Numerical Verification Methods for Dynamical Systems described by ODEs

本研究有两个目的。通过对目前已经发展起来的自验证计算方法的研究，我们选择了各种技术作为研究动力系统的有用工具。我们发展了新的理论方案和技术来改进动力系统的自验证计算。常微分方程组初值问题的自验证计算有几种方法。其中著名的是Lohner方法和TM方法，它们都有基于泰勒展开及其误差估计的理论基础。在验证计算中扩大误差的主要原因是所谓的包络效应。这些方法使用QR因式分解来减小包络效应。我们将这一技术应用于已发展起来的用于偏微分方程组验证计算的Nakao方法，并试图在Nakao方法的基础上构造一种常微分方程组的数值验证方法。但我们发现，简单的应用程序并不能很好地工作。然后研究了用Nakao方法减小步进过程中的包络效应。2)如何用Nakao方法处理整个时间过程中出现的大尺寸矩阵。3)Nakao方法在常微分方程组中的应用，以便在终点处得到较窄的误差界。在改进动力系统自验证计算的新的理论方案和技术方面得到了一些结果

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