This paper introduces a fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices. The preconditioner is analyzed for symmetric positive definite (SPD) matrices. For those matrices, the coarse space is constructed based on approximating two local subspaces in each subdomain. These subspaces are obtained by approximating a number of eigenvectors corresponding to dominant eigenvalues of two judiciously posed generalized eigenvalue problems. The number of eigenvectors can be chosen to control the condition number. For general sparse matrices, the coarse space is constructed by approximating the image of a local operator that can be defined from information in the coefficient matrix. The connection between the coarse spaces for SPD and general matrices is also discussed. Numerical experiments show the great effectiveness of the proposed preconditioners on matrices arising from a wide range of applications. The set of matrices includes SPD, symmetric indefinite, nonsymmetric, and saddle-point matrices. In addition, we compare the proposed preconditioners to the state-of-the-art domain decomposition preconditioners.

翻译：本文提出了一种完全代数化的两级加性Schwarz预处理器，适用于一般稀疏大规模矩阵。该预处理器针对对称正定（SPD）矩阵进行了分析。对于这类矩阵，粗空间基于每个子域中两个局部子空间的近似构建而成。这些子空间通过逼近两个精心构造的广义特征值问题中主导特征值对应的若干特征向量获得，且特征向量的数量可调节以控制条件数。对于一般稀疏矩阵，粗空间则通过逼近一个可由系数矩阵信息定义的局部算子的像来构建。本文还讨论了SPD矩阵与一般矩阵粗空间之间的关联。数值实验表明，所提出的预处理器对来自广泛应用的矩阵具有显著效果，涵盖SPD矩阵、对称不定矩阵、非对称矩阵及鞍点矩阵。此外，我们将所提预处理器与当前最先进的区域分解预处理器进行了比较。