Numerische Mathematik

, Volume 134, Issue 3, pp 611–636 | Cite as

Discretization of functionals involving the Monge–Ampère operator

  • Jean-David Benamou
  • Guillaume Carlier
  • Quentin Mérigot
  • Édouard Oudet
Article

Abstract

Gradient flows in the Wasserstein space have become a powerful tool in the analysis of diffusion equations, following the seminal work of Jordan, Kinderlehrer and Otto (JKO). The numerical applications of this formulation have been limited by the difficulty to compute the Wasserstein distance in dimension \(\geqslant \)2. One step of the JKO scheme is equivalent to a variational problem on the space of convex functions, which involves the Monge–Ampère operator. Convexity constraints are notably difficult to handle numerically, but in our setting the internal energy plays the role of a barrier for these constraints. This enables us to introduce a consistent discretization, which inherits convexity properties of the continuous variational problem. We show the effectiveness of our approach on nonlinear diffusion and crowd-motion models.

Mathematics Subject Classification

49M25 52B55 

Notes

Acknowledgments

The authors gratefully acknowledge the support of the French ANR, through the projects ISOTACE (ANR-12-MONU-0013), OPTIFORM (ANR-12-BS01-0007) and TOMMI (ANR-11-BSO1-014-01).

References

  1. 1.
    Agueh, M.: Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Ph.D. thesis, Georgia Institute of Technology, USA (2002)Google Scholar
  2. 2.
    Agueh, M.: Existence of solutions to degenerate parabolic equations via the Monge–Kantorovich theory. Adv. Differ. Equ. 10(3), 309–360 (2005)MathSciNetMATHGoogle Scholar
  3. 3.
    Agueh, M., Bowles, M.: One-dimensional numerical algorithms for gradient flows in the p-Wasserstein spaces. Acta Appl. Math. 125(1), 121–134 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows: in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich (2005)Google Scholar
  5. 5.
    Blanchet, A., Calvez, V., Carrillo, J.A.: Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model. SIAM J. Numer. Anal. 46(2), 691–721 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blanchet, A., Carlier, G.: Optimal transport and Cournot–Nash equilibria. arXiv:1206.6571 (2012, arXiv preprint)
  7. 7.
    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Burger, M., Carrillo, J.A., Wolfram, M.T., et al.: A mixed finite element method for nonlinear diffusion equations. Kinet. Related Models 3(1), 59–83 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Caffarelli, L.A.: Boundary regularity of maps with convex potentials. Commun. Pure Appl. Math. 45(9), 1141–1151 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Carlier, G., Lachand-Robert, T., Maury, B.: A numerical approach to variational problems subject to convexity constraint. Numer. Math. 88(2), 299–318 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Carrillo, J.A., Moll, J.S.: Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms. SIAM J. Sci. Comput. 31(6), 4305–4329 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    CGAL. Computational Geometry Algorithms Library. http://www.cgal.org
  13. 13.
    Choné, P., Le Meur, H.V.: Non-convergence result for conformal approximation of variational problems subject to a convexity constraint. Numer. Funct. Anal. Optim. 5–6(22), 529–547 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ekeland, I., Moreno-Bromberg, S.: An algorithm for computing solutions of variational problems with global convexity constraints. Numer. Math. 115(1), 45–69 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gutiérrez, C.E.: The Monge–Ampère Equation, vol. 44. Birkhauser, Basel (2001)MATHGoogle Scholar
  16. 16.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kinderlehrer, D., Walkington, N.J.: Approximation of parabolic equations using the Wasserstein metric. ESAIM Math. Model. Numer. Anal. 33(04), 837–852 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lachand-Robert, T., Oudet, É.: Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16(2), 368–379 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Maury, B., Roudneff-Chupin, A., Santambrogio, F.: A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mérigot, Q., Oudet, E.: Handling convexity-like constraints in variational problems. SIAM J. Numer. Anal 52(5), 2466–2487 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mirebeau, J.M.: Adaptive, anisotropic and hierarchical cones of discrete convex functions. arXiv:1402.1561 (2014, arXiv preprint)
  23. 23.
    Oberman, A.M.: Wide stencil finite difference schemes for the elliptic Monge–Ampère equation and functions of the eigenvalues of the hessian. Discrete Contin. Dyn. Syst. Ser. B 10(1), 221–238 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Oberman, A.M.: A numerical method for variational problems with convexity constraints. SIAM J. Sci. Comput. 35(1), A378–A396 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Oliker, V., Prussner, L.: On the numerical solution of the equation \(\frac{\partial ^2z}{\partial x^2} + \frac{\partial ^2 z}{\partial y^2} - \left(\frac{\partial ^2 z}{\partial x\partial y}\right)^2=f\). Numer. Math. 54(3), 271–293 (1988)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 44. Cambridge University Press, Cambridge (1993)CrossRefMATHGoogle Scholar
  28. 28.
    Trudinger, N.S., Wang, X.J.: The affine plateau problem. J. Am. Math. Soc. 18(2), 253–289 (2005)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Visintin, A.: Strong convergence results related to strict convexity. Commum. Partial Differ. Equ. 9(5), 439–466 (1984). doi:10.1080/03605308408820337 MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhou, B.: The first boundary value problem for Abreu’s equation. Int. Math. Res. Not. IMRN 7, 1439–1484 (2012)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jean-David Benamou
    • 1
  • Guillaume Carlier
    • 2
  • Quentin Mérigot
    • 3
  • Édouard Oudet
    • 4
  1. 1.INRIARocquencourtFrance
  2. 2.Ceremade, Université Paris-DauphineParisFrance
  3. 3.Ceremade, Université Paris-Dauphine, CNRSParisFrance
  4. 4.Laboratoire Jean KuntzmannUniv. Grenoble AlpesGrenobleFrance

Personalised recommendations