Benamou, J.D. & Brenier, Y. Journal of Optimization Theory and Applications (2001) 111: 255. doi:10.1023/A:1011926116573
A time-dependent minimization problem for the computation of a mixed L2-Wasserstein distance between two prescribed density functions is introduced in the spirit of Ref. 1 for the classical Wasserstein distance. The optimum of the cost function corresponds to an optimal mapping between prescribed initial and final densities. We enforce the final density conditions through a penalization term added to our cost function. A conjugate gradient method is used to solve this relaxed problem. We obtain an algorithm which computes an interpolated L2-Wasserstein distance between two densities and the corresponding optimal mapping.
Monge-Kantorovitch mass transfer problemWasserstein distanceleast-square distanceoptimal controlconjugate gradient algorithm