Error bounds for discretized optimal transport and its reliable efficient numerical solution

TL;DR

This paper develops error bounds for discretized optimal transport problems and introduces an active-set strategy that efficiently predicts solutions, confirmed by numerical experiments demonstrating convergence and scalability.

Contribution

It provides new error estimates for discretized optimal transport and proposes an active-set method leveraging optimality conditions within a multilevel framework.

Findings

Convergence rates match theoretical predictions.

Effective problem size grows linearly with discretization variables.

Numerical experiments validate the proposed method's efficiency.

Abstract

The discretization of optimal transport problems often leads to large linear programs with sparse solutions. We derive error estimates for the approximation of the problem using convex combinations of Dirac measures and devise an active-set strategy that uses the optimality conditions to predict the support of a solution within a multilevel strategy. Numerical experiments confirm the theoretically predicted convergence rates and a linear growth of effective problem sizes with respect to the variables used to discretize given data.