Quantitative Stability and Error Estimates for Optimal Transport Plans
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Optimal transport maps and plans between two absolutely continuous measures μ and ν can be approximated by solving semi-discrete 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, On Hölder continuity-in-time of the optimal transport map towards measures along a curve], we characterize how transport plans change under perturbation of both μ and ν. 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–Ampère equations and quantitative stability of Optimal Transport, Theorem 5.4] for semi-discrete methods, but the error notion is different.
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