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http://cimat.repositorioinstitucional.mx/jspui/handle/1008/1103| THE L1 THEORY OF OPTIMAL TRANSPORTAND FLOW MINIMIZATION PROBLEMS | |
| Emmanuel Rivera Guasco | |
| Acceso Abierto | |
| Atribución-NoComercial | |
| MATEMÁTICAS BÁSICAS | |
| In Monge’s transport problem we want to find the most efficient way of reshaping a sand pile into another so that they can be modeled by two non-negative densities and have the same volume. Here, efficient means that transportation has to be chosen so that it minimizes the average displacement. Surprisingly, it turned out to be a very difficult problem, so a relaxation of Monge’s problem and its dual formulation was crucial. This relaxation allows to prove that Monge’s transport problem admits a unique solution whenever the cost function is strictly convex. In particular, if we have a quadratic cost, then the solution of Monge’s problem is written as the gradient of a convex function. However, in the case of the L 1 transport problem, we no longer have strict convexity, and therefore we cannot guarantee the existence of an optimal transport map. This problem aroused interest among the mathematical community since it required a more geometrical approach. In this case, we have to consider an alternative variational problem, so we can decompose the support of an optimal plan into a family of rays with desirable geometric properties. Over these rays, the optimal plan behaves as a one-dimensional transport problem and the gradient of the kantorovich’s potential has Lipschitz regularity. The main goal of this manuscript is to focus on the study of the L 1 optimal transportation problem, and its relation with a flow minimization problem. To show the relationship between these two problems, we introduce the notion of transport intensity and transport flow to prove that any vector measure can be decomposed into a cycle and a flow induced by a measure on paths. From this decomposition, we can prove that an optimal solution for Beckmann’s problem is necessarily induced by an optimal transport plan for the L 1 -Kantorovich’s problem and such a solution does not depend on the choice of the optimal plan. | |
| 26-03-2021 | |
| Trabajo de grado, maestría | |
| OTRAS | |
| Versión aceptada | |
| acceptedVersion - Versión aceptada | |
| Aparece en las colecciones: | Tesis del CIMAT |
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