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Foundations of Computational Mathematics

, Volume 16, Issue 4, pp 899–939 | Cite as

Integrators on Homogeneous Spaces: Isotropy Choice and Connections

  • Hans Munthe-Kaas
  • Olivier VerdierEmail author
Article

Abstract

We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an isotropy choice. We show that equivariant isotropy choices combined with equivariant Lie group integrators produce equivariant homogeneous space integrators. Moreover, we show that the RKMK, Crouch–Grossman, or commutator-free methods are equivariant. To show this, we give a novel description of Lie group integrators in terms of stage trees and motion maps, which unifies the known Lie group integrators. We then proceed to study the equivariant isotropy maps of order zero, which we call connections, and show that they can be identified with reductive structures and invariant principal connections. We give concrete formulas for connections in standard homogeneous spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar decomposition manifolds. Finally, we show that the space of matrices of fixed rank possesses no connection.

Keywords

Homogeneous spaces Symmetric spaces Lie group integrators Connection Runge–Kutta Skeleton Stiefel manifold Lax pair  Grassmannian Projective space Polar decomposition Constant rank matrices 

Mathematics Subject Classification

22F30 53C30 14M15 65Pxx 37Mxx 

Notes

Acknowledgments

This research was supported by the Spade Ace Project and by the J. C. Kempe memorial fund.

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Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.Department of Mathematics and StatisticsUmeå UniversityUmeåSweden

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