Numerical solutions of the mass transfer problem

¹ Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada; dubucs@dms.umontreal.ca
² Department of Mathematics, Columbia Union College, 7600 Flower Avenue, Takoma Park, MD, 20912, USA; ikagabo@cuc.edu

Received: 10 December 2003
Accepted: 7 November 2005

Abstract

Let μ and ν be two probability measures on the real line and let c be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure ξ whose marginals coincide with μ and ν, and whose total cost ∫∫ c(x,y)dξ(x,y) is minimum. In this paper we present three algorithms to solve numerically this Monge-Kantorovitch problem when the commodity being shipped is one-dimensional and not necessarily confined to a bounded interval. We illustrate these numerical methods and determine the convergence rate.