The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamic formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle \emph{a priori} a vast class of cost functions and geometries. Several disretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space. In this article, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional one for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien and Solomon fit in this framework.

中文翻译：

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动态最优传输离散化的无条件收敛

最优输运的动力学公式，也称为 Benamou-Brenier 公式或计算流体动力学公式，相当于将最优输运问题写为 PDE 约束下凸函数的优化，并且可以处理 \emph{aprii}类成本函数和几何。已经提出了这个问题的几种离散化，导致对平坦空间和黎曼流形的计算，扩展到均值场博弈和 Wasserstein 空间中的梯度流。在本文中，我们提供了一个框架，该框架保证时空离散问题的解在网格细化下收敛到二次最优传输的无限维问题之一。收敛性不依赖于空间和时间步长之间的比率，并且可以将任意正度量作为输入处理，而底层空间可以是黎曼流形。Gladbach、Kopfer 和 Maas 提出的有限体积离散化，以及本作者与 Claici、Chien 和 Solomon 合作研究的曲面三角化离散化都适合这个框架。