Optimal Trajectories for Optimal Transport in Nonuniform Environments

In this work, we solve a discrete optimal transport problem in a nonuniform environment. The key 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 theoretical result of ours is to provide verifiable sufficient conditions of optimality of the solution of the Euler-Lagrange equation. We propose new algorithms to solve the problem, and illustrate our results and performance of the algorithms on several numerical examples.