A Multilevel Method for the Solution of Time Dependent Optimal Transport

In this paper we present a new computationally efficient numerical scheme for the minimizing flow for the computation of the optimal $L_2$mass transport mapping using the fluid approach. We review the method and discuss its numerical properties. We then derive a new scaleable, efficient discretization and a solution technique for the problem and show that the problem is equivalent to a mixed form formulation of a nonlinear fluid flow in porous media. We demonstrate the effectiveness of our approach using a number of numerical experiments.