Discretization of functionals involving the Monge–Ampère operator
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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 Classification49M25 52B55
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).
- 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
- 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
- 6.Blanchet, A., Carlier, G.: Optimal transport and Cournot–Nash equilibria. arXiv:1206.6571 (2012, arXiv preprint)
- 12.CGAL. Computational Geometry Algorithms Library. http://www.cgal.org
- 22.Mirebeau, J.M.: Adaptive, anisotropic and hierarchical cones of discrete convex functions. arXiv:1402.1561 (2014, arXiv preprint)