Techniques for continuous optimal transport problem

In this work, we consider and compare several different numerical methods to solve the classic continuous optimal transport problem between two probability densities, while minimizing the cost function given by the squared Euclidean distance. Classically, the problem reduces to having to solve a second order elliptic PDE (the Monge-Ampére PDE). An alternative is to consider the so-called fluid dynamics formulation of Benamou and Brenier. One of our goals in this work is to compare two numerical methods used for the fluid-dynamics formulation with a direct discretization of the Monge-Ampére PDE. Finally, we introduce a very natural new class of problems, which we call separable, for which we devise very accurate methods. We give implementation details of all the different methods, and extensive testing on many different problems which we created in order to provide a fairly complete arrays of the typical difficulties one encounters, and to highlight the benefits of different methods. With the same level of attention to implementation details, some insight into the relative merits of the different techniques emerges and we can draw conclusions and provide recommendations on what techniques to adopt for this problem.