Taylor Tube Method for Validated IVP

TL;DR

This paper introduces the Taylor Tube method, generalizing the Euler Tube for validated IVP algorithms, improving accuracy and potentially increasing speed when combined with bisection.

Contribution

It extends the Euler Tube approach to higher-degree Taylor Tubes, enhancing the accuracy and efficiency of validated IVP algorithms.

Findings

Higher-degree Taylor Tubes improve accuracy.

Combining Taylor Tubes with bisection can lead to speedup.

The method forms the basis of a complete validated IVP algorithm.

Abstract

We recently introduced a novel architecture for the design of validated IVP algorithms. This architecture forms the basis of our complete validated algorithm for IVP. A key subroutine in our algorithm is the \textbf{Euler Tube}: it gave a technique for refining end- and full-enclosures and is also key to deriving a complexity bound of our IVP solver. In this paper, we generalize it to \textbf{Taylor Tube} of degree $p\ge 1$. As expected, higher-degree Taylor Tubes improve accuracy. But surprisingly, our experiments show that it can also lead to an overall speedup when combined with bisection.