Numerical Verification Methods for Dynamical Systems described by ODEs

本研究主要有两个方面.通过对已有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效工具.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，常微分方程初值问题的自验证计算有几种方法。其中，Lohner方法和TM方法是比较著名的方法，它们的理论基础是泰勒展开式及其误差估计。在验证计算中扩大误差的主要原因是所谓的包裹效应。这些方法使用QR分解以减少包裹效应。我们将这种技术应用到Nakao方法中，该方法已被开发用于验证偏微分方程的计算，并尝试构建基于Nakao方法的常微分方程的数值验证方法。但我们发现，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）用Nakao方法减少逐步过程中的包裹效应; 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）将Nakao方法应用于常微分方程的边值问题，以获得较窄的端点误差界，并得到了一些改进自适应方法的新理论格式和技巧的结果。动力系统的验证计算

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