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Integrators on Homogeneous Spaces: Isotropy Choice and Connections
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2016 (English)In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 16, no 4, 899-939 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2016. Vol. 16, no 4, 899-939 p.
Keyword [en]
Homogeneous spaces, Symmetric spaces, Lie group integrators, Connection, Runge-Kutta, Skeleton, iefel manifold, Lax pair, Grassmannian, Projective space, Polar decomposition, Constant rank matrices
National Category
Mathematics Computer and Information Science
URN: urn:nbn:se:umu:diva-124832DOI: 10.1007/s10208-015-9267-7ISI: 000380266700003OAI: diva2:1010388
Available from: 2016-10-03 Created: 2016-08-26 Last updated: 2016-10-03Bibliographically approved

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Verdier, Olivier
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