Quantitative Stability and Error Estimates for Optimal Transport Plans

Understand

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.