Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem
Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem
复制标题
二次 Monge-Kantorovich 问题的最小凸扩展和有限差分离散化
DOI：
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发表时间：
2017
影响因子：
1.9
通讯作者：
V. Duval
中科院分区：
文献类型：
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作者：
J. Benamou;V. Duval
提出了一种适用于具有第二边值条件的MA方程的Monge-Ampère（MA）格基约化方法，其目标是凸集.这产生了一个快速的自适应方法数值求解两个绝对连续的措施，其中第二个凸支持之间的最优运输（OT）问题。所提出的数值方法实际上捕获了一个特定的Brenier解，该解在某种意义上是最小的。我们证明了收敛的网格步长为零的方法，并与数值实验表明，它是能够重现的OT问题的微妙性质。
We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.