  We consider problems of minimizing functionals $\mathcal{F}$ of probability
measures on the Euclidean space. To propose an accelerated gradient descent
algorithm for such problems, we consider gradient flow of transport maps that
give push-forward measures of an initial measure. Then we propose a
deterministic accelerated algorithm by extending Nesterov's acceleration
technique with momentum. This algorithm do not based on the Wasserstein
geometry. Furthermore, to estimate the convergence rate of the accelerated
algorithm, we introduce new convexity and smoothness for $\mathcal{F}$ based on
transport maps. As a result, we can show that the accelerated algorithm
converges faster than a normal gradient descent algorithm. Numerical
experiments support this theoretical result.
