Discretization of functionals involving the Monge-Ampère operator
arXiv:1408.4536
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 >= 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.
20 pages
Cited by in corpus (5)
- Scaling Algorithms for Unbalanced Transport Problems
- Numerical Study of a Particle Method for Gradient Flows
- Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks
- Entropic Wasserstein Gradient Flows
- Solvability of a class of singular fourth order equations of Monge-Ampère type