Barrett, John W.; Prigozhin, Leonid
A mixed formulation of the Monge-Kantorovich equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 6 , p. 1041-1060
Zbl 1132.35333 | MR 2377106 | 1 citation dans Numdam
doi : 10.1051/m2an:2007051
URL stable : http://www.numdam.org/item?id=M2AN_2007__41_6_1041_0

Classification:  35D05,  35J85,  49J40,  65N12,  65N30,  82B27
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.

Bibliographie

[1] L. Ambrosio, Optimal transport maps in Monge-Kantorovich problem, Proceedings of the ICM (Beijing, 2002) III. Higher Ed. Press, Beijing (2002) 131-140. Zbl 1005.49030

[2] L. Ambrosio, 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 1047.35001

[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000). MR 1857292 | Zbl 0957.49001

[4] S. Angenent, S. Haker and A. Tannenbaum, Minimizing flows for the Monge-Kantorovich problem. SIAM J. Math. Anal. 35 (2003) 61-97. Zbl 1042.49040

[5] G. Aronson, L.C. Evans and Y. Wu, Fast/slow diffusion and growing sandpiles. J. Diff. Eqns. 131 (1996) 304-335. Zbl 0864.35057

[6] C. Bahriawati and C. Carstensen, Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5 (2005) 333-361. Zbl 1086.65107

[7] J.W. Barrett and L. Prigozhin, Dual formulations in critical state problems. Interfaces Free Boundaries 8 (2006) 347-368. Zbl 1108.35098

[8] J.W. Barrett and L. Prigozhin, Partial L 1 Monge-Kantorovich problem: variational formulation and numerical approximation. (Submitted).

[9] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. Zbl 0968.76069

[10] G. Bouchitté, G. Buttazzo and P. Seppecher, Shape optimization solutions via Monge-Kantorovich equation. C.R. Acad. Sci. Paris 324-I (1997) 1185-1191. Zbl 0884.49023

[11] L.A. Caffarelli and R.J. Mccann, Free boundaries in optimal transport and Monge-Ampère obstacle problems. Ann. Math. (to appear).

[12] R. De Arcangelis and E. Zappale, The relaxation of some classes of variational integrals with pointwise continuous-type gradient constraints. Appl. Math. Optim. 51 (2005) 251-277. Zbl 1100.49015

[13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). MR 463994 | Zbl 0322.90046

[14] L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, C.B.M.S. 74. AMS, Providence RI (1990). MR 1034481 | Zbl 0698.35004

[15] L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current Developments in Mathematics. Int. Press, Boston (1997) 65-126. Zbl 0954.35011

[16] L.C. Evans and W. Gangbo, Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem. Mem. Amer. Math. Soc. 137 (1999). MR 1464149 | Zbl 0920.49004

[17] M. Farhloul, A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121-132. Zbl 0909.65086

[18] M. Farhloul and H. Manouzi, On a mixed finite element method for the p-Laplacian. Can. Appl. Math. Q. 8 (2000) 67-78. Zbl 0982.65126

[19] M. Feldman, 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 0924.35176

[20] G.B. Folland, Real Analysis: Modern Techniques and their Applications (Second Edition). Wiley-Interscience, New York (1984). MR 767633 | Zbl 0549.28001

[21] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986). MR 851383 | Zbl 0585.65077

[22] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Massachusetts (1985). MR 775683 | Zbl 0695.35060

[23] A. Pratelli, 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 1099.49030

[24] L. Prigozhin, Variational model for sandpile growth. Eur. J. Appl. Math. 7 (1996) 225-235. Zbl 0913.73079

[25] L. Prigozhin, 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] L. Rüschendorf and L. Uckelmann, Numerical and analytical results for the transportation problem of Monge-Kantorovich. Metrika 51 (2000) 245-258. Zbl 1016.60017

[27] G. Strang, L 1 and L approximation of vector fields in the plane. Lecture Notes in Num. Appl. Anal. 5 (1982) 273-288. Zbl 0523.49014

[28] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58. AMS, Providence RI (2003). MR 1964483 | Zbl 1106.90001