A mixed formulation of the Monge-Kantorovich equations

Mathematical Modelling and Numerical Analysis

Volumn 41, Issue 6, 2007, Pages 1041-1060

  Barrett, John W ^a   Prigozhin, Leonid ^b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

Author keywords

Convergence analysis; Existence; Finite elements; Mixed methods; Monge Kantorovich problem; Optimal transportation

Indexed keywords

EID: 37449033900     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2007051     Document Type: Article

References (28)

1. L. Ambrosio, Optimal transport maps in Monge-Kantorovich problem, Proceedings of the ICM (Beijing, 2002) III. Higher Ed. Press, Beijing (2002) 131-140.
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.
3. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000).
4. S. Angenent, S. Haker and A. Tannenbaum, Minimizing flows for the Monge-Kantorovich problem. SIAM J. Math. Anal. 35 (2003) 61-97.
5. G. Aronson, L..C Evans and Y. Wu, Fast/slow diffusion and growing sandpiles. J. Diff. Eqns. 131 (1996) 304-335.
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.
7. J.W. Barrett and L. Prigozhin, Dual formulations in critical state problems. Interfaces Free Boundaries 8 (2006) 347-368.
8. 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.
10. G. Bouchitté, G. Buttazzo and P. Seppecher, Shape optimization solutions via Monge-Kantorovich equation. CR. Acad. Sci. Paris 324-1 (1997) 1185-1191.
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.
13. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).
14. L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, C.B.M.S. 74. AMS, Providence RI (1990).
15. L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current Developments in Mathematics. Int. Press, Boston (1997) 65-126.
16. L.C. Evans and W. Gangbo, Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem. Mem. Amer. Math. Soc. 137 (1999).
17. M. Farhloul, A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121-132.
18. M. Farhloul and H. Manouzi, On a mixed finite element method for the p-Laplacian. Can. Appl. Math. Q. 8 (2000) 67-78.
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.
20. G.B. Folland, Real Analysis: Modern Techniques and their Applications (Second Edition). Wiley-Interscience, New York (1984).
21. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986).
22. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Massachusetts (1985).
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.
24. L. Prigozhin, Variational model for sandpile growth. Eur. J. Appl. Math. 7 (1996) 225-235.
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.
27. ∞ approximation of vector fields in the plane. Lecture Notes in Num. Appl. Anal. 5 (1982) 273-288.
28. C Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58. AMS, Providence RI (2003).