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 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 Holder 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-Ampere equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.