We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates, we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton–Jacobi equation.

我们提出了动态最优传输问题的离散化，当时间和空间步长消失时，我们可以获得传输成本值到其连续值的收敛率。这种收敛结果不需要对度量进行任何规律性假设，尽管实验表明速率并不陡峭。通过对偶性差距的分析，我们还获得了在温和规律性假设下最佳势和速度场梯度的收敛速率。为了获得这样的速率，我们将动态最优输运问题的对偶公式离散化，并使用与离散 Hamilton-Jacobi 方程引起的误差相关的成熟文献。