Numerical Verification Methods for Dynamical Systems described by ODEs

常微分方程描述的动力系统数值验证方法

基本信息

批准号：
17540106
负责人：
YAMAMOTO Nobito
金额：
$ 1.73万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540106/
关键词：
numerical verification method dynamical system ODE non-linear system chaos 精度補償

项目摘要

The present research has two purposes.1. Investigating self-validated computation methods which have been developed so far, we select various techniques as useful tools for study of dynamical systems.2. We develop new theoretical scheme and techniques for improvement of self-validated computation of dynamical systems.There are several methods for self-validated computation of initial value problems of ODEs. Among them, Lohner method and TM method are well-known, which have their theoretical base on Taylor expansion and its error estimation. The main reason which extends the error in the validated computation is so-called wrapping effect. These methods use QR factorization in order to reduce the wrapping effect. We applied this technique to the Nakao's method which has been developed for validated computation of PDEs and tried to construct a numerical verification methods for ODEs based on the Nakao's method. But we found that a straightforward application does not work well. Then we have investigated1) Reduction of the wrapping effect in the step-wise procedure by the Nakao's method.2) How to handle the large size matrices which appear in the whole time procedure by the Nakao's method3) Applications of the Nakao's method on boundary value problems to ODEs in order to get narrow error bound at the end pointand obtained some results on new theoretical scheme and techniques for improvement of self-validated computation of dynamical systems

本研究有两个目的：1．通过研究迄今为止已开发的自我验证的计算方法，我们选择了各种技术作为动力系统研究的有用工具。 2．我们开发新的理论方案和技术来改进动力系统的自验证计算。常微分方程初值问题的自验证计算有多种方法。其中，著名的Lohner方法和TM方法，其理论基础是泰勒展开式及其误差估计。验证计算中误差扩大的主要原因是所谓的包装效应。这些方法使用 QR 分解来减少包裹效应。我们将此技术应用于为验证偏微分方程计算而开发的 Nakao 方法，并尝试构建基于 Nakao 方法的 ODE 数值验证方法。但我们发现简单的应用程序效果不佳。然后我们研究了1）Nakao方法在逐步过程中减少包裹效应2）Nakao方法如何处理整个时间过程中出现的大尺寸矩阵3）Nakao方法在边界值上的应用问题到 ODE 以获得终点处的窄误差范围，并在改进动力系统自验证计算的新理论方案和技术方面取得了一些成果