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

Alexandre Anahory Simoes, Leonardo Colombo, Fátima Silva Leite

TL;DR

The paper addresses the problem of efficiently interpolating Riemannian cubic polynomials on Stiefel manifolds by comparing two numerical schemes: (i) the adjusted de Casteljau algorithm based on quasi-geodesics, and (ii) a symplectic integrator constructed from discretization maps. The study specializes to $n=3$ with $k=1$ (where $\mathbf{St}_{3,1}$ is sphere-like) and $k=2$ (where a genuine quasi-geodesic arises), highlighting differences between geodesic and quasi-geodesic behavior. Numerical experiments show that the adjusted de Casteljau method is fast but less accurate than the retraction-based symplectic integrator, which can achieve higher accuracy at increased computational cost and is better suited for boundary-value problems via shooting. The results illuminate trade-offs between speed and fidelity and point to future work on multi-chart intrinsic implementations and more efficient intrinsic discretization strategies.

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.

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

TL;DR

The paper addresses the problem of efficiently interpolating Riemannian cubic polynomials on Stiefel manifolds by comparing two numerical schemes: (i) the adjusted de Casteljau algorithm based on quasi-geodesics, and (ii) a symplectic integrator constructed from discretization maps. The study specializes to with (where is sphere-like) and (where a genuine quasi-geodesic arises), highlighting differences between geodesic and quasi-geodesic behavior. Numerical experiments show that the adjusted de Casteljau method is fast but less accurate than the retraction-based symplectic integrator, which can achieve higher accuracy at increased computational cost and is better suited for boundary-value problems via shooting. The results illuminate trade-offs between speed and fidelity and point to future work on multi-chart intrinsic implementations and more efficient intrinsic discretization strategies.

Abstract

In this paper we compare two numerical methods to integrate Riemannian cubic polynomials on the Stiefel manifold . 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 together with and . 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.
Paper Structure (5 sections, 1 theorem, 11 equations)

This paper contains 5 sections, 1 theorem, 11 equations.

Table of Contents

  1. Introduction
  2. The Stiefel manifold
  3. Background & notations
  4. Retractions and quasi-geodesics on Stiefel manifolds
  5. The adjusted de Casteljau Algorithm

Key Result

proposition thmcounterproposition

Let $S_0$ and $S_1$ be two distinct points in $\mathbf{St}_{n,k}$ so that, for $i=0,1$, $P_i=S_iS_i^\top \in \mathbf{Gr}_{n,k}$. Then, if the quasi-geodesic $\beta: [0,1]\mapsto \mathbf{St}_{n,k}$ defined by is a smooth curve connecting $S_0$ to $S_1$.

Theorems & Definitions (3)

  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • remark thmcounterremark