We introduce and analyse a mixed formulation of the Monge-Kantorovich equations, which express optimality conditions for the mass transportation problem with cost proportional to distance. 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 the optimal transport density and the associated Kantorovich potential are computed for a coupling problem and problems involving obstacles and regions of cheap transportation.
Mots-clés : Monge-kantorovich problem, optimal transportation, mixed methods, finite elements, existence, convergence analysis
@article{M2AN_2007__41_6_1041_0,
author = {Barrett, John W. and Prigozhin, Leonid},
title = {A mixed formulation of the {Monge-Kantorovich} equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1041--1060},
publisher = {EDP-Sciences},
volume = {41},
number = {6},
year = {2007},
doi = {10.1051/m2an:2007051},
mrnumber = {2377106},
zbl = {1132.35333},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2007051/}
}
TY - JOUR AU - Barrett, John W. AU - Prigozhin, Leonid TI - A mixed formulation of the Monge-Kantorovich equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 1041 EP - 1060 VL - 41 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2007051/ DO - 10.1051/m2an:2007051 LA - en ID - M2AN_2007__41_6_1041_0 ER -
%0 Journal Article %A Barrett, John W. %A Prigozhin, Leonid %T A mixed formulation of the Monge-Kantorovich equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 1041-1060 %V 41 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2007051/ %R 10.1051/m2an:2007051 %G en %F M2AN_2007__41_6_1041_0
Barrett, John W.; Prigozhin, Leonid. A mixed formulation of the Monge-Kantorovich equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1041-1060. doi : 10.1051/m2an:2007051. http://archive.numdam.org/articles/10.1051/m2an:2007051/
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