Mixed L 2-Wasserstein Optimal Mapping Between Prescribed Density Functions

A time-dependent minimization problem for the computation of a mixed L ²-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 L ²-Wasserstein distance between two densities and the corresponding optimal mapping.