  This work is concerned with the efficient optimization method for solving a
large class of optimal mass transport problems. An inexact primal-dual
algorithm is presented from the time discretization of a proper dynamical
system, and by using the tool of Lyapunov function, the global (super-)linear
convergence rate is established for function residual and feasibility
violation. The proposed algorithm contains an inner problem that possesses
strong semismoothness property and motivates the use of the semismooth Newton
iteration. By exploring the hidden structure of the problem itself, the linear
system arising from the Newton iteration is transferred equivalently into a
graph Laplacian system, for which a robust algebraic multigrid method is
proposed and also analyzed via the famous Xu--Zikatanov identity. Finally,
numerical experiments are provided to validate the efficiency of our method.
