Core Concepts
This paper compares two numerical methods for integrating Riemannian cubic polynomials on the Stiefel manifold: the adjusted de Casteljau algorithm and a symplectic integrator constructed through discretization maps. The authors provide a numerical comparison of the two methods and discuss the benefits of each approach.
Abstract
The paper focuses on comparing two numerical methods for generating Riemannian cubic polynomials on Stiefel manifolds:
The adjusted de Casteljau algorithm, which uses quasi-geodesics to modify the standard de Casteljau algorithm for Riemannian manifolds.
A symplectic integrator constructed through discretization maps, which provides a numerical scheme for approximating the Hamiltonian flow associated with Riemannian cubic polynomials.
The authors consider two specific cases: the Stiefel manifold St3,1, which is diffeomorphic to the sphere, and the Stiefel manifold St3,2, which has a pure quasi-geodesic different from a geodesic.
The key findings are:
The adjusted de Casteljau algorithm provides a reasonably good approximation of the Riemannian cubic polynomials, with relative mean errors around 0.08% for the sphere and 0.45% for St3,2.
The retraction-based symplectic integrators have an error that decreases as the time step is reduced, making them suitable for simulating dynamics near the initial point.
Retraction-based integrators require more computational effort than the adjusted de Casteljau algorithm, but can achieve higher accuracy.
The adjusted de Casteljau algorithm is better suited for solving boundary value problems, while the retraction-based methods are more suitable for initial value problems.
The authors discuss the trade-offs between the two methods and suggest future research directions, such as improving the performance of retraction-based integrators by exploiting the geometric structure of the manifold.
Stats
The paper provides the following key figures and statistics:
Relative mean error of the adjusted de Casteljau algorithm: 0.08% for the sphere, 0.45% for St3,2
Comparison of mean error between the adjusted de Casteljau algorithm and retraction-based symplectic integrators for the sphere and St3,2
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