Geometric integration on symmetric spaces

TL;DR

This paper develops geometric numerical integrators for differential equations on symmetric spaces, leveraging their structure for efficient algorithms applicable to spaces like spheres, Grassmannians, and hyperbolic spaces.

Contribution

It introduces a unified framework for geometric integration on symmetric spaces using canonical operations and exponential maps, with simple, low-cost algorithms for specific examples.

Findings

Algorithms are simple and cost only O(n) operations per step for spheres and hyperbolic spaces.

The integrators preserve geometric structures inherent to symmetric spaces.

Concrete examples demonstrate the effectiveness of the proposed methods.

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 O(n) operations per step.