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A mixed formulation of the Monge-Kantorovich equations. (English) Zbl 1132.35333

Authors’ summary: 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.

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
49J40 Variational inequalities
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
82B27 Critical phenomena in equilibrium statistical mechanics

Software:

mfem
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References:

[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 · 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 · 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 · 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 · 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 · 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 · 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 · Zbl 1108.35098
[8] J.W. Barrett and L. Prigozhin , Partial \(L^1\) Monge-Kantorovich problem: variational formulation and numerical approximation . (Submitted). · Zbl 1219.35371
[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 · 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 · 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). · Zbl 1196.35231
[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 · Zbl 1100.49015
[13] I. Ekeland and R. Temam , Convex Analysis and Variational Problems . North-Holland, Amsterdam ( 1976 ). MR 463994 | Zbl 0322.90046 · 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 · 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 · 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 · 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 · 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 . Article | Zbl 0982.65126 · 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 · 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 · 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 · Zbl 0585.65077
[22] P. Grisvard , Elliptic Problems in Nonsmooth Domains . Pitman, Massachusetts ( 1985 ). MR 775683 | Zbl 0695.35060 · 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 · Zbl 1099.49030
[24] L. Prigozhin , Variational model for sandpile growth . Eur. J. Appl. Math. 7 ( 1996 ) 225 - 235 . Zbl 0913.73079 · 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 ). arXiv
[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 · Zbl 1016.60017
[27] G. Strang , \(L^1\) and \(L^{\infty }\) approximation of vector fields in the plane . Lecture Notes in Num. Appl. Anal. 5 ( 1982 ) 273 - 288 . Zbl 0523.49014 · Zbl 0523.49014
[28] C. Villani , Topics in Optimal Transportation , Graduate Studies in Mathematics 58. AMS, Providence RI ( 2003 ). MR 1964483 | Zbl 1106.90001 · Zbl 1106.90001
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