  The time-fractional optimal transport (OT) and mean-field planning (MFP)
models are developed to describe the anomalous transport of the agents in a
heterogeneous environment such that their densities are transported from the
initial density distribution to the terminal one with the minimal cost. We
derive a strongly coupled nonlinear system of a time-fractional transport
equation and a backward time-fractional Hamilton-Jacobi equation based on the
first-order optimality condition. The general-proximal primal-dual hybrid
gradient (G-prox PDHG) algorithm is applied to discretize the OT and MFP
formulations, in which a preconditioner induced by the numerical approximation
to the time-fractional PDE is derived to accelerate the convergence of the
algorithm for both problems. Numerical experiments for OT and MFP problems
between Gaussian distributions and between image densities are carried out to
investigate the performance of the OT and MFP formulations. Those numerical
experiments also demonstrate the effectiveness and flexibility of our proposed
algorithm.
