Geometric integration on symmetric spaces

Hans Munthe-Kaas

doi:10.3934/jcd.2023015

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2024, Volume 11, Issue 1: 43-58. Doi: 10.3934/jcd.2023015

This issue Previous Article Euler–Maruyama approximations of the stochastic heat equation on the sphere Next Article Pseudo-Hamiltonian system identification

Geometric integration on symmetric spaces

Hans Munthe-Kaas^{1, 2, ,}

1.
UIT The Arctic University of Norway
2.
University of Bergen, Norway

^*Corresponding author: Hans Munthe-Kaas
^*Corresponding author: Hans Munthe-Kaas

Received: August 2023

Revised: December 2023

Early access: January 3, 2024

Published online: January 3, 2024

The author is supported by the Isaac Newton Institute programme "Geometry, compatibility and structure preservation in computational differential equations" (2019) and Research Council of Norway project 302831 "Computational Dynamics and Stochastics on Manifolds" (CODYSMA).

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Abstract

We consider geometric numerical integration algorithms for differential equations evolving on symmetric spaces. The integrators are constructed from canonical operations on the symmetric space, its Lie triple system (LTS), and the exponential from the LTS to the symmetric space. Examples of symmetric spaces are $ n $-spheres and Grassmann manifolds, the space of positive definite symmetric matrices, Lie groups with a symmetric product, and elliptic and hyperbolic spaces with constant sectional curvatures. We illustrate the abstract algorithm with concrete examples. In particular for the $ n $-sphere and the $ n $-dimensional hyperbolic space the resulting algorithms are very simple and cost only $ {\mathcal O}(n) $ operations per step.

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Mathematics Subject Classification: Primary: 65L05; Secondary: 53C35, 22E70.

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References

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