最优输运计划的定量稳定性和误差估计
Quantitative Stability and Error Estimates for Optimal Transport Plans
摘要：
两个绝对连续测度$μ$和$ν$之间的最优输运映射和计划可以通过求解半离散或全离散最优输运问题来近似。这两个问题是由Dirac测度对$μ$或对$μ$和$ν$两者的逼近引起的。扩展的想法[吉利，在霍尔德连续时间的最优输运地图对措施沿曲线]，我们描述了如何运输计划的变化扰动下的$μ$和$ν$。我们应用这种见解来证明半离散和全离散算法的误差估计的误差只产生于近似措施。我们得到加权$L^2$误差估计的收敛速度$O(h^{1/2})$的两种类型的算法。这与半离散方法[Berman，收敛速度for discretized蒙日-安培方程and quantitative stability of最优输运，Theorem 5.4]中的速率一致，但误差概念不同。
英文摘要：
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.
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