SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A2295-A2319, January 2021.
In this paper, we address the numerical solution of the optimal transport problem on undirected weighted graphs, taking the shortest path distance as transport cost. The optimal solution is obtained from the long-time limit of the gradient descent dynamics. Among different time stepping procedures for the discretization of this dynamics, a backward Euler time stepping scheme combined with the inexact Newton--Raphson method results in a robust and accurate approach for the solution of the optimal transport problem on graphs. It is found experimentally that the algorithm requires solving between $\mathcal{O}(1)$ and $\mathcal{O}(m^{0.36})$ linear systems involving weighted Laplacian matrices, where $m$ is the number of edges. These linear systems are solved via algebraic multigrid methods, resulting in an efficient solver for the optimal transport problem on graphs.

中文翻译：

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图上最优输运问题的快速迭代求解

SIAM 科学计算杂志，第 43 卷，第 3 期，第 A2295-A2319 页，2021 年 1 月。
在本文中，我们解决了无向加权图上最优运输问题的数值解，以最短路径距离作为运输成本。最优解是从梯度下降动力学的长时间限制中获得的。在此动力学离散化的不同时间步长程序中，后向欧拉时间步长方案与不精确的 Newton--Raphson 方法相结合，为解决图上的最佳传输问题提供了一种稳健而准确的方法。实验发现，该算法需要在 $\mathcal{O}(1)$ 和 $\mathcal{O}(m^{0.36})$ 之间求解涉及加权拉普拉斯矩阵的线性系统，其中 $m$ 是边缘。这些线性系统通过代数多重网格方法求解，