We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values and plans by solving finite dimensional discretizations of the corresponding Kantorovich problem. Then we approximate optimal maps by means of the usual barycentric projection or by an analogous procedure available in general spaces without a linear structure. We prove the convergence of all these approximants in full generality and show that our convergence results are sharp.

翻译：我们研究了在一般紧致度量空间中Monge-Kantorovich问题的逼近问题，证明了最优值、最优运输方案和最优映射可以通过完全离散方法有效逼近。首先，通过求解相应Kantorovich问题的有限维离散化问题，我们逼近了最优值和最优运输方案。随后，利用通常的重心投影或一般空间中无需线性结构的类似方法，实现了对最优映射的逼近。我们证明所有这些逼近项在完全一般情形下的收敛性，并表明收敛结果是精确的。