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.

翻译：本文中，我们解决了非均匀环境中的离散最优传输问题。关键挑战在于构建成本矩阵，这需要找到两点之间的最优路径；为此我们建立并求解了相应的欧拉-拉格朗日方程。我们的主要理论成果是为欧拉-拉格朗日方程的解提供了可验证的最优性充分条件。我们提出了解决该问题的新算法，并通过若干数值算例展示了理论结果与算法性能。