Momentum Accelerated Multigrid Methods

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.