FastAGMGar: An aggregation-based algebraic multigrid method

Abstract

In this paper, we solve large sparse symmetric positive definite linear systems with the Krylov subspace method preconditioned by an aggregation-based algebraic multigrid (AGMG) scheme. We study the AGMGar (stands for AGMG with guaranteed convergence rate) method and present a new method called FastAGMGar. This method is developed by relaxing the aggregation requirement employed in AGMGar. Additionally, an integral correction method is introduced to improve the Jacobi smoother. The applicability of AGMGar and FastAGMGar methods to non-M-matrices is investigated, and their limitations are also examined and mitigated. To improve the performance of solving linear systems with non-M-matrices as coefficient matrices, the original aggregation algorithm is modified by only accepting the aggregate that contains nodes corresponding to negative couplings. Moreover, to reduce the high setup cost of AGMGar caused by the low coarsening ratio, different approaches are considered to compute the coarse-grid matrices based on the coarsening ratio. The numerical results demonstrate the effectiveness of these improvements. Furthermore, compared with classical AGMG and AGMGar, the newly proposed FastAGMGar not only features a shorter setup time but also maintains competitive efficiency in the solution phase. Consequently, this method showcases superior performance, with the shortest total CPU time for all test problems.