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。研究到目前为止已经开发的自validated计算方法，我们选择各种技术作为研究动力系统的有用工具。2。我们开发了新的理论方案和技术来改善动力系统的自validated计算。这是多种自验化计算ODE的初始值问题的方法。其中，Lohner方法和TM方法是众所周知的，它们的理论基础是Taylor扩展及其误差估计。在经过验证的计算中扩展误差的主要原因是所谓的包装效果。这些方法使用QR分解来减少包装效果。我们将此技术应用于Nakao的方法，该方法已开发用于验证PDE的计算，并试图根据Nakao的方法构建ODE的数值验证方法。但是我们发现直接应用程序无法正常工作。然后我们已经研究了1）Nakao的方法减少了逐步程序中的包装效果。2）如何处理Nakao的方法中出现在整个时间过程中出现的大型矩阵3）Nakao方法在边界价值问题上应用的方法在ode上的边界价值问题上，以在最终的范围内进行狭窄的误差，以在最终的范围内限制新的误差。