Canonical duality approach in the approximation of optimal Monge mass transfer mapping

TLDR: In this article, a variational method was used to transform the 1-dimensional mass transfer problem into a sequence of minimization problems, which can then be converted into a nonlinear differential equation with constraints by applying the canonical duality method.

Abstract: This paper mainly addresses the Monge mass transfer problem in the 1-D case. Through an ingenious approximation mechanism, one transforms the Monge problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational method. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the duality method gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of optimal mapping for the Monge problem according to the theoretical results.