Bingwei Zhang

Bingwei Zhang, Thomas Chen, Kai Hormann et al.

Range functions are a fundamental tool for certified computations in geometric modeling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ($m=2$). For ``superior'' convergence order (i.e., $m>2$), we exploit the Cornelius--Lohner framework in order to introduce new bivariate range functions based on Taylor, Lagrange, and Hermite interpolation. In particular, we focus on practical range functions with cubic and quartic convergence order. We implemented them in Julia and provide experimental validation of their performance in terms of efficiency and efficacy.

Bingwei Zhang, Chee Yap

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.