  In this paper, we propose an efficient and flexible algorithm to solve
dynamic mean-field planning problems based on an accelerated proximal gradient
method. Besides an easy-to-implement gradient descent step in this algorithm, a
crucial projection step becomes solving an elliptic equation whose solution can
be obtained by conventional methods efficiently. By induction on iterations
used in the algorithm, we theoretically show that the proposed discrete
solution converges to the underlying continuous solution as the grid size
increases. Furthermore, we generalize our algorithm to mean-field game problems
and accelerate it using multilevel and multigrid strategies. We conduct
comprehensive numerical experiments to confirm the convergence analysis of the
proposed algorithm, to show its efficiency and mass preservation property by
comparing it with state-of-the-art methods, and to illustrates its flexibility
for handling various mean-field variational problems.
