  This paper presents a methodology and numerical algorithms for constructing
accelerated gradient flows on the space of probability distributions. In
particular, we extend the recent variational formulation of accelerated
gradient methods in (wibisono, et. al. 2016) from vector valued variables to
probability distributions. The variational problem is modeled as a mean-field
optimal control problem. The maximum principle of optimal control theory is
used to derive Hamilton's equations for the optimal gradient flow. The
Hamilton's equation are shown to achieve the accelerated form of density
transport from any initial probability distribution to a target probability
distribution. A quantitative estimate on the asymptotic convergence rate is
provided based on a Lyapunov function construction, when the objective
functional is displacement convex. Two numerical approximations are presented
to implement the Hamilton's equations as a system of $N$ interacting particles.
The continuous limit of the Nesterov's algorithm is shown to be a special case
with $N=1$. The algorithm is illustrated with numerical examples.
