This article reviews the use of first order convex optimization schemes to
solve the discretized dynamic optimal transport problem, initially proposed by
Benamou and Brenier. We develop a staggered grid discretization that is well
adapted to the computation of the $L^2$ optimal transport geodesic between
distributions defined on a uniform spatial grid. We show how proximal splitting
schemes can be used to solve the resulting large scale convex optimization
problem. A specific instantiation of this method on a centered grid corresponds
to the initial algorithm developed by Benamou and Brenier. We also show how
more general cost functions can be taken into account and how to extend the
method to perform optimal transport on a Riemannian manifold.

本文回顾了使用一阶凸优化方法来解决 Benamou 和 Brenier 最初提出的离散动态最优输运问题的方法，介绍了适用于计算基于均匀空间网格上定义的分布之间的 $L^2$ 最优输运测地线的交错网格离散化方法，展示了如何使用近端分裂方法来解决结果导致的大规模凸优化问题。同时，还介绍了如何考虑更一般的成本函数，以及如何扩展该方法以在 Riemann 流形上执行最优输运。