Nonlinear Schwarz methods to compute geodesics on manifolds

TL;DR

This paper explores a nonlinear Schwarz method for computing geodesics on Riemannian manifolds, improving convergence by viewing the leapfrog algorithm as a domain decomposition approach and applying nonlinear preconditioning.

Contribution

It introduces a novel interpretation of the leapfrog algorithm as a Schwarz alternating method and enhances its convergence using nonlinear preconditioning techniques.

Findings

The method effectively computes geodesics on manifolds.

Preliminary experiments show promising convergence improvements.

The approach offers a new perspective on domain decomposition for geometric problems.

Abstract

We consider the leapfrog algorithm by Noakes for computing geodesics on Riemannian manifolds. The main idea behind this algorithm is to subdivide the original endpoint geodesic problem into several local problems, for which the endpoint geodesic problem can be solved more easily by any local method (e.g., the single shooting method). The algorithm then iteratively updates a piecewise geodesic to obtain a global geodesic between the original endpoints. From a domain decomposition perspective, we show that the leapfrog algorithm can be viewed as a classical Schwarz alternating method. Thanks to this analogy, we use techniques from nonlinear preconditioning to improve the convergence properties of the method. Preliminary numerical experiments suggest that this is a promising approach.