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.
Highlights
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
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
We focus on Haar type multiresolution analysis (MRA) on Rn; the constructions of this kind of MRA are associated with the symmetries of the spaces; the approximations are related to the geometrical properties of the space
Summary
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. 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); a general procedure is given, but the examples are in a two-dimensional cube and use the dyadic partition of the cube for approximation. We focus on Haar type MRA on Rn; the constructions of this kind of MRA are associated with the symmetries of the spaces; the approximations are related to the geometrical properties of the space. Abstract and Applied Analysis 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.
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