Numerical solution of the three dimensional Optimal Transport Problem

In this thesis we analyze a model introduced in [14, 17] and its extensions [14, 15]. These works conjectured a new formulation of Optimal Transport, an expanding area of mathematics whose aim is the identification of the most efficient strategy to reallocate resources from one place to another. The numerical approximation of the equations of the model represents a simple yet effective numerical approach to solve Optimal Transport problems. However, the numerical scheme was analyzed only in the two dimensional case. The aim of this thesis is to exploit the model to solve three dimensional Optimal Transport problems, where few examples of numerical solution are known from the literature. We present all the non trivial challenges required by the three dimensional extension, together with an ample series of numerical experiments, that confirms the conjectured equivalence with the Optimal Transport problem. The results show that the numerical scheme is robust and efficient, with ample space for improvement from the computational point of view.