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A Mixed Formulation of the Monge-Kantorovich Equations

Published online by Cambridge University Press:  15 December 2007

John W. Barrett  and
Leonid Prigozhin
John W. Barrett
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK. [email protected]
Leonid Prigozhin
Affiliation:
Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 84990, Israel.
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Abstract

We introduce and analyse a mixed formulation of theMonge-Kantorovich equations, which express optimality conditions forthe mass transportation problem with cost proportional to distance.Furthermore, we introduce and analyse the finite elementapproximation of this formulation using the lowest orderRaviart-Thomas element. Finally, we present some numericalexperiments, where both the optimal transport density and theassociated Kantorovich potential are computed for a coupling problemand problems involving obstacles and regions of cheaptransportation.

Keywords

Monge-Kantorovich problemoptimal transportationmixed methodsfinite elementsexistenceconvergence analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 6 , November 2007 , pp. 1041 - 1060
Copyright
© EDP Sciences, SMAI, 2007

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References

L. Ambrosio, Optimal transport maps in Monge-Kantorovich problem, Proceedings of the ICM (Beijing, 2002) III . Higher Ed. Press, Beijing (2002) 131–140.
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.
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000).
Angenent, S., Haker, S. and Tannenbaum, A., Minimizing flows for the Monge-Kantorovich problem. SIAM J. Math. Anal. 35 (2003) 61–97. CrossRef
Aronson, G., Evans, L.C. and Fast, Y. Wu/slow diffusion and growing sandpiles. J. Diff. Eqns. 131 (1996) 304–335. CrossRef
Bahriawati, C. and Carstensen, C., Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5 (2005) 333–361. CrossRef
Barrett, J.W. and Prigozhin, L., Dual formulations in critical state problems. Interfaces Free Boundaries 8 (2006) 347–368.
J.W. Barrett and L. Prigozhin, Partial L 1 Monge-Kantorovich problem: variational formulation and numerical approximation. (Submitted).
Benamou, J.-D. and Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375–393. CrossRef
G. Bouchitté, G. Buttazzo and P. Seppecher, Shape optimization solutions via Monge-Kantorovich equation. C.R. Acad. Sci. Paris 324-I (1997) 1185–1191.
L.A. Caffarelli and R.J. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems. Ann. Math. (to appear).
De Arcangelis, R. and Zappale, E., The relaxation of some classes of variational integrals with pointwise continuous-type gradient constraints. Appl. Math. Optim. 51 (2005) 251–277. CrossRef
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).
L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, C.B.M.S. 74. AMS, Providence RI (1990).
L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current Developments in Mathematics. Int. Press, Boston (1997) 65–126.
L.C. Evans and W. Gangbo, Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem. Mem. Amer. Math. Soc. 137 (1999).
Farhloul, M., A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121–132. CrossRef
Farhloul, M. and Manouzi, H., On a mixed finite element method for the p-Laplacian. Can. Appl. Math. Q. 8 (2000) 67–78.
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.
G.B. Folland, Real Analysis: Modern Techniques and their Applications (Second Edition). Wiley-Interscience, New York (1984).
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Massachusetts (1985).
Pratelli, A., Equivalence between some definitions for the optimal mass transport problem and for transport density on manifolds. Ann. Mat. Pura Appl. 184 (2005) 215–238. CrossRef
Prigozhin, L., Variational model for sandpile growth. Eur. J. Appl. Math. 7 (1996) 225–235.
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).
Rüschendorf, L. and Uckelmann, L., Numerical and analytical results for the transportation problem of Monge-Kantorovich. Metrika 51 (2000) 245–258.
Strang, G., L 1 and L ∞ approximation of vector fields in the plane. Lecture Notes in Num. Appl. Anal. 5 (1982) 273–288.
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58. AMS, Providence RI (2003).