用于计算流形上测地线的非线性Schwarz方法
Nonlinear Schwarz methods to compute geodesics on manifolds
AI总结：
本文将Noakes蛙跳算法解释为经典Schwarz交替方法，并利用非线性预处理技术改进其收敛性，数值实验验证了该思路的有效性。
摘要：
我们考虑Noakes提出的用于计算黎曼流形上测地线的蛙跳算法。该算法的主要思想是将原始端点测地线问题细分为若干局部问题，对于这些局部问题，端点测地线问题可以通过任意局部方法（例如单射法）更容易地求解。该算法随后迭代更新分段测地线，以获得原始端点之间的全局测地线。从区域分解的角度来看，我们证明蛙跳算法可以被视为经典的Schwarz交替方法。借助这一类比，我们使用非线性预处理技术来改进该方法的收敛性质。初步数值实验表明这是一种有前景的方法。
英文摘要：
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.
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