Nontrivial Solutions for the Heat Convection Problem," to appear in the Journal of Mathematical Fluid Mechanics). In that work, the authors considered two-dimensional Rayleigh-Bénard convection and proposed an approach to prove existence of steady-state solutions based on an infinite dimensional fixed-point theorem using a Newton-like operator with spectral approximation and constructive error estimates. We numerically verified several exact non-trivial solutions which correspond to solutions bifurcating from the trivial solution. This paper shows more detailed results of verification for given Prandtl and Rayleigh numbers. In particular, we found a new and interesting solution branch which was not obtained in the previous study, and it should enable us to present important information to clarify the global bifurcation structure. All numerical examples discussed are take into account of the effects of rounding errors in the floating point computations.
International conference proceedings, English

熱対流問題の解に対する計算機援用証明,
渡部 善隆; 中尾 充宏; 山本 野人; 西田 孝明
数学解析の計算機上での理論的展開とその遂行可能性, 京都大学数理解析研究所講究録, Kyoto University, 1286, 17-26, Sep. 2002
Research institution, Japanese

Rayleigh-Benard対流の定常解に対する精度保証付き数値計算 II
渡部 善隆; 山本 野人; 中尾 充宏; 西田 孝明
微分方程式の離散化手法と数値計算アルゴリズム, 京都大学数理解析研究所講究録, Kyoto University, 1265, 71-80, May 2002
Research institution, Japanese

Verified numerical computations for an inverse elliptic eigenvalue problem with finite data
MT Nakao; Y Watanabe; N Yamamoto
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS, KINOKUNIYA CO LTD, 18, 2, 587-602, Jun. 2001, Peer-reviwed, We consider a numerical enclosure method for solutions of an inverse Dirichlet eigenvalue problem. When the finite number of prescribed eigenvalues are given, we reconstruct a potential function, with guaranteed error bounds, for which the corresponding elliptic operator exactly has those eigenvalues including the ordering property. All computations are executed with numerical verifications based upon the finite and infinite fixed point theorems using interval arithmetic. Therefore, the results obtained are mathematically correct. We present numerical examples which confirm us the enclosure algorithm works on real problems.
Scientific journal, English

A simple method for error bounds of eigenvalues of symmetric matrices
N Yamamoto
LINEAR ALGEBRA AND ITS APPLICATIONS, ELSEVIER SCIENCE INC, 324, 1-3, 227-234, Feb. 2001, Peer-reviwed, We propose a simple method for validated computation of eigenvalues of symmetric matrices. The method is based on LDLT decomposition and its error estimation. The indices of eigenvalues with respect to magnitude can also be obtained by this method. (C) 2001 Elsevier Science Inc. All rights reserved.
Scientific journal, English

Numerical verification Numerical verification method for solutions of the perturbed Gelfand equation
Teruya Minamoto; Nobito Yamamoto; Mitsuhiro T. Nakao
Methods and Applications of Analysis, 7, 1, 251--262, 2001, Peer-reviwed
Scientific journal, English

A guaranteed bound of the optimal constant in the error estimates for linear triangular element
Mitsuhiro T. Nakao; Nobito Yamamoto
Computing Supplement, Springer Wien NewYork, 15, 163-173, 2001, Peer-reviwed
Scientific journal, English

A guaranteed bound of the optimal constant in the error estimates for linear triangular elements part II: Details
MT Nakao; N Yamamoto
PERSPECTIVES ON ENCLOSURE METHODS, SPRINGER-VERLAG WIEN, 265-276, 2001, Peer-reviwed, In the previous paper([6]), we formulated a numerical method to get a guaranteed bound of the optimal constant in the error estimates with linear triangular elements in R-2. We describe, in this paper, detailed computational procedures for obtaining a rigorous upper bound of that constant with sufficient sharpness. The numerical verification method for solutions of nonlinear elliptic problems is successfully applied to the present purpose. A constructive error estimate for the triangular element with Neumann boundary condition plays an important role to implement the actual verified computations. Particularly, some special kind of techniques axe utilized to improve the computational cost for the algorithm. As a result, we obtained a sufficiently sharp upper bound from the practical viewpoint.
International conference proceedings, English

有限要素法の近似能力について
山本野人
応用数学合同研究集会、龍谷大学（2000年12月21日）, Dec. 2000
Japanese

``Error estimates of finite element solutions by spectrum method with verified computation''
Nobito Yamamoto
Matsuyama Workshop on Numerical Analysis, October 5, 2000, Oct. 2000
English

''Verified Numerical Computation for an Inverse Elliptic Eigenvalue Problem''
Yoshitaka Watanabe; Nobito Yamamoto; Mitsuhiro T. Nakao
9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2000) jointly International Conference on Interval Methods in Science and Engineering, September 18-22, 2000, Karlsruhe, Germany, Sep. 2000
English

''Verified Computation for PDE with Spectral Methods''
Nobito Yamamoto
9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2000) jointly International Conference on Interval Methods in Science and Engineering, September 18-22, 2000, Karlsruhe, Germany, Sep. 2000
English

楕円型逆固有値問題の精度保証付き数値計算
渡部 善隆; 山本 野人; 中尾 充宏
第4回環瀬戸内応用数理研究部会シンポジウム講演プログラム・要旨集, 武雄センチュリーホテル, 佐賀県武雄市, 64-65, Jun. 2000
Japanese

Rayleigh-B\'enard 対流の定常分岐解に対する精度保証付き数値計算
渡部 善隆; 山本 野人; 中尾 充宏
九州大学数値解析学セミナー, 九州大学大学院数理学研究院, May 2000
Japanese

Numerical verification method for solutions of the perturbed Gelfand eguation
Minamoto, T; Yamamoto, N; Nakao, M.T
Methods and Applications of Analysis, 7, 1, 251-262, 2000
English

Validated computation for a linear elliptic problem with a parameter
Nobito Yamamoto; Mitsuhiro T. Nakao; Yoshitaka Watanabe
Advances in Numerical Mathematics; Proceedings of the Fourth Japan-China Joint Seminar on Numerical Mathematics, held in Chiba, Japan, August 24-28, 1998 (H. Kawarada, M. Nakamura, Z. Shi, eds.), GAKUTO International Series Mathematical Sciences and Appli, 155-162, 1999, Peer-reviwed
International conference proceedings, English

Verification Methods of Generalized Eigenvalue Problems and its Applications
Watanabe Yoshitaka; Yamamoto Nobito; Nakao Mitsuhiro T
Transactions of the Japan Society for Industrial and Applied Mathematics, 一般社団法人 日本応用数理学会, 9, 3, 137-150, 1999, Peer-reviwed, We consider numerical verification methods to obtain the maximum absolute value of generalized eigenvalue problems. We present four kinds of methods and compare the performance in various situations as well as give evaluation of the advantage and disadvantage of these methods. All numerical results have been calculated by the interval arithmetic software for considering the rounding error occuring in the calculation. Finally, we will present an application to an eigenvalue problem appeared in some a priori error estimates for the finite element solution of the Stokes equations.
Scientific journal, Japanese

A Numerical Verification Method of Solutions for the Navier-Stokes Equations.
Yoshitaka Watanabe; Nobito Yamamoto; Mitsuhiro T. Nakao
Reliable Computing, 5, 3, 347-357, 1999, Peer-reviwed
Scientific journal, English

Numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem
N Yamamoto
SIAM JOURNAL ON NUMERICAL ANALYSIS, SIAM PUBLICATIONS, 35, 5, 2004-2013, Oct. 1998, Peer-reviwed, In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a solution, we transform it by an approximate linearization of the operator appearing in the equation. Then we calculate the radii of the transformed set in order to check sufficient conditions for Banach's fixed-point theorem. This method is applied to elliptic problems and numerical examples are given.
Scientific journal, English

On the Best Constant in the Error Bound for the H1 0-Projection into Piecewise Polynomial Spaces
Mitsuhiro T. Nakao; Nobito Yamamoto; Seiji Kimura
Journal of Approximation Theory, Academic Press Inc., 93, 3, 491-500, 1998, Peer-reviwed, Explicit a priori error bounds for the approximation by the H1 0-projection into piecewise polynomial spaces are given. In particular, for the quadratic approximation, the optimal constant is derived, and a nearly optimal value for the cubic is obtained. These constants play an important role in the numerical verification method of finite element solutions for nonlinear elliptic equations. © 1998 Academic Press.
Scientific journal, English

Numerical Verification of Solutions for Nonlinear Elliptic Problems Using anL∞Residual Method
Mitsuhiro T. Nakao; Nobito Yamamoto
Journal of Mathematical Analysis and Applications, 217, 1, 246-262, 01 Jan. 1998, Peer-reviwed, We consider a numerical enclosure method with guaranteedL∞error bounds for the solution of nonlinear elliptic problems of second order. By using an a posteriori error estimate for the approximate solution of the problem with a higher orderC0-finite element, it is shown that we can obtain the guaranteedL∞error bounds with high accuracy. A particular emphasis is that our method needs no assumption of the existence of the solution of the original nonlinear equation, but it follows as the result of computation itself. A numerical example that confirms the effectiveness of the method is presented. © 1998 Academic Press.
Scientific journal, English

A Posteriori and Constructive A Priori Error Bounds for Finite Element Solutions of the Stokes Equations.
Mitsuhiro T. Nakao; Nobito Yamamoto; Yoshitaka Watanabe
Journal of Computational and Applied Mathematics, 91, 137-158, 1998, Peer-reviwed
Scientific journal, English

Constructive L2 error estimates for finite element solutions of the stokes equations
Mitsuhiro T. Nakao; Nobito Yamamoto; Yoshitaka Watanabe
Reliable Computing, 4, 2, 115-124, 1998, Peer-reviwed
Scientific journal, English

Verified Computations of Solutions for Nondifferentiable Elliptic Equations Related to MHD Equilibria
Yoshitaka Watanabe; Nobito Yamamoto; Mitsuhiro T. Nakao
Nonlinear Analysis, Theory, Methods & Applications, 28, 3, 577-587, 1997, Peer-reviwed
Scientific journal, English

Guaranteed error bounds for finite element solutions of the Stokes problem
MT Nakao; N Yamamoto; Y Watanabe
SCIENTIFIC COMPUTING AND VALIDATED NUMERICS, AKADEMIE VERLAG GMBH, 90, 258-264, 1996, Peer-reviwed
International conference proceedings, English

A Posteriori Error Estimate for Finite Element Solutions of Stokes Equation
Nakao Mitsuhiro T; Yamamoto Nobito; Watanabe Yoshitaka
RIMS Kokyuroku, Kyoto University, 928, 20-31, Nov. 1995
Japanese

NUMERICAL VERIFICATIONS FOR SOLUTIONS TO ELLIPTIC-EQUATIONS USING RESIDUAL ITERATIONS WITH A HIGHER-ORDER FINITE-ELEMENT
N YAMAMOTO; MT NAKAO
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, ELSEVIER SCIENCE BV, 60, 1-2, 271-279, Jun. 1995, Peer-reviwed, The verifications of solutions to weakly nonlinear elliptic equations by the method described e.g. by Nakao (1988, 1989), etc. are sometimes hardly accomplished when the right-hand sides of the equations are very large. To overcome such difficulties, a residual iteration technique with approximate solution was introduced by Nakao (1993). In the present paper, we propose an a posteriori method for the residual iteration, and show that a remarkable improvement in efficiency and in accuracy of the verification can be obtained when we use a higher order finite element.
Scientific journal, English

NUMERICAL VERIFICATIONS OF SOLUTIONS FOR ELLIPTIC-EQUATIONS IN NONCONVEX POLYGONAL DOMAINS
N YAMAMOTO; MT NAKAO
NUMERISCHE MATHEMATIK, SPRINGER VERLAG, 65, 4, 503-521, Aug. 1993, In this paper, methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied. In order to verify solutions using computer, it is necessary to determine some constants which appear in a priori error estimations. We propose some methods for determination of these constants. In numerical examples, calculating these constants for an L-shaped domain, we verify the solution of a nonlinear elliptic equation.
Scientific journal, English

NUMERICAL VERIFICATIONS OF SOLUTIONS FOR ELLIPTIC-EQUATIONS WITH STRONG NONLINEARITY
MT NAKAO; N YAMAMOTO
NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, MARCEL DEKKER INC, 12, 5-6, 535-543, 1991, Numerical methods for automatic proof of the existence and the local uniqueness of weak solutions of elliptic boundary value problems with strongly nonlinear terms are proposed. They are based on the infinite dimensional fixed point theorems and the explicit error estimates for finite element approximations. We present detailed verification procedures and numerical examples for the typical model problem: - DELTA-u = e(U).
Scientific journal, English

中尾充宏氏の業績 ー 偏微分方程式の精度保証付き数値計算 ー
山本野人
The Mathematical Society of Japan, Apr. 2013, 数学, 65, 2, 200-207, Japanese, Introduction other, 0039-470X, 10031177312, AN00125036

Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems (vol 43, pg 388, 2010)
Mitsuhiro T. Nakao; Yoshitaka Watanabe; Nobito Yamamoto; Takaaki Nishida; Myoungnyoun Kim
SPRINGER/PLENUM PUBLISHERS, Jul. 2010, JOURNAL OF SCIENTIFIC COMPUTING, 44, 1, 107-107, English, Others, 0885-7474, WOS:000278180600005

On symbolic computation in numerical verification for ODEs (Numerical Analysis : Theory, Methods and Applications)
Yamamoto Nobito; Matsuda Nozomu
Kyoto University, Apr. 2009, RIMS Kokyuroku, 1638, 159-168, Japanese, 1880-2818, 110007055257, AN00061013

Effect of preconditioning in edge-based finite-element method
Hajime Igarashi; Nobito Yamamoto
This paper discusses mathematical properties of preconditioned finite-element matrices based on vector potential formulation (A method) and vector and scalar potential formulation (A-V method) for eddy-current problems. Numerical results show that A-V method with preconditioning is stable at all frequencies in contrast to A method. In this paper, this property is mathematically discussed by considering the diagonal scaling which is one of the simple preconditioning methods. In addition, regularization of A method is discussed., IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC, Jun. 2008, IEEE TRANSACTIONS ON MAGNETICS, 44, 6, 942-945, English, 0018-9464, 120001377328, WOS:000258183400069

Validated Computation of Bessel functions
YAMAMOTO Nobito
2005, Proc. NOLTA2005, 20001292011

Enclosing Potential Functions of an Inverse Elliptic Eigenvalue Problem with Finite Data (Numerical Solution of Partial Differential Equations and Related Topics II)
Watanabe Yoshitaka; Yamamoto Nobito; Nakao Mitsuhiro T
京都大学, Apr. 2001, RIMS Kokyuroku, 1198, 239-244, Japanese, 1880-2818, 110000165254

Numerical verifications for eigenvalues of second-order elliptic operators
MT Nakao; N Yamamoto; K Nagatou
In this paper, we consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is based on Nakao's method [9] using the Newton-like operator and the error estimates for the C-0 finite element solution. We construct, in computer, a set containing solutions which satisfies the hypothesis of Schauder's fixed point theorem for compact map on a certain Sobolev space. Moreover, we propose a method to verify the eigenvalue which has the smallest absolute value. A numerical example is presented., KINOKUNIYA CO LTD, Oct. 1999, JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS, 16, 3, 307-320, English, 0916-7005, WOS:000085274500001

An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness
K Nagatou; N Yamamoto; MT Nakao
We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kantorovich type. Numerical examples are presented., TAYLOR & FRANCIS INC, 1999, NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, 20, 5-6, 543-565, English, 0163-0563, 1532-2467, WOS:000081842300009

一般化固有値問題の精度保証付きプログラム
渡部 善隆; 山本 野人; 中尾 充宏
九州大学大型計算機センター, Jun. 1998, 九州大学大型計算機センター広報, 31, 2, 51-60, Japanese, 0389-7885, 120005475804

Numerical Verifications of Solutions for the Navier-Stokes Equations
Watanabe Yoshitaka; Yamamoto Nobito; Nakao Mitsuhiro T
Kyoto University, Apr. 1998, RIMS Kokyuroku, 1040, 100-105, Japanese, 1880-2818, 110002338064

実対称行列の正定値性判定プログラム
渡部 善隆; 山本 野人; 中尾 充宏
九州大学大型計算機センター, Mar. 1998, 九州大学大型計算機センター広報, 31, 1, 1-10, Japanese, 0389-7885, 120005475799

A priori 誤差評価定数の計算機による数値評価について(科学技術における数値計算の理論と応用II)
中尾 充宏; 山本 野人
京都大学, Apr. 1997, 数理解析研究所講究録, 990, 224-234, Japanese, 1880-2818, 110004093657, AN00061013

Numerical Verfications for eigenvalues of elliptic operators and its Application
Yamamoto Nobito; Nagatou Kaori; Nakao Mitsuhiro T.
Kyushu University, 1997, Research reports on computer science, Computer Center, Kyushu University, 14, 53-60, Japanese, 0910-352X, 110000516449, AN10121075

State of the art for the numerical computations with guaranteed accuracy
NAKAO Mitsuhiro T.; YAMAMOTO Nobito
日本計算工学会, 29 May 1996, Proceedings of the conference on computational engineering and science, 1, 1, 135-138, Japanese, 1342-145X, 10016693779, AN10581224

A Priori Error Estimate for Finite Element Solutions of the Stokes Equations
Nakao Mitsuhiro; Yamamoto Nobito; Watanabe Yoshitaka
Kyoto University, Apr. 1996, RIMS Kokyuroku, 944, 41-49, Japanese, Meeting report, 1880-2818, 110004679074

楕円型方程式の解の数値的検証法へのKrawczyk法の適用(数値計算における品質保証とその応用 : 感度解析から証明まで) 
山本 野人; 中尾 充宏
京都大学, Nov. 1995, 数理解析研究所講究録, 928, 8-13, Japanese, 1880-2818, 110004707103, AN00061013

非線形楕円型問題に対する有限要素解の最大値ノルムによる精度保証(数値計算アルゴリズムの現状と展望II)
中尾 充宏; 山本 野人
京都大学, Jun. 1995, 数理解析研究所講究録, 915, 130-135, Japanese, 1880-2818, 110004720883, AN00061013

非凸領域における楕円型方程式の解の数値的検証法(数値計算アルゴリズムの現状と展望)
山本 野人; 中尾 充宏
京都大学, Jul. 1994, 数理解析研究所講究録, 880, 127-133, Japanese, 1880-2818, 110004757410, AN00061013

Numerical verifications for solutions of elliptic equations using residual iterations with higher order elements
Yamamoto Nobito; Nakao Mitsuhiro T.
Kyoto University, Apr. 1993, RIMS Kokyuroku, 831, 149-157, Japanese, 1880-2818, 110004838010, AN00061013

Numerical Verifications of Solutions for Nondifferentiable Elliptic Equations
Watanabe Yoshitaka; Yamamoto Nobito; Nakao Mitsuhiro T
Kyoto University, Apr. 1993, RIMS Kokyuroku, 831, 141-148, Japanese, Meeting report, 1880-2818, 110004838009

微分不能項を持つ楕円型境界値問題の解に対する数値的検証法
渡部 善隆; 山本 野人; 中尾 充宏
1992, 電子情報通信学会技術研究報告, NLP-92-44, Japanese, Meeting report

第2版 現代数理科学事典
山本野人
Dictionary or encycropedia, Japanese, 丸善出版, 2009

精度保証付き数値計算---コンピュータによる無限への挑戦---
中尾 充宏; 山本 野人
Japanese, 日本評論社, Jun. 1998

精度保証法による保存系 ODE の扱いについて
山本野人
Oral presentation, Japanese, 第50回数値解析シンポジウム
14 Jun. 2024
12 Jun. 2024- 14 Jun. 2024

保存量を持つ常微分方程式 系の精度保証付き計算につい て
山本野人; 新田光輝
Oral presentation, Japanese, 日本応用数理学会2022年度年会
08 Sep. 2022

離散力学系非双曲型不動点近傍でのLyapunov関数の精度保証による構成について
皆本 啓吾; 新田 光輝; 山本 野人
Oral presentation, Japanese, 日本応用数理学会第17回 研究部会連合発表会, 日本応用数理学会, zoom, Domestic conference
05 Mar. 2021

非双曲型平衡点近傍での Lyapunov 関数の精度保証法による構成について
新田 光輝; 山本 野人
Oral presentation, Japanese, 2020 年度応用数学合同研究集会, 日本数学会, zoom, Domestic conference
20 Dec. 2020

離散力学系における不動点近傍での安定多様体の捕捉について
山本野人; 皆本 啓吾; 新田 光輝
Oral presentation, Japanese, 日本応用数理学会 2020年度 年会, 日本応用数理学会, zoom, Domestic conference
08 Sep. 2020

Fixed-point theorems as tools of verified numerics on ODEs
Nobito Yamamoto
Keynote oral presentation, English, ICIAM 2019, Invited, スペイン, International conference
19 Jul. 2019

Numerical verification methods for limit cycles in dynamical systems
Nobito Yamamoto; Tomohiro Hiwaki
Invited oral presentation, English, International Workshop on Numerical Verification and its Applications 2014 (INVA 2014), Invited, 早稲田大学, International conference
16 Mar. 2014

Numerical verification methods for limit cycles in dynamical systems
Nobito Yamamoto; Tomohiro Hiwaki
Invited oral presentation, English, The International Workshop on Numerical Verification and its Applications 2014, Invited, Shin'ichi Oishi, Waseda University, Tokyo, Japan, Poincar ́e map is a general tool to treat limit cycles in dynamical systems. In or- der to prove existence of a limit cycle by validated computation, Zgliczyn ́ski ver- ified existence of a fixed point of a Poincar ́e map using a fixed point theorem[5]. However it was not an easy work to specify ’first return time’ Ts, a time period between an initial point x0 on the Poincar ́e section Γ and x1 := φ(Ts, x0) ∈ Γ, where φ(t, x0) denotes a point on the trajectory from x0 at time t. Of course one have to verify that there is no point φ(t, x0) ∈ Γ for any t ∈ (0, Ts). Zgliczyn ́ski proposed a way to handle the situation and showed numerical examples to ap- peal effectiveness of his method.
Hereafter we propose another way in which one has not to construct a Poincar ́e map any longer., International conference
15 Mar. 2014

LyapunovTracing による常微分方程式の精度保証法について
樋脇 知広; 渡辺 真伊智; 山本 野人; 松江 要
Oral presentation, Japanese, 日本応用数理学会2013年度年会, Domestic conference
10 Sep. 2013

リミットサイクルの吸引域に含まれる領域の精度保証法による同定
樋脇知広; 山本野人
Oral presentation, Japanese, 2012 年度日本数学会秋季総合分科会, Domestic conference
21 Sep. 2012

Saddle-saddle connection の精度保証付き数値検証
松江要; 山本野人
Oral presentation, Japanese, 日本応用数理学会 2012 年度年会, Domestic conference
31 Aug. 2012

Validated Computation of Global Solutions to ODEs,
N. Yamamoto; M. Harikae
Invited oral presentation, English, the Japan-German Workshop on Computer-Assisted proofs and Verification Methods,, Invited, Karisruhe,Germany, International conference
18 Sep. 2011

Interval Multiple-Precision Arithmetic with center-radius form
N. Yamamoto; N. Matsuda
Oral presentation, English, SCAN2010, 14th GAMM-IMACS, International Symposium on Scientific Comput- ing, Computer Arithmetic and Validated Numerics,, ENS de Lyon, Francs, International conference
27 Sep. 2010

常微分方程式の精度保証付き計算の技法と利用法
山本野人
Invited oral presentation, Japanese, ちばＮ体2010, 天体力学Ｎ体力学研究会
Mar. 2010

Numerical verification on existence of periodic solutions to ODEs''
K.Shioda; N.Yamamoto
Oral presentation, English, International Workshop on Numerical verification and its App
Mar. 2008

常微分方程式の数値解に関する精度保証の技法について
小森喬; 山本野人
Oral presentation, Japanese, 日本応用数理学会研究部会連合講演会「科学技術計算と数値解析」研究部会,日本応用数理学会研究部会連合講演会「科学技術計算と数値解析」研究部会
Mar. 2008

A numerical verification method for ODEs based on the Nakao Theory
N.Yamamoto; T.Komori
Oral presentation, English, DMHF2007,DMHF2007
Oct. 2007

高次補間に基づく常微分方程式の精度保証法について
小森喬; 山本野人
Oral presentation, Japanese, 日本数学会2007年度秋期総合分科会,日本数学会2007年度秋期総合分科会
Sep. 2007

A theorem for numerical verification of local uniquness
Nobito Yamamoto
Oral presentation, English, ICIAM 07, Zurich,Switzerland
Jul. 2007

高次補間に基づく常微分方程式の精度保証法について
山本野人; 小森喬
Oral presentation, Japanese, 第36回数値解析シンポジウム,第36回数値解析シンポジウム
Jun. 2007

常微分方程式初期値問題に対する精度保証付き計算の新手法
山本 野人; 小森 喬
Oral presentation, Japanese, 2007年日本応用数理学会連合発表会
Mar. 2007

時系列パタンに対するラベリングを行なう神経回路モデルについて
田中 一穂; 矢野 慎一郎; 山本 野人
Oral presentation, Japanese, 2007年日本応用数理学会連合発表会
Mar. 2007

常微分方程式の精度保証法に関する新しい計算技法について
山本野人; 小森喬; 足立英輔
Oral presentation, Japanese, 応用数学合同研究集会
Dec. 2006

時系列パタンに対するラベリングを行なう神経回路モデルについて
田中一穂; 矢野慎一郎; 山本野人
Oral presentation, Japanese, 応用数学合同研究集会
Dec. 2006

A Numerical Verification Method for ODEs based on Nakao's Theory
N.Yamamoto
Oral presentation, English, NOLTA2006, Bologna,Italy
Sep. 2006

A numerical verification method for nonlinear two-point boundary value problems
N.Yamamoto
Oral presentation, English, ICNAAM2006,Crete,Greece
Sep. 2006

A numerical verification method for ODEs with narrow error bounds
N.Yamamoto
Oral presentation, English, SCAN2006, Duisburg,Germany
Sep. 2006

常微分方程式初期値問題の精度保証付き計算について
小森喬; 山本野人
Oral presentation, Japanese, 日本応用数理学会研究部会連合講演会「科学技術計算と数値解析」研究部会
Mar. 2006

Validated computation of Bessel functions using multiple-precisi on arithmetic
Yamamoto,N
Public symposium, English, Workshop on Numerical Analysis of Flow Problems and Validated Computations, 長崎
Nov. 2005

熱対流問題の分岐点に対する計算機援用証明
渡部善隆; 中尾充宏; 山本野人; 西田孝明
Oral presentation, Japanese, 日本数学会秋季総合分科会,日本数学会秋季総合分科会
Sep. 2005

A theorem for numerical verification of uniqueness
Yamamoto,N
Public symposium, English, Dagstuhl Seminar, Algebraic and Numerical Algorithms and Computer-assisted Proofs(Seminar No 05391)
Sep. 2005

Validated Computation of Bessel Functions
N. Yamamoto; N. Matsuda
Oral presentation, English, NOLTA2005
Sep. 2005

常微分方程式の精度保証に関する話題
山本野人
Oral presentation, Japanese, 第54回理論応用力学講演会 NCTAM2005
Jan. 2005

非凸領域におけるPoisson方程式に対するFEM近似の誤差評価に ついて
山本野人; 弦間健太
Oral presentation, Japanese, 応用数理合同研究集会
Dec. 2004

Bessel関数の精度保証ライブラリ
山本野人; 松田望
Oral presentation, Japanese, 応用数理合同研究集会
Dec. 2004

Numerical Error Estomation with Guaranteed Accuracy for Finite Element Method
Yamamoto,N
Keynote oral presentation, English, International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN2004), Fukuoka, Japan, International conference
Oct. 2004

有限要素法の誤差評価定数に関する考察 -- 精度保証法のための 定数評価--
山本 野人; 弦間健太
Oral presentation, Japanese, 日本応用数理学会年会
Sep. 2004

常微分方程式（初期値問題）の精度保証付き計算について
山本 野人; 小森喬
Oral presentation, Japanese, 日本応用数理学会年会
Sep. 2004

自己随伴でない楕円型微分作用素の固有値の非存在範囲に対する精度保証付き計算、
山本 野人; 小森 喬; 内田 貴紀
Oral presentation, Japanese, 日本数学会年会
Mar. 2003

精度保証付き数値計算の基盤と応用 -- 有限要素法の射影誤差限界に対する定量的見積もり ---
山本 野人
Keynote oral presentation, Japanese, 日本数学会年会, Domestic conference
Mar. 2003

円環領域における楕円型偏微分方程式の精度保証法
山本 野人; 佐久間 祐幸
Oral presentation, Japanese, 応用数学合同研究集会
Dec. 2002

Bessel関数の境界条件への適合に関する精度保証付き計算
山本 野人
Oral presentation, Japanese, 研究集会「微分方程式の数値解法と線形計算」、 京都大学数理解析研究所
Nov. 2002

積分方程式の解の一意性の数値的検証法について
小森喬; 山本野人
Oral presentation, Japanese, 日本応用数理学会2002年度年会
Sep. 2002

Numerical Enclosures for Nontrivial Solutions of the Heat Convection Problems
Watanabe,Y; Yamamoto,N; Nakao,M.T; Nishida,T
Oral presentation, English, SCAN2002, Universite Pierre et Marie Curie
Sep. 2002

On verified computation of PDE using Spectral methods with Bessel functions
Yamamoto,N
Oral presentation, English, SCAN2002, Universite Pierre et Marie Curie
Sep. 2002

Calculation with guaranteed accuracy of the constants in the error estimation of FEM
Yamammoto, N
Keynote oral presentation, English, The Sixth Japan-China Joint Seminar on Numerical Mathematics, University of Tsukuba, International conference
Aug. 2002

Rigorous calculation for error estimation of FEM-solutions to Poisson equation
Yamamoto,N
Oral presentation, English, GENERAL INFORMATION FOR ICFS 2002,Waseda Univ.
Mar. 2002

日本数学会

日本応用数理学会

現象解析のツールとしての精度保証付き計算法の開発
山本 野人
Principal investigator
01 Apr. 2021 - 31 Mar. 2023

精度保証による力学系解析ツールの開発
YAMAMOTO, Nobito
Principal investigator, 精度保証法の技術に基づき、力学系の解析のためのツールを開発して応用に資する。
01 Apr. 2018 - 31 Mar. 2021

精度保証法によるLyapunov関数構成法の研究
山本 野人
Principal investigator, 力学系の不動点もしくは周期軌道近傍での解の挙動解析のために、Lyapunov関数を精度保証の技法を用いて構成する方法について研究する。
01 Apr. 2015 - 01 Mar. 2018

Development of computer assisted analysis for complicated nonlinear phenomena
NAKAO Mitsuhiro; EI Shin-ichiro; TABATA Masahisa; NAGATOU Kaori; MURASHIGE Sunao; YAMAMOTO Nobito; WATANABE Yoshitaka; OISHI Shinichi
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (S), We were working on the development and applications of the numerical verification methods for solutions of nonlinear partial differential equations, in particular, we succeeded in finding a new and very efficient verification principle for nonlinear evolutional problems. Also we extended and improved the existing verification methods for solutions of elliptic problems as well as we proved the effectiveness of the computer assisted proofs by applying our methods to resolve the actual nonlinear problems for which any theoretical approaches seem to be not useful to apply., 20224001
2008 - 2011

Establishment of Verified Numerical Computation
OISHI Shin'ichi; NAKAO Mitsuhiro; NISHIDA Takaaki; SHIBATA Yoshihiro; YAMAMOTO Nobito; WATANABE Yoshitaka
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Waseda University, Grant-in-Aid for Specially Promoted Research, Establishment of Verified Numerical Computation We have studied verified numerical computations for partial differential equations and systems of linear equations using digital computers. Calculating sum of a vector and dot product of two vectors with guaranteed high accuracy is ubiquitous in scientific computing. We have developed such algorithms for accurate sum and dot product, which are known to be the fastest so far. As applications, we have applied the fast and accurate algorithms to sparse matrix computations, computational geometry and so forth. Moreover, we have succeeded in proving the existence and uniqueness of a solution of a partial differential equation, and in calculating an error bound of its approximate solution., 17002012
2005 - 2009

Finite Element Methods for Huge Domain and Domain Decomposition Methods with Related Topics
USHIJIMA Teruo; TAKEDA Tatsuoki; KAKO Takashi; YAMAMOTO Nobito; TABATA Masahisa; FUJIMA Shoiti
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, The University of Electro-Communications, Grant-in-Aid for Scientific Research (B), 1.Main Results Obtained in the Joint Works around the Head Investigator : Demonstration of the possibility for determination of the mapping of wing through finite element computation. Error estimate for the solutions of FSM(=Fundamental Solution Method)approximate problems to reduced wave problems in a domain exterior to a disc. Confirmation of the effectiveness of an FEM-FSM combined method applied to 2D exterior reduced wave problem, and its application to linear water wave problems in an exterior water region with constant water depth, where the abbreviation FEM stands for Finite Element Method. 2.Remarkable Progress Obtained in the Works by Investigators : (1)Establishment of a method solving linear systems determining discrete vector potentials(by J.Watanabe). (2)Application of multi layer neural networks to various types of inverse problems(in computer tomography, in data assimilation, in parameter evaluation, in time series prediction)(by T.Takeda). (3)Application of finite element analysis for stationary wave transmission phenomena in unbounded domains to the problem of voice generation with successfully captured formants(by T.Kako). (4)Development of a new method for determination of upper bounds for error estimation constants appeared in finite element computation of Poisson equations in non-convex polygonal domains(by N.Yamamoto). (5)Theoretical study on the effect of stationary non homogeneous spatial structure to qualitative properties of solutions in the case of non-linear reaction-diffusion equations through numerical simulation and asymptotic analysis(by K.Nakamura). (6)Mathematical and numerically experimental analysis of characteristic futures of approximation methods for various types of partial differential equations obtained through Runge-Kutta type formulas(by T.Koto). (7)Overcome of the difficulties in finite element numerical solution methods in flow problems through upwinding technique and approximation way of characteristic curves(by M.Tabata). (8)Finite element analysis of non-stationary field of eddy current based on moving coordinate system(by H.Kanayama). (9)Flux free finite element method applied to two phase fluid problems(by K Ohmori). (10)Parallel computation through mortar domain decomposition method(by S.Fujima). (11)Purely theoretical analysis and numerical analysis of numerical instability problems arising with association of steep change of phenomena, in such as shock waves(by H.aiso). 3.Invitation of Foreign Cooperative Researcher : (1)Professor Han Hou-de of Applied Mathematics Department, Tsinghua University, Beijing, China from July 26 to August 16,2002. (2)Professor Yu De-hao of Institute of Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China from September 12 to October 3,2004. 4.Research Meetings : (1)Yokohama Research Meeting was held an KKR Hotel Port Pear Yokohama from January 8 to 10,2003. (2)Chofu Research Meeting was held at the University of Electro-Communications from February 19 to 20,2004. (3)Chofu Symposium 2005 was held at the University of Electro-Communication from February 17 to 19,2005., 14340031
2002 - 2004

Construction of a Practical Computation Code for Heat Convection Problems with Slow Flow
TABATA Masahisa; FUKUMOTO Yasuhide; HONDA Satoru; NAKAO Mitsuhiro; SUZUKI Atsushi; YAMAMOTO Nobito
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, KYUSHU UNIVERSITY, Grant-in-Aid for Scientific Research (B), (1) We have built a finite element scheme for solving numerically heat convection problems with slow flow like Earth's mantle convection in geophysics and melting glass convection in glass product furnaces. We have shown unconditional stability of the scheme and the convergence rate of the finite element solutions. These problems are modeled by Rayleigh-Benard equations with infinite Prandtl number, whose viscosity is strongly dependent on temperature. The obtained scheme is practically useful for three-dimensional problems. In order to reduce computation load we have employed the tetrahedral linear element for every unknown functions, velocity, pressure and temperature, and used stabilized finite element method. (2) We have constructed a computation code for the scheme mentioned above and implemented it on parallel computers. The Earth's mantle convection problem is solved in a spherically symmetric domain. By virtue of this property we have divided the domain into the union of congruent subdomains, which have allowed us to keep only stiffness matrices in a representative subdomain in solving Stokes equations by a preconditioned iterative method. As a result the required memory has reduced drastically. We could get speeding up of about 20 times in using 24 CPUs of Fujitsu GP7000, a shared memory type computer at Computing and Communications Center, Kyushu University. Using this code, we have studied the viscosity ratio dependency of stationary temperature fields and flow patterns. When the ratio increases, the heads of plumes flatten and the number of plumes increases. (3) We have presented a numerical verification method for solutions of the Navier-Stokes equations, and succeeded in the verification for low Reynolds number problems. Performing accuracy guaranteed computation, we have given a computer aided proof to the existence of bifurcation branches for two-dimensional heat convection problems. (4) Using a code for the convection in a three-dimensional sphere, we have studied the relation between the existence of continents and mantle convection. We have shown numerically that plumes arive under continents in some tens of billion years., 11554003
1999 - 2001

Development of practical methods for rigorous calculation with guaranteed accuracy
YAMAMOTO Nobito
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Our objective in this study which is fonded by Grant-in-Aid for Scientific Research is development of practical methods for rigorous calculation with, guaranteed accuracy. Through the period of this study over 4 years, we have obtained some results on the following. 1. Verified computation of the maximum eigenvalue of Newton operators in infinite dimensional spaces 2. Verified computation methods for eigenvalues of symmetric band matrices together with their indices 3. Extension of the above methods to general eigenvalue problems 4. Methods for verification of uniqueness of solutions to fixed point equations 5. Research on a bifurcation diagram of Perturbed Gelfand Equation with guaranteed accuracy 6. Rigorous calculation of constants appearing in error estimations of FEM 7. Research on methods for transaction of rounding errors using Fortran 90 and quadruple-precision floating point numbers 8. Numerical verification of solutions to the Navier-Stockes equation using spectral methods 9. Estimation methods for influence of rounding error by interval arithmetic 10. Estimation of ability of approximation of FEM. Consequently we can conclude that practical methods for verified computation of eigenvalue problems. are developed. On the methods for PDEs, they are also developed but there are some difficulties concerning mathematical matters in practical use for non-professional users., 09640278
1997 - 2000

最適制御問題と最良近似問題の研究
川崎 英文; 田中 靖子; 笛田 薫; 山本 野人; 中尾 充宏; 柳川 尭
日本学術振興会, 科学研究費助成事業, 九州大学, 基盤研究(C), 1、川崎は、不等式相制約から導かれる最大型関数の1次、2次の方向微分公式を与え、さらに、片側相条件は自明な例外を除いて常に包絡線を生成する事を示した。また、古賀(富山大助手)との共同研究で、不等式相制約をもつ変分問題に対するLegendre型の最適性条件を導いた。これらの結果を研究集会「離散と連続の数理」(数理研、10月)と「情報・統計科学シンポジウム」(九州大学、12月、特別講演)で発表した。 2、柳川は、森川、遠藤らと多次元離散型データ解析のための確率モデルについて共同研究を行った。特に反応がいくつかの順序カテゴリーに表される場合の用量反応関係モデルを開発し、毒性の無影響量決定問題に適用した。Sydney Statistical Congress(Sydney、8月)をはじめとする国際会議において3件日本数学会(都立大、9月)等の国内学会で6件の講演をおこなった。 3、中尾、山本は、関数方程式の解に対する数値的検証法の研究に関して、3件の研究成果を得た。 (1)変分不等式の解に対する数値的検証。(2)Stokes方程式の有限要素解のa posteriori型誤差評価。(3)楕円型作用素の固有値評価の精度保証付き計算。 これらに関してICCAM(Belgium、7月)をはじめとする国際会議で3件、応用数理学会(東京大学、9月)等の国内学会で10件の講演をおこなった。 4、笛田は、統計的推測理論の研究、乱数、モンテカルロシミュレーションに関する以下の研究で成果を得、日本統計学会研究部会で講演をおこなった。 (1)凸和距離から導かれる統計量の漸近正規性。(2)サンプル数が少ない場合に順位統計量の正確な分布を計算するための、組み合わせ生成アルゴリズム。, 08640294
1996 - 1996

偏微分方程式の精度保証付き計算のための総合的手法の開発
山本 野人
日本学術振興会, 科学研究費助成事業, 九州大学, 奨励研究(A), 今年度における精度保証付き計算法の研究の中で得られた、次のような新しい成果を報告する。 1.丸め誤差を処理するための演算手法の開発 既存の有理数演算用のパッケージをもとにして、区間演算を利用して丸め誤差を処理するプログラムを開発した。すなわち、 (1)区間型の変数および演算を導入した。 (2)加減算の度に連分数展開を用いて、有理数を与えられた桁数に丸め、その誤差を含むように区間幅を広げるルーチンを作成した。このプログラムによって、丸め誤差の影響までも考慮した厳密な計算が可能となった。 2.残差反復を用いた誤差の改善 残差反復法と誤差の事後評価の方法を開発し、これを高次の有限要素空間を用いた楕円型方程式の解の数値的検証法に応用したところ、収束と誤差評価とに劇的な改善が見られた。 3.MHD方程式の解析 自由境界を持つMHD方程式の解の数値的検証を行なった。これは微分不可能な項を持つため、Newton型反復を適用するにあたって特別の工夫を要した。 今後の研究計画としては、まず、これまでの結果をさらに発展させて、有理数演算及び区間演算、あるいは区間演算を応用した完全精度計算を用いた精度保証計算用の演算パッケージを開発することが挙げられる。次に、問題によって必要となる区間係数の扱いや誤差評価の方法などについての複雑な手順を上述の演算パッケージで計算可能になるように工夫する。これは同じ計算量で最大の精度が得られるような理論と演算双方での工夫を意味するだけでなく、応用の簡便さという視点から、できるだけ明解で適用範囲の広い手法の開発をも意味している。, 07740161
1995 - 1995

最適化と最良近似
川崎 英文; 笛田 薫; 山本 野人; 中尾 充宏
日本学術振興会, 科学研究費助成事業, 九州大学, 一般研究(C), 1、川崎は古賀さゆり(博士2年)と共同で、不等式相制約を持つ変分問題に対する2次の最適性条件の研究をおこない、相制約から包絡線が生成されることを明らかにした。この内、2次の最適性条件に関する研究はProceedings of APORS'94に掲載された。また、包絡線に関する結果を研究集会「非線形解析学と凸解析学の研究」(9月、京大数理研)で発表した。 2、川崎は微分不可能最適化の観点から、動節点を持つ折れ線近似問題の研究を行い、最良近似解の必要条件を与えた。さらに、節点の個数が2個の場合については、最良近似解の分類に成功した。この内、必要条件に関する研究はProceedings of APORS'94に掲載された。また、最良近似と最適化に関する講演を第5回RAMPシンポジウム(9月、東北大学)とオペレーションズ・リサーチ学会大阪研究部会(12月、大阪)でおこなった。いずれも招待講演である。 3、川崎は研究集会「最適化における離散と連続構造」(京大数理研、11月)の研究代表者をつとめた。研究集会の講演数は26件であった。 4、中尾・山本は共同で非線形偏微分方程式に解に対する数値検証法に関して、高次有限要素を用いた残差反復法による検証の効率化と高精度化をおこなった。この結果はJournal of Computational and Applied Mathematicsに掲載された。 5、中尾は2階双曲型偏微分方程式に対する解の数値的検証法を定式化し、その数値例を与えた。この結果は、Interval Computationsに掲載された。 6、山本は中尾らと共同で、自由境界を持つMHD方程式の解の数値的検証を行った。これは微分不可能な項を持つため、Newton型反復を適用するにあたって特別な工夫を要した。この結果はNonlinear Analysisに掲載予定である。, 07640316
1995 - 1995

関数方程式の解に対する精度保証付き数値計算法
中尾 充宏; 山本 野人; 大塚 寛; 川崎 英文; 小西 貞則; 田中 俊一
日本学術振興会, 科学研究費助成事業, 九州大学, 一般研究(C), これまでに得られた楕円型方程式に関する結果を、より実用度の高いものに改良・拡張するための検討を行なった。また、原理的な検証定式化が行なわれている放物型および双曲型方程式に対して、その適用性を高めることを試みた。具体的な検討結果は以下の通り。 (1)最大値ノルムの意味での精度保証が可能な方法を検討し、数値的に構成可能なa priori誤差評価を得るとともに、高次要素を用いて高精度で最大値ノルム型のa posteriori誤差評価を得る方法を見い出した。 (2)パラメータに依存する非線形微分方程式系に対し、turning pointやbifurcation pointの近傍における特異性の影響を克服した検証方式を定式化し、またそれらの点自体を数値的に精度保証する方法を実現した。 (3)方程式の中に未知関数についてのフレッシェ微分が不能な項を含む場合にも、ニュートン的方法による検証な可能なことを、電磁流体の平衡系方程式を例にとって明らかにした。 (4)高次有限要素を用いて近似解のa posteriori誤差評価を行い、その結果に残差反復を適用することにより、検証能力が飛躍的に向上することを見い出した。 (5)非線形楕円型方程式の球対称解の漸近挙動を特徴づける積分方程式について、その解を精度保証することにより、理論的に解明困難な問題に対し数値的解決を与えた。 (6)空間2次元および3次元の非線形放物型問題に対する数値的検証法を定式化し、その検証例を与えた。 (7)Stokes方程式の有限要素解に対するa posteriori誤差評価の方法を見いだし、Navier-Stokes方程式の解の数値的検証定式化への見通しを得た。 (8)検証プログラムの高速化と効率化について検討し、検証手順の簡易化手法を見いだし、これによりにより検証プログラムの実行効率と検証精度の向上が計れた。, 06640321
1994 - 1994

偏微分方程式の解の数値的検証法
山本 野人
日本学術振興会, 科学研究費助成事業, 九州大学, 奨励研究(A), 今年度における精度保証付き計算法の研究の中で得られた、次のような新しい成果を報告する。 1.丸め誤差を処理するための演算手法の開発 既存の有理数演算用のパッケージをもとにして、区間演算を利用して丸め誤差を処理するプログラムを開発した。すなわち、 (1) 区間型の変数および演算を導入した。 (2) 加減算の度に連分数展開を用いて、有理数を与えられた桁数に丸め、その誤差を含むように区間幅を広げるルーチンを作成した。 このプログラムによって、丸め誤差の影響までも考慮した厳密な計算が可能となった。 2.非線形偏微分方程式の球対称解の漸近挙動に関する応用 conformal scalar curvature equationと呼ばれる非線形偏微分方程式の球対称解は、原点での値に依って三種の異なる漸近挙動を取ることが知られているが、どのタイプを取るかの判定法は一般には与えられていなかった。報告者は、積分方程式に対する精度保証付き計算法を考案し、これを用いてPohozaevの恒等式にあらわれる量を厳密に計算することで、漸近挙動の判定を行なう方法を提案した。 今後の研究計画としては、まず、これまでの結果をさらに発展させて、有理数演算及び区間演算、あるいは区間演算を応用した完全精度計算を用いた精度保証計算用の演算パッケージを開発することが挙げられる。次に、問題によって必要となる区間係数の扱いや誤差評価の方法などについての複雑な手順を上述の演算パッケージで計算可能になるように工夫する。これは同じ計算量で最大の精度が得られるような理論と演算双方での工夫を意味するだけでなく、応用の簡便さという視点から、できるだけ明解で適用範囲の広い手法の開発をも意味している。具体的には、上記の球対称解を扱う場合での積分方程式への変換及び数値積分の手法の応用を発展させていくことなどを考えている。, 05740136
1993 - 1993

非線形最適化の基礎理論とその応用
川崎 英文; 山本 野人; 末吉 豊; 横田 佳之; 坂内 悦子; 中尾 充宏
日本学術振興会, 科学研究費助成事業, 九州大学, 一般研究(C), 1、川崎は微分不可能計画法の分野で1次の包洛線効果について研究をおこない、sup型関数の方向微分の公式を与えた。その応用として、1個の動節点をもつ折れ線による最良近似問題を考察し、新しい交代定理を導いた。これらの結果を、シンポジウム「非線形解析学と数理経済の研究」(10月、京大数理研)と、日本オペレーションズ.リサーチ学会(10月、筑波大)で発表した。 2、川崎は古賀さゆり(院生)との共同研究により、不等式相条件をもつ変分問題の弱極値に対するLegendre条件を導き、その結果をシンポジウム「最適化理論と数理構造」(12月、京大数理研)で発表した。 3、中尾と山本は、非線形楕円型方程式の解の数値検証法について検討し、従来手法の適用領域の拡張及び検証効率、精度の改良をおこなった。 4、坂内は、Hamming association scheme H(d,q)がmodular不変性を持つ事を示し、有限巡回群上のassociation schemeのmodular不変性を完全に決定した。これらに関し、国際シンポジウム「Algebraic Combinatorics」(11月、九大)、「Shanghai Conference:Designs,Codes and Finite Geometries」(5月、上海交通大学)等で4件の講演をおこなった。 5、高田は河野俊丈との共同研究により、量子群の表現に付随した3次元多様体のWitten不変量を構成し、framed linkの不変量に関するSymmetryを与え、それを利用してホモロジー3球面についての不変量の値の周期性を与えた。これに関連して、「The Second Japan-Korea Seminar on Knots and links」(8月、大阪)で講演をおこなった。, 05640273
1993 - 1993

微分方程式に対する精度保証付き数値計算法
中尾 充宏; 大塚 寛; 山本 野人; 川崎 英文; 田中 俊一; 古川 長太
日本学術振興会, 科学研究費助成事業, 九州大学, 一般研究(C), 本年度は特に、楕円型境界値問題と放物型初期境界値問題の厳密解を、計算機によってその存在と精度の保証付きで求める方法(数値的検証法)について検討し、従来手法の改良拡張に関し以下の成果を得た。 1.パラメータに依存しturning pointを持つような微分方程式に対する数値的検証法の定式化 従来の検証法では、turning pointの近傍では線形化作用素の特異性のために検証不能となったが、この点を克服する手法を見いだし、生物数学に現われる非線形常微分方程式の2点境界値問題に適用しその十分な有効性を確認した。 2.非凸領域での楕円型境界値問題の解の検証法 非凸領域ではPoisson方程式の解の滑らかさが落ちるため、その有限要素解の構成的a priori誤差評価が困難であり、したがってこれまでの検証定式化は適用できなかった。今回、計算機を用いてPoisson方程式の有限要素解のa priori誤差評価を与える方法を見いだし、平面上のL-shape domainの場合適用し検証数値例を与えた。 3.空間多次元の放物型方程式の解に対する検証法 空間1次元の場合は既に定式化と基本的検証数値例とが与えられているが、多次元の場合にそのまま適用することはできなかった。今回その点を改良し原理的には空間3次元まで適用可能とし、2次元に対する検証例を与えた。 4.残差反復法による楕円型境界値問題に対する検証能力の向上 従来の検証法では検証の原理的要因から、解の大きさがある程度以上になると、それにともなって誤差が増大し検証実行時のニュートン的反復列が発散して検証不能となる場合があった。この難点を克服するための種々の残差方程式への変換技法について検討し有効な方法を見いだした。なお本項については今後も継続して検討する予定である。, 04804006
1992 - 1992

確率測度の無限次元級数解析と距離解析
佐藤 坦; 田中 輝雄; 中尾 充宏; 山本 野人; 川崎 英文; 柳川 尭
日本学術振興会, 科学研究費助成事業, 九州大学, 一般研究(C), 今年度の研究成果として筆頭に挙げられるのは、なんと言っても数年来の懸案であった準不変測度の作用群に関する連続性を、博士課程大学院生水町仁との共同研究で、非常に一般的な形で解決したことであろう。すなわちGを確率空間(X,B,μ)に作用する距離付け可能な局所コンパクト群、m_GをGの左不変ハール測定、μ_g(A)=μg^<-1>A),A∈B,g∈Gとする。このとき全変分ノルムの意味でlim_||μ_g-μ||=0となるための必要十分条件がμ《μg^*μであることを証明した。μ_gのこのような連続性はエルゴード理論の研究では、よく必要になるものであり、これまで多くの研究があるが、それらのいずれもが(X,B,μ)になんらかの位相的または測定論的な仮定をおくものであった。それに対して今回得た結果は(X,B)が可測空間でありさえすればよく、ほぼ最終的な結果であると考えられる。早速応用としてループ群上のウィーナー測度の0-1法則を証明した。今後多くの応用が期待される。 他方距離空間上で「徑比不等式」をみたす確率測度について大域密度定理を証明したのが[2]である。今後フラクタル解析などへの応用を目指している。 代表者佐藤が創始した「確率移動の絶対連続性」の研究について、日ソ確率論シンポジウム(キエフ)での報告が[3]である。これについては今年度さらに新しい結果を得た。 局所凸空間上のガウス測度についての台湾大学での講義録が[1]である。ガウス測度の性質を「カメロン・マルチン空間の準不変性とエルゴード性」の観点からまとめたのが特徴である。 また、分担者たちもそれぞれ成果を得た。, 04640169
1992 - 1992

Computing Science and Complex systems
TANAKA Syunichi; OHTSUKA Hiroshi; KAWASAKI Hidefumi; NAKAO Mitsuhiro; YANAGAWA Takashi; FURUKAWA Nagata
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Kyushu University, Grant-in-Aid for General Scientific Research (B), The purpose of this project is to study possible relations between computing science and complex systems such as brain and market. As yet no secure theoretical foundation for studying these systems is available, we tried to search the one. The candidate is Formal Mathematics and Science based on the axiomatic set theory. Nonstandard Analysis is an important example in mathematics. In information and languages, Milner's Process Calculus and Barwise's Situation Theory (for natural languages) are formalized on non-well-founded set theory (developed by Aczel). These independent developments suggest the unity of various formal methods. Neural network theory is a dynamical aspects of a complex system. Infinitesimals and hyperfinite integers of nonstandard analysis may give us new perspective on large finite dynamical systems. Details of above observations are contained in the accompanying report., 02452010
1990 - 1991

関数方程式に対する自己検証的数値計算法
中尾 充宏; 大塚 寛; 川崎 英文; 山本 野人; 河原 康雄; 藤野 精一
日本学術振興会, 科学研究費助成事業, 九州大学, 一般研究(C), 本年度は特に偏微分方程式の解の存在、一意性および存在範囲の特定を計算機によって数値的に検証する方法として、非線形楕円型境界値問題と放物型初期値境界値問題を対象に検討した。これまでの研究成果をもとに、検証可能な方程式の範囲の拡大を図り、得られた検証法を、実際に物理学や生物数学上に登場する具体的方程式に対し適用することにより、その有効性を評価すると共に、手法の改良を行った。研究内容と成果は以下の通りである。 1.非線形楕円型境界値問題の検証を行う場合、従来のL^2理論に基く方法では高々多項式オ-ダ-の非線形性にしか対応できなかったが、非線形項のTaylor展開を考えることによりこれを克服できることがわかった。例えば指数関数的な非線形性を持つ方程式の検討にもL^2理論で対応できることを明らかにし、具体的適用例として方程式:-Δu=λe^uの解の検証を行った。 2.生物数学に現れる反応拡散系の定常問題を記述する非線形楕円型方程式:-Δu=λu(1ーu)(uーa)を対象として解の検証を試み、検証方式の実用性の評価を行った。その過程において従来方式の問題点が明らかにされ、その点を改良することにより効率良い検証アルゴリズムが得られ、有効性が高められた。 3.非線形発展方程式に対する検証法について検討した。先ず準線形放物形方程式に対する初期値境界値問題の解をコンパクト作用素の不動点として定式化し、RoundingとRounding・errorの概念に基づく検証条件を明らかにし、具体的な近似空間を設定して検証手順と検証例とを与えた。 4.非線形常微分方程式の2点境界値問題に対しても、より効率の良い検証法を開発した。, 02804007
1990 - 1990

フックス型微分方程式の代数幾何、微分幾何及び位相幾何的研究
吉田 正章; 山本 野人; 佐々木 武; 茂手木 公彦; 塩浜 勝博; 山崎 正
日本学術振興会, 科学研究費助成事業, 九州大学, 一般研究(C), 研究代表者(吉田)は分担者佐々木武氏等の協力のもとに、線型偏微分方程式系(解空間は有限次元)の幾何学的理論を推進させた。即n変数階数r+1の線型微分方程式系の解をならべてI次元複素射影空間への埋め込みを作ることにより、微分方程式を射影部分多様体を対応させ、r=n+2の場合に、微分方程式の係数の微分不変式と射影多様体の射影不変式の関係を明確にした。この場合の幾何は射影超曲面の等角幾何である。更に数種の代数多様体のモジュライを記述する微分方程式系(ガウス・マニン接続)を前述の不変量を計算することによって具体的に求め、微分方程式論的群論的代数幾何的及び数論的性質をくわしく調べた。特に代数多様体がある種のK3曲面のとき我々の研究した微分方程式は、最近ゲルファント達の提唱している超幾何方程式の一般化の一番実り多い例の解析になっている。 分担者山崎正は、一般次のジ-ゲル保形型式に対応するディリクレ級数の解析的性質を、重さ0の非正則なアイゼンシュタイン級数とのコンボル-ションを取ることにより導出した。即昔Rankinが楕円モジュラ-に対してやった“Symmetric scquare"がこの場合に拡張出来ることを示した。 分担者塩浜勝博は、非負曲率完備多様体上の局所凸集合の位相に関するBurago-Zalgallerの結果を更に精密化し、Gromoll-CheegerのSoul Theoremを一般化して、凸集合の位相構造に関する最終結果を得た。 分担者茂手木公彦は3次元球面の中の結び目をそれを内部に含むソリッドト-ラスに関してねじった際に結び目のTypeが変るか否かについて調べ、ソリッドト-ラスが自明でない時はいつでも変り、そうでないときも変らないのは高々有限ケのTwistであることなどを示した。また、twistに関して不変な性質についても調べた。, 01540145
1989 - 1989