Computation of Optimal Transport on Discrete Metric Measure Spaces
  In this paper we investigate the numerical approximation of an analogue of
the Wasserstein distance for optimal transport on graphs that is defined via a
discrete modification of the Benamou--Brenier formula. This approach involves
the logarithmic mean of measure densities on adjacent nodes of the graph. For
this model a variational time discretization of the probability densities on
graph nodes and the momenta on graph edges is proposed. A robust descent
algorithm for the action functional is derived, which in particular uses a
proximal splitting with an edgewise nonlinear projection on the convex subgraph
of the logarithmic mean. Thereby, suitable chosen slack variables avoid a
global coupling of probability densities on all graph nodes in the projection
step. For the time discrete action functional $\Gamma$--convergence to the time
continuous action is established. Numerical results for a selection of test
cases show qualitative and quantitative properties of the optimal transport on
graphs. Finally, we use our algorithm to implement a JKO scheme for the
gradient flow of the entropy in the discrete transportation distance, which is
known to coincide with the underlying Markov semigroup, and test our results
against a classical backward Euler discretization of this discrete heat flow.
