利用最优输运计算黎曼流形的割迹
Computing the Cut Locus of a Riemannian Manifold via Optimal Transport
摘要：
本文给出了紧致黎曼流形上一点的割迹作为Monge-Kantorovich方程的最优输运密度解的零点集的一个新刻画，给出了费用等于测地距离的最优输运问题的一个PDE形式.结合这一结果与最优输运数值求解器的基础上，所谓的动态Monge-Kantorovich方法，我们提出了一个新的框架的数值逼近的割迹的一个点在一个流形。我们展示了所提出的方法的适用性，解决了几个例子上的二维表面嵌入在$R^{3}$和讨论的优点和局限性。
英文摘要：
In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver based on the so-called dynamical Monge-Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in $R^{3}$ and discuss advantages and limitations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
