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

TL;DR

This paper introduces a canonical duality approach to approximate solutions for the 1-D Monge mass transfer problem, transforming it into a sequence of solvable minimization and differential equations with proven existence and uniqueness.

Contribution

It develops a novel approximation mechanism using canonical duality to solve the Monge mass transfer problem in one dimension, providing theoretical guarantees.

Findings

Sequence of nonlinear differential equations with solutions

Existence and uniqueness of solutions established

Constructed approximation of optimal mapping

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.