Research on optimal transport

Optimal transport. Joint work with R. H. Nochetto (UMD, USA).

Link to our paper: Quantitative stability and error estimates for optimal transport plans.

In the paper Quantitative stability and error estimates for optimal transport plans, we study the errors induced by the approximation of measures, with emphasis on quadratic costs. Our work obtains the first error estimate for the error of transport maps in the fully discrete shcemes where both continuous measures are approximated with discrete measures.

We extend the ideas in (N. Gigli 2011 Proc. Edinb. Math. Soc. (2)) for transport maps to transport plans, which leads to a quantitative characterization on how transport plans change under the perturbation of measures $\mu$ and $\nu$. This leads to estimates of errors in both semidiscrete and fully discrete OT algorithms induced by the approximation of continuous measures. We obtain weighted $L^2$ error estimates for both cases, and obtain convergence rates of order $O(h^{1/2})$, which coincide with the rate in (R. J. Berman 2021 Found. Comput. Math.) for a semidiscrete method on a different notion of error.