  This article details a novel numerical scheme to approximate gradient flows
for optimal transport (i.e. Wasserstein) metrics. These flows have proved
useful to tackle theoretically and numerically non-linear diffusion equations
that model for instance porous media or crowd evolutions. These gradient flows
define a suitable notion of weak solutions for these evolutions and they can be
approximated in a stable way using discrete flows. These discrete flows are
implicit Euler time stepping according to the Wasserstein metric. A bottleneck
of these approaches is the high computational load induced by the resolution of
each step. Indeed, this corresponds to the resolution of a convex optimization
problem involving a Wasserstein distance to the previous iterate. Following
several recent works on the approximation of Wasserstein distances, we consider
a discrete flow induced by an entropic regularization of the transportation
coupling. This entropic regularization allows one to trade the initial
Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to
deal with numerically. We show how KL proximal schemes, and in particular
Dykstra's algorithm, can be used to compute each step of the regularized flow.
The resulting algorithm is both fast, parallelizable and versatile, because it
only requires multiplications by a Gibbs kernel. On Euclidean domains
discretized on an uniform grid, this corresponds to a linear filtering (for
instance a Gaussian filtering when $c$ is the squared Euclidean distance) which
can be computed in nearly linear time. On more general domains, such as
(possibly non-convex) shapes or on manifolds discretized by a triangular mesh,
following a recently proposed numerical scheme for optimal transport, this
Gibbs kernel multiplication is approximated by a short-time heat diffusion.
