  We design fast numerical methods for Hamilton-Jacobi equations in density
space (HJD), which arises in optimal transport and mean field games. We
overcome the curse-of-infinite-dimensionality nature of HJD by proposing a
generalized Hopf formula in density space. The formula transfers optimal
control problems in density space, which are constrained minimizations
supported on both spatial and time variables, to optimization problems over
only spatial variables. This transformation allows us to compute HJD
efficiently via multi-level approaches and coordinate descent methods.
