Interfaces and Free Boundaries

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Volume 11, Issue 2, 2009, pp. 201–238
DOI: 10.4171/IFB/209

Published online: 2009-06-30

Partial L1 Monge–Kantorovich problem: variational formulation and numerical approximation

John W. Barrett[1] and Leonid Prigozhin[2]

(1) Imperial College London, United Kingdom
(2) Ben Gurion University of the Negev, Beer-Sheba, Israel

We consider the Monge–Kantorovich problem with transportation cost equal to distance and a relaxed mass balance condition: instead of optimally transporting one given distribution of mass onto another with the same total mass, only a given amount of mass, m, has to be optimally transported. In this partial problem the given distributions are allowed to have different total masses and m should not exceed the least of them. We derive and analyze a variational formulation of the arising free boundary problem in optimal transportation. Furthermore, we introduce and analyse the finite element approximation of this formulation using the lowest order Raviart–Thomas element. Finally, we present some numerical experiments where both approximations to the optimal transportation domains and the optimal transport between them are computed.

Keywords: Monge–Kantorovich problem, optimal transportation, free boundary, variational formulation, finite elements, augmented Lagrangian, convergence analysis

Barrett John, Prigozhin Leonid: Partial L1 Monge–Kantorovich problem: variational formulation and numerical approximation. Interfaces Free Bound. 11 (2009), 201-238. doi: 10.4171/IFB/209