An efficient linear programming method for Optimal Transportation

TL;DR

This paper introduces a fast linear programming approach for solving the Optimal Transportation problem, improving computational efficiency and accuracy for various complex applications.

Contribution

It presents a grid refinement method that makes linear programming solutions feasible for larger problems and extends to partial OT and barycenter computations.

Findings

Method achieves high accuracy in complex OT scenarios

Computational efficiency surpasses traditional LP discretization

Successfully applied to nonconvex domains and barycenter problems

Abstract

An efficient method for computing solutions to the Optimal Transportation (OT) problem with a wide class of cost functions is presented. The standard linear programming (LP) discretization of the continuous problem becomes intractible for moderate grid sizes. A grid refinement method results in a linear cost algorithm. Weak convergence of solutions is stablished. Barycentric projection of transference plans is used to improve the accuracy of solutions. The method is applied to more general problems, including partial optimal transportation, and barycenter problems. Computational examples validate the accuracy and efficiency of the method. Optimal maps between nonconvex domains, partial OT free boundaries, and high accuracy barycenters are presented.