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 $\Gamma$-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. Convergence hinges on a regularity assumption that we investigate in detail. 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.

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最优传输域分解的渐近分析

大的最优传输问题可以通过域分解来解决，即通过迭代地独立和并行地解决小部分问题。已在非正则化和熵正则化设置中显示了在合适假设下收敛到全局最小化器，并且其计算效率已通过实验证明。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 $\Gamma$-convergence 推导出算法在无限细分割单元的极限下的渐近描述，从而实现这种理解。耦合的极限轨迹由乘积空间上的连续方程描述，其中动量是纯水平的并由成本函数的梯度驱动。收敛取决于我们详细研究的规律性假设。极限轨迹的全局最优性仍然是一个有趣的开放问题，即使全局最优性是在有限尺度上建立的。我们的结果提供了关于域分解算法在有限分辨率下以及与粗到细方案相结合的效率的见解。