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.

翻译：我们研究了Noakes提出的用于计算黎曼流形上测地线的蛙跳算法。该算法的核心思想是将原始端点测地线问题分解为若干局部问题，每个局部问题可通过任意局部方法（例如单打靶法）更易求解。算法通过迭代更新分段测地线，最终获得原始端点间的全局测地线。从区域分解的角度，我们证明蛙跳算法可视为经典Schwarz交替法。基于这一类比，我们运用非线性预处理技术改进该方法的收敛特性。初步数值实验表明这是一种具有前景的研究路径。