Abstract and Applied Analysis

Volume 2018, Article ID 1764175, 9 pages

https://doi.org/10.1155/2018/1764175

## Multiresolution Analysis Applied to the Monge-Kantorovich Problem

^{1}Facultad de Matemáticas, Universidad Veracruzana, Xalapa, VER, Mexico^{2}Escuela Superior de Ingeniera Mecánica y Eléctrica, Instituto Politécnico Nacional, Mexico City, Mexico

Correspondence should be addressed to Armando Sánchez-Nungaray; xm.vu@zehcnasmra

Received 17 February 2018; Accepted 4 April 2018; Published 3 June 2018

Academic Editor: Turgut Öziş

Copyright © 2018 Armando Sánchez-Nungaray et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We give a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problem that approximates the original MK problem for each of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem.

#### 1. Introduction

The optimal transport problem was first formulated by Monge in 1781 and concerned finding the optimal way in the sense of minimal transportation cost of moving a pile of soil from one site to another. This problem was given a modern formulation in the work of Kantorovich in 1942 and so is now known as the Monge-Kantorovich problem.

On the other hand, a big advantage over schemes of approximation was given in the seminal article [1]; it introduced approximation schemes for infinite linear program; in particular, it showed that under suitable assumptions the program’s optimum value can be finite-dimensional linear programs and that, in addition, every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution for the original problem. An example given in this article is the Monge-Kantorovich mass transfer (MK) problem on the space itself, where the underlying space is compact.

In [2], a general method of approximation for the MK problem is given, where and are Polish spaces; however, this method is noneasier implementation. Later, in [3], a scheme of approximation of MK problem is provided, which consists in giving a sequence of finite transportation problems underlying original MK problem (the space is compact); nevertheless, a general procedure is given, but the examples are in a two-dimensional cube and use the dyadic partition of the cube for approximation. Our objective is to give other families to approximate this kind of problems, based on Haar type multiresolution analysis (MRA). The advantage of using this technique is that we select a multiresolution analysis that is constructed according to the symmetries of the underlying space. Therefore, the new schemes of approximation take a lot of characteristics of the space into consideration. Note that the dyadic partition of the cube is a particular case of the MRA type Haar using translations and dilations of the underlying space.

The MRA is an important method to approximate functions in different context (signal processing, differential equations, etc.). In particular, we focus on Haar type MRA on ; the constructions of this kind of MRA are associated with the symmetries of the spaces; thus the approximations are related to the geometrical properties of the space. In this construction, the symmetries that we use are general dilations, rotations, reflections, translations, and so forth; for more details, see [4–6].

The main objective of this paper is giving a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problems that approximate the original MK problem for each level of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem.

This paper is organized as follows. In Section 2, we introduce the basic elements of the Haar type multiresolution analysis and we give some examples of this kind of MRA. In Section 3, we present the approximation of probability measures using Haar type MRA. In Section 5, we apply the Haar type multiresolution analysis to MK problem for each level of this MRA; thus for each level, this new problem is equivalent to transport problem. Moreover, we prove that the optimal solution of MK problems is equal to the limit of the optimal solutions of underlying transportation problems when the level of the MRA goes to infinity. Finally, we give an illustrative example of this method.

#### 2. Haar Type Multiresolution Analysis

We introduce the basic concepts of Haar type multiresolution analysis, following Gröchenig and Madych in [4] and Guo et al. in [5]. Similar results have been obtained independently by Krishtal et al. to be published in [6].

Let be a lattice such that for any . The classical multiresolution analysis (MRA) associated with a sequence of dilations , where , is a sequence of closed subspaces of , which satisfies the following conditions:(i).(ii).(iii).(iv).(v)There exists such that , , is an orthonormal basis for .

Let be a finite subgroup of such that for all and . The operator generator by the dilations , , and the translations , form a group. The relation allows us to define the operation to given by and we obtain a new group denoted by . The -invariant spaces are the closed subspaces of such that whenever , , and . This leads us to the following version of (v): () There exists such that , and , is an orthonormal basis for .

In consequence, we have the following concept.

*Definition 1. *Let be dilatation set, let be a finite subgroup of with , and let be a complete lattice such that . The multiresolution analysis associated with the dilation set and the group or -MRA is a collection of closed subspaces of , which satisfies conditions (i), (ii), (iii), (iv), and ().

The classical MRA is considered as an -MRA when is the trivial group. Note that the space is not generated by the -translations of the single scaling function ; however, the relation and the conditions imply that the functions , with , are the generators of . Also, we have the following set:Note that and .

We consider the scaling function , where is the characteristic function of , , and . The region satisfieswhere for and is the action of on . In addition, the symbol denotes the translation and scaling of the region by and , respectively. Thus, the function is denoted by . And so we have the following relation:

Finally, we define as the orthogonal projection from to which is given byfor all .

We show some examples of the multiresolution analysis associated with the dilation set and the finite group :(1)We take a matrix and the group , , of symmetries of unit square; thus, and for . Let be the triangular region with vertices in , , and ; we denote for . If , then the system is an orthogonal basis for its closed span . The space is the subspace of , consisting of all square integrable functions that are constant on each -translate of the triangles , . Thus, the spaces , , consist of all functions in , which are constant in each -translate of the triangles , , in consequence . Hence, is an -MRA with as a scaling function. Figure 1 is reproduced from Krishtal et al. (2007) [under the Creative Commons Attribution License/public domain].(2)Let be the group generated by the matrix . This group has order of 6 and is the counter-clockwise rotation by radians. Consider the hexagon with vertices in set Let be the triangle with vertices in , , and . We define for . We take and we define . The set of all functions that are constant on -translates of triangles , , is the space . Moreover, the elements of translate the center of the hexagon to the point of , and so we have a partition of . Let and the function , with . The MRA is obtained by the system where the spaces are -dilatation of , for . Figure 2 can be found in [6].