Algebraic Multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two-level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine- and coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two-level convergence for symmetric and positive-definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non-diagonally dominant systems coming from anisotropic diffusion.

翻译：代数多重网格（AMG）方法通常是解决由离散偏微分方程及其他问题产生的大规模稀疏线性系统的稳健高效求解器，其性能依赖于启发式图算法。基于约化的代数多重网格（AMGr）算法试图通过提供依赖于给定矩阵细网格与粗网格自由度划分性质的具体两层收敛界来形式化这些启发式方法。MacLachlan 与 Saad（SISC 2007）证明了，对于对角占优的对称正定矩阵，AMGr 方法能够给出稳健的两层收敛保证，其收敛因子受粗化参数的函数约束。然而，将 AMGr 算法应用于非对角占优矩阵时，不仅收敛因子界不再成立，实际性能也显著下降。本文对经典 AMGr 算法进行了改进，通过引入连接强度、稀疏近似逆（SPAI）技术以及插值截断与重缩放策略，在保持算法成本可控的同时，提升了非对角占优矩阵上的稳健性。我们给出了数值结果，验证了该方法在经典各向同性扩散问题及各向异性扩散产生的非对角占优系统中的稳健性。