The $L^1$ optimal transport density $\mu^*$ is the unique $L^\infty$ solution of the Monge-Kantorovich equations. It has been recently characterized also as the unique minimizer of the $L^1$ -transport energy functional E. In the present work we develop and we prove convergence of a numerical approxi- mation scheme for $\mu^*$ . Our approach relies upon the combination of a FEM- inspired variational approximation of E with a minimization algorithm based on a gradient flow method.

翻译：$L^1$ 最优输运密度 $\mu^*$ 是 Monge-Kantorovich 方程的唯一 $L^\infty$ 解，且近期也被刻画为 $L^1$ 输运能量泛函 E 的唯一极小元。本文提出并证明了一种针对 $\mu^*$ 的数值逼近方案的收敛性。我们的方法基于将受有限元方法（FEM）启发的变分逼近E与基于梯度流方法的极小化算法相结合。