Computing Optimal Trajectories for Optimal Transport in Nonuniform Environments

TL;DR

This paper develops methods to compute optimal transport trajectories in complex environments by solving Euler-Lagrange equations, formulating verifiable optimality conditions, and validating solutions with new algorithms, demonstrated through numerical examples.

Contribution

It introduces verifiable optimality conditions and algorithms for checking the optimality of trajectories in nonuniform environments, enhancing the accuracy of discrete optimal transport solutions.

Findings

Algorithms successfully validate optimality of computed trajectories.

Numerical examples demonstrate effectiveness in 2D and 3D environments.

Proposed methods improve reliability of cost matrix formation.

Abstract

In this work, we solve a discrete optimal transport problem in a nonuniform environment. To solve the optimal transport problem, we build the cost matrix and then use classical solvers for discrete optimal transport. The challenge is to form the cost matrix, which requires finding the optimal path between two points, and for this task we formulate and solve the associated Euler-Lagrange equations. A main contribution of ours is to provide verifiable sufficient conditions of optimality of the solution of the Euler-Lagrange equation and to propose new algorithms to to check optimality a-posteriori, thus validating the (exact) computation of the cost matrix. We illustrate our results and performance of the algorithms on several numerical examples in 2 and 3 dimensions.