Quantitative Stability and Error Estimates for Optimal Transport Plans

Optimal transport maps and plans between two absolutely continuous measures $\mu$ and $\nu$ can be approximated by solving semi-discrete 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, On H\"older continuity-in-time of the optimal transport map towards measures along a curve], we characterize how transport plans change under perturbation of both $\mu$ and $\nu$. 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--Amp\`ere equations and quantitative stability of Optimal Transport, Theorem 5.4] for semi-discrete methods, but the error notion is different.