Research of Computer aided Nonlinear Analysis with Flexibility

计算机辅助柔性非线性分析研究

• 批准号: 07455164
• 负责人: HORIUCHI Kazuo
• 金额: $ 4.29万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07455164/
• 关键词: Uncertainty; Fuzzy Mapping; Fixed Point Theorem; Nonlinear Problem of Circuit Systems; Interval Arithmetic; Numerical Method with Guaranteed Accuracy; Chaos; Bifurcation Phenomena; 非線形システム; 非線形回路; 区間解析; オブジェクト指向ソフトウェア

We had the purpose to establish fundamental theories and to constitute the elements of computer aided mathematical analyzing software. We developed our research as we planned as the following :(1) We proposed the fixed point theorem for fuzzy map which is obtained by modeling the system with uncertain property.(2) We make the theory to prove numerically the existence of solution for nonlinear operator equations.(3) Based on C++ and an object oriented language in which rational number arithmetic is implemented, we constituted 3 prototypes of object oriented software which can deal with various objects corresponding to interval analysis, automatic differentiation, function representation and so on.(4) By extending the methods to prove numerically the existence of bifurcation point, we developed the theory to cancel singular points. We also applied our theory to various types of bifurcation phenomena and indicated that we can prove the existence of actual bifurcation phenomena.(5) We prop … More osed the theory to prove the existence of homoclinic orbits or heteroclinic orbits and prove their existence for actual examples.(6) We proposed an algorithm to prove the existence of all solutions in a bounded region for finite dimensional nonlinear equations and proved that this algorithm stops within finite steps under the certain conditions.(7) We proposed a method to prove the existence of all solutions with high speed in a bounded region for finite dimensional nonlinear equations with separability, whose example is VLSI circuit. (8) We could change the speed of calculation by the accuracy. Concretely, We proposed the method in which we can obtain the calculated results with super high speed when we demand its low accuracy and in which we can obtain the results with high speed even when we demand its high accuracy.(9) We realized the obtained techniques of numerical method with guaranteed accuracy on our prototypes of the software. We applied our software to various nonlinear functional equations and indicated its usefulness.(10) We combined the numerical method in the case that we demand its low accuracy and the numerical one in the case that we demand its high accuracy, by which we proposed the numerical method with high speed at our request of accuracy. We realized this method on our software and indicated its usefulness by the example of nonlinear circuit systems.(11) We integrated the above organized investigations and remade a prototype of the software for computer aided nonlinear analysis which can correspond to the changes of the problem or the accuracy. We also indicated its usefulness by applying it to nonlinear circuit problems. Less

我们的目的是建立基本理论，并构成计算机辅助数学分析软件的要素。我们按照计划进行了以下研究：(1)提出了模糊映射的不动点定理，该定理是通过对具有不确定性的系统进行建模得到的。(2)从数值上证明了非线性算子方程解的存在性。(3)基于c++和实现有理数运算的面向对象语言，构建了3个可处理区间分析、自动微分、函数表示等不同对象的面向对象软件原型。(4)通过推广分岔点存在性的数值证明方法，建立了奇异点消去理论。我们还将我们的理论应用于各种类型的分岔现象，并表明我们可以证明实际分岔现象的存在。(5)进一步证明了同斜轨道和异斜轨道的存在性，并通过实例证明了它们的存在性。(6)提出了一种证明有限维非线性方程在有界区域内所有解存在的算法，并证明了该算法在一定条件下在有限步内停止。(7)以VLSI电路为例，给出了有限维可分非线性方程在有界区域内高速解的存在性证明方法。我们可以通过精度来改变计算的速度。具体来说，我们提出了在精度要求低的情况下可以超高速得到计算结果，在精度要求高的情况下也可以高速得到计算结果的方法。(9)在软件样机上实现了数值方法得到的技术，保证了精度。我们将该软件应用于各种非线性泛函方程，并说明了它的实用性。（10）在精度要求低的情况下，将数值方法与精度要求高的情况下，将数值方法结合起来，在精度要求高的情况下，提出了速度较快的数值方法。在软件上实现了该方法，并以非线性电路系统为例说明了该方法的实用性。（11）综合上述有组织的调查，重新制作了一个能对应问题变化或精度要求的计算机辅助非线性分析软件原型。我们还通过将其应用于非线性电路问题来说明它的实用性。少

项目成果

Kiyotaka Yamamura and Masakazu Mishina: ""An algorithm for finding all solutions of piecewise-linear resistive circuits"" Proceedings of 12th European Conference on Circuit Theory and Design. 123-126 (1995)

Kiyotaka Yamamura 和 Masakazu Mishina：“一种寻找分段线性电阻电路所有解的算法”第 12 届欧洲电路理论与设计会议论文集。

Yusuke Nakaya, Shin'ichi Oishi: ""Finding All Solutions of Nonlinear Systems of Equations Using Linear Programmingwith Guaranteed Accuracy"" Proc.GAMM/IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN

Yusuke Nakaya、Shinichi Oishi：“使用保证精度的线性规划查找非线性方程组的所有解”Proc.GAMM/IMACS 国际科学计算、计算机算术和验证数值研讨会 (SCAN

Y. Nakaya and S. Oishi: "A Numerical Method for Checking Nonexistence of Solution Nonlinear Equations Using Optimization" Proc. 1997 International Symposium on Nonlinear theory and its Applications. 313-316 (1997)

Y. Nakaya 和 S. Oishi：“使用优化检查非线性方程解不存在的数值方法”Proc。

堀内 和夫: "集合値写像概念による双方向通進路の変動理論" 電子情報通信学会基礎・境界ソサイエティ大会. A-2-9. (1997)

Kazuo Horiuchi：“基于集值映射概念的双向路径变分理论”IEICE 基础和边界学会会议 A-2-9。

相馬 隆郎,大石 進一,堀内 和夫: "精度保証付き数値計算を用いた常微分方程式の近似解の任意精度反復改良" 電子情報通信学会技術研究報告. CAS96-54. 99-106 (1996)

Takao Soma、Shinichi Oishi、Kazuo Horiuchi：“使用保证精度的数值计算对常微分方程的近似解进行任意精度迭代改进” CAS96-54 (1996)。