$L^\alpha$-Regularization of the Beckmann Problem

TL;DR

This paper explores $L^ ext{alpha}$ regularization in the Beckmann optimal transport problem, ensuring unique, integrable solutions and introducing a semi-smooth Newton method for efficient numerical resolution.

Contribution

It introduces an $L^ ext{alpha}$ regularization approach that guarantees uniqueness and integrability of solutions, along with a semi-smooth Newton scheme for numerical computation.

Findings

Regularization yields unique, integrable solutions.

Development of a semi-smooth Newton numerical scheme.

Approximation results for vanishing regularization.

Abstract

We investigate the problem of optimal transport in the so-called Beckmann form, i.e. given two Radon measures on a compact set, we seek an optimal flow field which is a vector valued Radon measure on the same set that describes a flow between these two measures and minimizes a certain linear cost function. We consider $L^\alpha$ regularization of the problem, which guarantees uniqueness and forces the solution to be an integrable function rather than a Radon measure. This regularization naturally gives rise to a semi-smooth Newton scheme that can be used to solve the problem numerically. Besides motivating and developing the numerical scheme, we also include approximation results for vanishing regularization in the continuous setting.