Asymptotic analysis of domain decomposition for optimal transport

TL;DR

This paper provides an asymptotic analysis of domain decomposition algorithms for optimal transport, revealing the limiting behavior of the algorithm as the partition becomes infinitely fine and offering insights into its efficiency.

Contribution

It derives a Gamma-convergence-based asymptotic description of the algorithm's limit trajectory, advancing theoretical understanding of its convergence speed in geometric settings.

Findings

Limit trajectories described by a continuity equation with horizontal momentum.

Convergence depends on a regularity assumption that is analyzed in detail.

Provides insights into the algorithm's efficiency at finite resolutions and with coarse-to-fine schemes.

Abstract

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.