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Discretization of functionals involving the Monge-Ampère operator

Benamou, Jean-David; Carlier, Guillaume; Mérigot, Quentin; Oudet, Edouard (2016), Discretization of functionals involving the Monge-Ampère operator, Numerische Mathematik, 134, 3, p. 611-636. 10.1007/s00211-015-0781-y

Type
Article accepté pour publication ou publié
External document link
https://arxiv.org/abs/1408.4536v1
Date
2016
Journal name
Numerische Mathematik
Volume
134
Number
3
Publisher
Springer
Pages
611-636
Publication identifier
10.1007/s00211-015-0781-y
Metadata
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Author(s)
Benamou, Jean-David
Carlier, Guillaume
Mérigot, Quentin
Oudet, Edouard
Abstract (EN)
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 >= 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.
Subjects / Keywords
Monge–Ampère equation

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