Numerical Verification Methods for Dynamical Systems described by ODEs

本研究有两个目的。考察迄今为止发展起来的自验证计算方法，我们选择了各种技术作为研究动力系统的有用工具。我们开发了新的理论方案和技术来改进动力系统的自验证计算。对于微分方程的初值问题，有几种自验证计算方法。其中以Lohner方法和TM方法最为著名，这两种方法的理论基础都是Taylor展开及其误差估计。导致验证计算误差扩大的主要原因是所谓的包裹效应。这些方法使用QR分解来减少包裹效应。我们将该技术应用于已经发展起来的用于偏微分方程验证计算的Nakao方法，并试图在Nakao方法的基础上构建偏微分方程的数值验证方法。但是我们发现一个简单的应用程序并不能很好地工作。然后，我们研究了1)在逐步过程中使用中尾方法减少包裹效应。2)如何用Nakao方法处理在整个时间过程中出现的大尺寸矩阵3)将Nakao方法应用于微分方程的边值问题，以得到终点处较窄的误差界，并在改进动力系统自验证计算的新理论方案和技术上取得了一些成果

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