Yamamoto NOBITO (山本 野人)

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (S), Fund Type: -, Overall Grant Amount: - (direct: 65300000, indirect: 19590000)
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.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Specially Promoted Research, Fund Type: -, Overall Grant Amount: - (direct: 329700000, indirect: 98910000)
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.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (B), Fund Type: -, Overall Grant Amount: - (direct: 12800000, indirect: -)
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.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (B), Fund Type: -, Overall Grant Amount: - (direct: 12700000, indirect: -)
(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.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3200000, indirect: -)
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.

Offer Organization: 日本学術振興会, System Name: 科学研究費助成事業, Category: 基盤研究(C), Fund Type: -, Overall Grant Amount: - (direct: 2100000, indirect: -)
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。

Offer Organization: 日本学術振興会, System Name: 科学研究費助成事業, Category: 一般研究(C), Fund Type: -, Overall Grant Amount: - (direct: 1900000, indirect: -)
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。