Comparison of two numerical methods for Riemannian cubic polynomials on Stiefel manifolds

TL;DR

This paper compares two numerical methods for integrating Riemannian cubic polynomials on Stiefel manifolds, highlighting their advantages through specific cases where the manifold resembles a sphere or exhibits quasi-geodesics.

Contribution

It provides a comparative analysis of the adjusted de Casteljau algorithm and a symplectic integrator on Stiefel manifolds for the first time.

Findings

The adjusted de Casteljau algorithm performs well on spherical cases.

The symplectic integrator handles non-geodesic quasi-geodesics effectively.

Both methods have distinct advantages depending on the manifold's geometry.

Abstract

In this paper we compare two numerical methods to integrate Riemannian cubic polynomials on the Stiefel manifold $\textbf{St}_{n,k}$. The first one is the adjusted de Casteljau algorithm, and the second one is a symplectic integrator constructed through discretization maps. In particular, we choose the cases of $n=3$ together with $k=1$ and $k=2$. The first case is diffeomorphic to the sphere and the quasi-geodesics appearing in the adjusted de Casteljau algorithm are actually geodesics. The second case is an example where we have a pure quasi-geodesic different from a geodesic. We provide a numerical comparison of both methods and discuss the obtained results to highlight the benefits of each method.