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].
3. Approximation of Measures Using Multiresolution Analysis
3.1. Absolutely Continuous Measures with respect to Lebesgue Measures
We consider the following conditions:(A1) is a compact subset of .(A2)The measure is absolutely continuous with respect to , where is the Lebesgue measure.
Condition (A2) guarantees the existence of functions , where is the Radon-Nikodym derivative with respect to Lebesgue measure .
In this analysis, we also assume the following extra condition:(A3)The functions also belong in .
Notice that
Let be an -MRA on ; the elements of are given by scaling functions , the latice , finite group , and dilatation .
Remark 2. Note that the classical multiresolution analysis of Haar on is a particular case of -MRA on , where the complete lattice is , the group is trivial, and the dilation associated is . In this case, we have that scaling function is , where is the fundamental domain associated with the action of on .
We denote by the projection from into , which is given by for all . Moreover, if the function has support in , then the above sum is finite, since is compact.
From now on, we shall only functions with support in the compact set and we have ; that is, each function can be considered in , where if .
We know that the -MRA is dense in ; thus we have that Using the fact that is compact, we obtain that
From the above equation, we define the approximation of the measure to the level bywhich are measures in for each . Now we want to prove that these measures are probability measures.
Definition 3. The expectation with respect to Lebesgue measure of function is defined by where is a Lebesgue measurable function. Also, the conditional expectation of given , with being a Lebesgue measurable set, is defined by .
In particular, the expectation on the Lebesgue measure satisfies the following property.
Theorem 4. Consider that is fixed and . Then , where is the projection of the level associated with -MRA .
Proof. We take ; now we calculate ; thus We have that the functions , where is the translation and dilation of the fundamental region and ; this value does not depend on . From this, we have that Using the above equations, we have thatMoreover, using the fact that for , it is clear that
As immediate consequence of the previous theorem, we get the following result.
Corollary 5. We suppose that the measure on satisfies ; then the sequences of probability measures given by (15) converge to in and sense.
3.2. Non-Absolutely-Continuous Measures with respect to Lebesgue Measure
Now, we consider that is a probability measure on the compact set , which is unnecessarily absolutely continuous measure with respect to Lebesgue measure . Note that each element of sequence (15) can be written byfor all Borel measurable sets on . Moreover, these approximations are well defined for every measure on .
Definition 6. Let be a sequence of probability measures on a metric space . We say that converges weakly to and denote if as for all bounded continuous functions on , where .
Theorem 7. Given a measure on the compact set , the sequence of measures on defined in (21) converge weakly to measure .
Proof. Let be a continuous and bounded real function. From the fact that is compact, we obtain that is absolutely continuous function on ; thus, given , there exists such that We take such that for all implies for all and . Moreover, we know that there exist such that In consequence, we obtain the following relations: we take an element to obtain the following relations: Therefore,
4. Discretization of the Monge-Kantorovich Problem Using Multiresolution Analysis
Let be the linear space of finite signed on and let be the set of all nonnegative measures in . Given , we denote by and the marginal of on and , respectively; that is, for all sets and , such that and are measurable, respectively.
The Monge-Kantorovich mass transfer problem is given as follows:
A measure is said to be a feasible solution for MK problem if it satisfies (28) and is finite. The MK problem is called consistent if the set of feasible solutions is nonempty, in which case its optimal value is defined as
It is said that the MK problem is solvable if there is a feasible solution that attains the optimal value. In this case, is called an optimal solution for the MK problem and the value is written as .
Note that since and are probability measures, a feasible solution for MK is also a probability measure. Moreover, if is a continuous function on and and are compact subsets on , then the product measure is a feasible solution. Therefore the MK problem is consistent, and so the MK problem is solvable in this case.
We assume the following conditions: and are compact subsets of and is a bounded continuous function.
Let and be -MRA on with scaling functions and , lattices and , finite groups and , and dilatations and , respectively. We can obtain a new -MRA on with scaling functions , lattice , finite group , and dilatation . The projections of these -MRA to the level are denoted by , , and .
Now we proceed to discretization of the MK-problem using the results of Section 2; then we have that , , and are the projections to level of the respective -MRA; thus, using these measures, we obtain the discretization of the MK-problem in the level , which is given by
We shall give explicit expressions for the discretization of the measures and the cost function associated with MK-problem using the -MRA, which are given bywhere , , and are nonnegative real numbers. Now we calculate ; thus and hence, using the above equation and the orthonormality of family , we obtain
We consider that is a fixed element in and we have that is the region associated with for ; then we obtain
On the other hand, we have that From the above equations, we have that the condition is equivalent to
Analogously, for all the elements of , we have thatUsing Theorem 4, we obtain that
In summary, we have that the problem given by (30) is equivalent to the following problem:
The problem , given the above system, has a feasible solution, so we know that is bounded. Then we have that the problem has an optimal solution; for more details, see Chapter 10 in [7].
Let be the optimal solution of MK problem given in (28). We use the optimal solution to induce a sequence of measures such that (see Section 3.2). Let be the optimal solution of problem given in (30).
For , we defined the following -algebras: where denotes the -algebra generated by the sets with and the lattices , , and are defined as in (3).
Definition 8. Let be a sub--algebra of , and let be a random variable. We say that the random variable is the conditional expectation of with respect to and denote it by if and only if (i).(ii) is -measurable.(iii), for all .
Remark 9. Given , we have that There are analogous results for and through the conditional expectations of and , respectively.
Proposition 10. Let be a feasible solution of the MK problem; then is a feasible solution of the -problem.
Proof. We suppose that satisfies (28). Then we have the following relations: where and a measurable set. Similarly, we obtain that . Therefore the result is as follows.
Remark 11. If is an optimal solution of the MK problem, then is a feasible solution of the -problem.
Proposition 12. Given and , one has where and are feasible solutions of the MK and problems, respectively.
Proof. Let be a generator element of -algebra . Then In above equation, we use the fact that for all in consequence for all . Analogously, we have that .
Definition 13. Let be a topological space, and let be -algebra on that contains the topology . Let be a collection of measures defined on . The collection is called tight if, for any , there is a compact subset of such that, for all measures , , where is the total variation measure of .
Proposition 14. Consider the sequence of probability measures, where is the optimal solution of for each . Then there exists a subsequent and a probability measure such that weakly. Moreover, the probability measure is an optimal solution of .
Proof. We know that and are compact sets, provided that the measures are tight for each . Now, using Prokhorov’s Theorem (for details, see Chapter 1 in [8]), there exists a subsequence of and a probability measure such that weakly.
Notice that is an optimal solution of ; thus, we obtain Using the Theorem, we have that and weakly. From this fact, it is clear that is a feasible solution of MK.
Now, we consider as the optimal solution of MK. from Proposition 10, we have that is feasible solution of . It follows that taking the limit when , we have On the other hand, is a feasible solution of MK; then which completes the proof of the theorem.
5. An Illustrative Example of This Method
In this section, we present an example of the ideas presented in the previous section. Let be the square with vertices in the points and let be a hexagon with vertices being in set We consider the function defined by . We claim to solve the following problem:where and are the normalized Lebesgue measures to and , respectively; that is, . We consider the following -MRA on : (1), where the fundamental region is and the scaling function , the lattice is , the finite group is the trivial group, and the dilation .(2), where the scaling function , with being the triangle with vertices in , , and and ; the lattice is , where ; the finite group is the rotation group of a regular hexagon with vertices in the unit circle; the scaling is given by , where .
We denote by and the squares and the triangles associated with level of the multiresolution analysis presented above. Since is a continuous function in each pair of sets , we have that is approximated to , where and are the centroids of and , respectively.
In particular, we present a discretization of MK problem given by (51) for the level ; the graphical description of the process of discretization is given in Figure 3 (which was generated using Mathematica 11.1.0).
Using and in and , respectively, notice that is the union of congruent squares with disjoined interiors. Similarly, the hexagon is the union by congruent equilateral triangles , with disjoined interiors. Note that and .
In consequence, a discrete approximation for the problem defined in (51) is as follows:The optimal solution of this linear programming problem is 4.02996. In Table 1 we present the solution for some levels.
6. Conclusions
The application of the Haar type multiresolution analysis (MRA) to the problem of Monge-Kantorovich allows us to obtain a discretization scheme for this problem at each level of the MRA; moreover, this MRA is based on the symmetries of the underlying space. This is an advantage because it provided us a natural method of discretization based on the geometry. Each level of the MRA induces a soluble finite linear programming problem. So, we obtain a sequence of optimal solutions of these transport problems and this sequence converges weakly to the optimal solution of the original problem.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was partially supported by the CONACYT project, México. The second author is grateful to Instituto Politécnico Nacional, México, for financial support for development of this work.