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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an:2007051/
[1] , Optimal transport maps in Monge-Kantorovich problem, Proceedings of the ICM (Beijing, 2002) III. Higher Ed. Press, Beijing (2002) 131-140. | Zbl
[2] , Lecture notes on optimal transport, in Mathematical Aspects of Evolving Interfaces, L. Ambrosio et al. Eds., Lect. Notes in Math. 1812 (2003) 1-52. | Zbl
[3] , and, Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000). | MR | Zbl
[4] , and, Minimizing flows for the Monge-Kantorovich problem. SIAM J. Math. Anal. 35 (2003) 61-97. | Zbl
[5] , and, Fast/slow diffusion and growing sandpiles. J. Diff. Eqns. 131 (1996) 304-335. | Zbl
[6] and, Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5 (2005) 333-361. | Zbl
[7] and, Dual formulations in critical state problems. Interfaces Free Boundaries 8 (2006) 347-368. | Zbl
[8] and, Partial Monge-Kantorovich problem: variational formulation and numerical approximation. (Submitted). | Zbl
[9] and, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. | Zbl
[10] , and, Shape optimization solutions via Monge-Kantorovich equation. C.R. Acad. Sci. Paris 324-I (1997) 1185-1191. | Zbl
[11] and, Free boundaries in optimal transport and Monge-Ampère obstacle problems. Ann. Math. (to appear).
[12] and, The relaxation of some classes of variational integrals with pointwise continuous-type gradient constraints. Appl. Math. Optim. 51 (2005) 251-277. | Zbl
[13] and, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). | MR | Zbl
[14] , Weak Convergence Methods for Nonlinear Partial Differential Equations, C.B.M.S. 74. AMS, Providence RI (1990). | MR | Zbl
[15] , Partial differential equations and Monge-Kantorovich mass transfer, Current Developments in Mathematics. Int. Press, Boston (1997) 65-126. | Zbl
[16] and, Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem. Mem. Amer. Math. Soc. 137 (1999). | MR | Zbl
[17] , A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121-132. | Zbl
[18] and, On a mixed finite element method for the -Laplacian. Can. Appl. Math. Q. 8 (2000) 67-78. | Zbl
[19] , Growth of a sandpile around an obstacle, in Monge Ampere Equation: Applications to Geometry and Optimization, L.A Caffarelli and M. Milman Eds., Contemp. Math. 226, AMS, Providence (1999) 55-78. | Zbl
[20] , Real Analysis: Modern Techniques and their Applications (Second Edition). Wiley-Interscience, New York (1984). | MR | Zbl
[21] and, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986). | MR | Zbl
[22] , Elliptic Problems in Nonsmooth Domains. Pitman, Massachusetts (1985). | MR | Zbl
[23] , Equivalence between some definitions for the optimal mass transport problem and for transport density on manifolds. Ann. Mat. Pura Appl. 184 (2005) 215-238. | Zbl
[24] , Variational model for sandpile growth. Eur. J. Appl. Math. 7 (1996) 225-235. | Zbl
[25] , Solutions to Monge-Kantorovich equations as stationary points of a dynamical system. arXiv:math.OC/0507330, http://xxx.tau.ac.il/abs/math.OC/ 0507330 (2005).
[26] and, Numerical and analytical results for the transportation problem of Monge-Kantorovich. Metrika 51 (2000) 245-258. | Zbl
[27] , and approximation of vector fields in the plane. Lecture Notes in Num. Appl. Anal. 5 (1982) 273-288. | Zbl
[28] , Topics in Optimal Transportation, Graduate Studies in Mathematics 58. AMS, Providence RI (2003). | MR | Zbl
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