  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-Amp\`ere
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.
