Computing Optimal Trajectories for Optimal Transport in Nonuniform Environments
Luca Dieci, Daniyar Omarov
TL;DR
This work tackles discrete optimal transport in nonuniform environments by defining pairwise costs as path-optimal integrals and constructing a cost matrix to drive OT solvers. It develops and analyzes Euler–Lagrange formulations for both length and energy costs, proves equivalence between the two geodesic notions, and provides verifiable sufficiency conditions via conjugate-point analysis using Riccati dynamics. The authors implement collocation-based BVP solvers with a homotopy continuation to obtain optimal trajectories, and they certify optimality a posteriori while solving OT with linear assignment or Sinkhorn methods. Numerical experiments in 2D/3D demonstrate accurate cost computations, consistent energy–length geodesics, and distinct transport patterns under the two cost notions, highlighting the method’s robustness and practical impact for nonuniform transport problems.
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.