  In this paper, we propose two momentum accelerated MG cycles. The main idea
is to rewrite the linear systems as optimization problems and apply momentum
accelerations, e.g., the heavy ball and Nesterov acceleration methods, to
define the coarse-level solvers for multigrid (MG) methods. The resulting MG
cycles, which we call them H-cycle (uses heavy ball method) and N-cycle (uses
Nesterov acceleration), share the advantages of both algebraic multilevel
iteration (AMLI)- and K-cycle (a nonlinear version of the AMLI-cycle). Namely,
similar to the K-cycle, our H- and N-cycle do not require the estimation of
extreme eigenvalues while the computational cost is the same as AMLI-cycle.
Theoretical analysis shows that the momentum accelerated cycles are essentially
special AMLI-cycle methods and, thus, they are uniformly convergent under
standard assumptions. Finally, we present numerical experiments to verify the
theoretical results and demonstrate the efficiency of H- and N-cycle.
