Quantitative stability and error estimates for optimal transport plans

Abstract

Optimal transport maps and plans between two absolutely continuous measures μ and ν can be approximated by solving semidiscrete 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 (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 μ and ν. 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 L2 error estimates for both types of algorithms with a convergence rate O(h1/2). 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.