Efficient Method to Solve the Monge–Kantarovich Problem Using Wavelet Analysis

Abstract

In this paper, we present and justify a methodology to solve the Monge–Kantorovich mass transfer problem through Haar multiresolution analysis and wavelet transform with the advantage of requiring a reduced number of operations to carry out. The methodology has the following steps. We apply wavelet analysis on a discretization of the cost function level j and obtain four components comprising one corresponding to a low-pass filter plus three from a high-pass filter. We obtain the solution corresponding to the low-pass component in level j−1 denoted by μj−1*, and using the information of the high-pass filter components, we get a solution in level j denoted by μ^j. Finally, we make a local refinement of μ^j and obtain the final solution μjσ.