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.

翻译：本文提出了一种动态最优输运问题的离散化方法，并证明了当时间步长与空间步长趋于零时，输运代价数值解向连续解的收敛速度。该收敛结果无需对测度施加任何正则性假设，尽管实验表明收敛速度并非最优。通过对偶间隙分析，我们在温和正则性假设下进一步获得了最优势函数梯度与速度场的收敛速率。为推导这些速率，我们对动态最优输运问题的对偶形式进行离散化，并利用哈密顿-雅可比方程离散化误差分析的成熟文献。