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.
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References
Agueh, M.: Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Ph.D. thesis, Georgia Institute of Technology, USA (2002)
Agueh, M.: Existence of solutions to degenerate parabolic equations via the Monge–Kantorovich theory. Adv. Differ. Equ. 10(3), 309–360 (2005)
Agueh, M., Bowles, M.: One-dimensional numerical algorithms for gradient flows in the p-Wasserstein spaces. Acta Appl. Math. 125(1), 121–134 (2013)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows: in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich (2005)
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)
Blanchet, A., Carlier, G.: Optimal transport and Cournot–Nash equilibria. arXiv:1206.6571 (2012, arXiv preprint)
Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)
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)
Caffarelli, L.A.: Boundary regularity of maps with convex potentials. Commun. Pure Appl. Math. 45(9), 1141–1151 (1992)
Carlier, G., Lachand-Robert, T., Maury, B.: A numerical approach to variational problems subject to convexity constraint. Numer. Math. 88(2), 299–318 (2001)
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)
CGAL. Computational Geometry Algorithms Library. http://www.cgal.org
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)
Ekeland, I., Moreno-Bromberg, S.: An algorithm for computing solutions of variational problems with global convexity constraints. Numer. Math. 115(1), 45–69 (2010)
Gutiérrez, C.E.: The Monge–Ampère Equation, vol. 44. Birkhauser, Basel (2001)
Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
Kinderlehrer, D., Walkington, N.J.: Approximation of parabolic equations using the Wasserstein metric. ESAIM Math. Model. Numer. Anal. 33(04), 837–852 (1999)
Lachand-Robert, T., Oudet, É.: Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16(2), 368–379 (2005)
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)
McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)
Mérigot, Q., Oudet, E.: Handling convexity-like constraints in variational problems. SIAM J. Numer. Anal 52(5), 2466–2487 (2014)
Mirebeau, J.M.: Adaptive, anisotropic and hierarchical cones of discrete convex functions. arXiv:1402.1561 (2014, arXiv preprint)
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)
Oberman, A.M.: A numerical method for variational problems with convexity constraints. SIAM J. Sci. Comput. 35(1), A378–A396 (2013)
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)
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 44. Cambridge University Press, Cambridge (1993)
Trudinger, N.S., Wang, X.J.: The affine plateau problem. J. Am. Math. Soc. 18(2), 253–289 (2005)
Visintin, A.: Strong convergence results related to strict convexity. Commum. Partial Differ. Equ. 9(5), 439–466 (1984). doi:10.1080/03605308408820337
Zhou, B.: The first boundary value problem for Abreu’s equation. Int. Math. Res. Not. IMRN 7, 1439–1484 (2012)
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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).
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Benamou, JD., Carlier, G., Mérigot, Q. et al. Discretization of functionals involving the Monge–Ampère operator. Numer. Math. 134, 611–636 (2016). https://doi.org/10.1007/s00211-015-0781-y
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DOI: https://doi.org/10.1007/s00211-015-0781-y