Large optimal transport problems can be approached via domain decomposition, i.e. by iteratively solving small partial problems independently and in parallel. Convergence to the global minimizers under suitable assumptions has been shown in the unregularized and entropy regularized setting and its computational efficiency has been demonstrated experimentally. An accurate theoretical understanding of its convergence speed in geometric settings is still lacking. In this article we work towards such an understanding by deriving, via \(\varGamma \)-convergence, an asymptotic description of the algorithm in the limit of infinitely fine partition cells. The limit trajectory of couplings is described by a continuity equation on the product space where the momentum is purely horizontal and driven by the gradient of the cost function. Global optimality of the limit trajectories remains an interesting open problem, even when global optimality is established at finite scales. Our result provides insights about the efficiency of the domain decomposition algorithm at finite resolutions and in combination with coarse-to-fine schemes.

大型最优传输问题可以通过域分解来解决，即通过独立和并行地迭代解决小的部分问题。在适当的假设下收敛到全局最小值已经在非正则化和熵正则化设置中得到证明，并且其计算效率已经通过实验证明。仍然缺乏对其在几何设置中收敛速度的准确理论理解。在本文中，我们通过\(\varGamma \)-收敛，算法在无限精细划分单元的限制下的渐近描述。耦合的极限轨迹由乘积空间上的连续性方程描述，其中动量是纯水平的，由成本函数的梯度驱动。极限轨迹的全局最优性仍然是一个有趣的开放性问题，即使在有限尺度上建立了全局最优性。我们的结果提供了有关域分解算法在有限分辨率下以及结合由粗到细方案的效率的见解。