论文标题
最佳运输计划的定量稳定性和错误估计
Quantitative Stability and Error Estimates for Optimal Transport Plans
论文作者
论文摘要
通过解决半污泥或完全污泥的最佳运输问题，可以近似两个绝对连续的措施和$ν$之间的最佳运输图和计划。这两个问题是从近似$μ$或$μ$和$ν$的。我们从[Gigli，在Hölder的连续性中延伸到最佳运输地图沿曲线的措施]的想法，我们表征了运输计划在$μ$和$ν$的扰动下如何变化。我们将这种见解应用于证明半差异和完全discrete算法的错误估计，这仅仅是由于近似度量而产生的错误。我们获得了两种类型的算法的加权$ l^2 $错误估计，其收敛速率$ o（h^{1/2}）$。这与[伯曼，离散的蒙格方程的收敛速率和最佳传输的定量稳定性，定理5.4]相吻合，而误差概念则不同。
Optimal transport maps and plans between two absolutely continuous measures $μ$ and $ν$ can be approximated by solving semi-discrete or fully-discrete optimal transport problems. These two problems ensue from approximating $μ$ or both $μ$ and $ν$ by Dirac measures. Extending an idea from [Gigli, On Hölder continuity-in-time of the optimal transport map towards measures along a curve], we characterize how transport plans change under perturbation of both $μ$ and $ν$. We apply this insight to prove error estimates for semi-discrete and fully-discrete algorithms in terms of errors solely arising from approximating measures. We obtain weighted $L^2$ error estimates for both types of algorithms with a convergence rate $O(h^{1/2})$. This coincides with the rate in [Berman, Convergence rates for discretized Monge--Ampère equations and quantitative stability of Optimal Transport, Theorem 5.4] for semi-discrete methods, but the error notion is different.
