Optimal transport maps and plans between two absolutely continuous measures |$\mu$| and |$\nu$| can be approximated by solving semidiscrete or fully discrete optimal transport problems. These two problems ensue from approximating |$\mu$| or both |$\mu$| and |$\nu$| by Dirac measures. Extending an idea from Gigli (2011, On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proc. Edinb. Math. Soc. (2), 54, 401–409), we characterize how transport plans change under the perturbation of both |$\mu$| and |$\nu$|⁠. We apply this insight to prove error estimates for semidiscrete 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 Theorem 5.4 in Berman (2018, Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.

中文翻译：

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最优运输计划的定量稳定性和误差估计

两个绝对连续测度之间的最优交通图和规划|$\mu$| 和|$\nu$| 可以通过求解半离散或完全离散的最优传输问题来近似。这两个问题源于逼近|$\mu$| 或两者|$\mu$| 和|$\nu$| 通过狄拉克措施。扩展 Gigli (2011, On HölderContinuous-in-time of the best transport map to measure on a curve. Proc. Edinb. Math. Soc. (2) , 54, 401–409) 中的想法，我们描述了交通计划如何|$\mu$|的扰动下的变化 和|$\nu$|⁠. 我们应用这一见解来证明半离散和完全离散算法的误差估计，这些误差估计仅由近似度量引起。我们得到加权|$L^2$| 两种算法的误差估计都具有收敛速度|$O(h^{1/2})$|⁠。这与 Berman 定理 5.4 中的比率（2018 年，离散化 Monge-Ampère 方程的收敛率和最优传输的定量稳定性。预印本可在 arXiv:1803.00785 中获得）用于半离散方法，但误差概念不同。