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Optimal transport maps and plans between two absolutely continuous measures$\mu$ and $\nu$ can be approximated by solving semi-discrete or fully-discreteoptimal transport problems. These two problems ensue from approximating $\mu$or both $\mu$ and $\nu$ by Dirac measures. Extending an idea from [Gigli, OnH\"older continuity-in-time of the optimal transport map towards measures alonga curve], we characterize how transport plans change under perturbation of both$\mu$ and $\nu$. We apply this insight to prove error estimates forsemi-discrete and fully-discrete algorithms in terms of errors solely arisingfrom approximating measures. We obtain weighted $L^2$ error estimates for bothtypes of algorithms with a convergence rate $O(h^{1/2})$. This coincides withthe rate in [Berman, Convergence rates for discretized Monge--Amp\`ereequations and quantitative stability of Optimal Transport, Theorem 5.4] forsemi-discrete methods, but the error notion is different.

Quantitative Stability and Error Estimates for Optimal Transport Plans