Abstracts and Downloads

(see also Bibliography)

Towards an optimal control of the wrapping effect

Abstract: In \cite{Kuehn98}, an algorithm was given to compute enclosures for the orbits of discrete dynamical systems. The enclosure sets are high order zonotopes, i.e., bodies which are the finite sum of parallelepipeds. A Flush-When-Full (FWF) strategy on the summands effectively avoids the wrapping effect by wrapping small summands much more often than large ones. In a competitive analysis, different on-line FWF strategies are evaluated according to their performance relative to an optimal off-line algorithm. Numerical evidence shows that the original FWF in \cite{Kuehn98} can be replaced by an optimized on-line strategy. [zipped PS 68kb]

Rigorously computed orbits of dynamical systems without the wrapping effect

Abstract: Introduced is a new method for rigorously computing orbits of discrete dynamical systems. High order zonotope enclosures of the orbit are computed using only matrix algebra. The wrapping effect can be made arbitrarily small by choosing the order high enough. The method is easy to implement and especially suited for parallel computing. It is compared to other well known strategies, and several examples are given. [zipped PS 233kb]

Cutting high dimensional cakes

Abstract:None. [zipped PS 36kb][DVI 4kb]

Zonotope Dynamics in Numerical Quality Control

Abstract: Mathematical rigorous error bounds for the numerical approximation of dynamical systems have long been hindered by the wrapping effect. We present a new method which constructs high order zonotope (special polytopes) enclosures for the orbits of discrete dynamical systems. The wrapping effect can made arbitrarily small by controlling the order of the zonotopes. The method induces in the space of zonotopes a dynamical system of amazing geometrical complexity. We emphasis the visualization of the zonotopes to better understand the involved dynamics.[zipped PS 102kb]

Rigorous error bounds for the initial value problem based on defect estimations

Abstract: A method is presented which gives rigorous local error bounds for the initial value problem of ODEs. The method is almost entirely based on rigorous defect estimations, and is therefore suited to work with any method which produces such estimates. Besides the defect, only a computationally negligible estimate for a second derivative is needed. The method also gives rigorous error bounds for the solution of the variational and the adjoint equation without any additional computations. This is especially useful for solving boundary value problems with the shooting method. As an application, periodic orbits of the Lorenz system are verified. [zipped PS 107kb]

Teaching Elementary Linear Algebra without Determinants

Abstract: The paper wants to introduce elementary linear algebra without using determinants. Gaussian elimination will be the main tool. In particular, the eigen value problem is solved with a slightly extended Gaussian elimination scheme.[zipped PS 54kb]

Last modified by Wolfgang Kühn on Tuesday, 6 March 2001