This article presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the adjacency graph of the subdomains, resulting in a two‐level block diagonal structure. A full binary tree structure $𝒯$ is then built to facilitate the construction of the preconditioner. A key property that is exploited is the observation that the difference between the inverse of the original matrix and that of its block diagonal approximation is often well approximated by a low‐rank matrix. This property and the block diagonal structure of the reordered matrix lead to a multicolor low‐rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom‐up traversal of the tree $𝒯$. All irregular matrix computations, such as ILU factorizations and related triangular solves, are restricted to leaf nodes where these operations can be performed independently. Computations in nonleaf nodes only involve easy‐to‐optimize dense matrix operations. In order to further reduce the number of iteration of the Preconditioned Krylov subspace procedure, we combine MCLR with a few classical block‐relaxation techniques. Numerical experiments on various test problems are proposed to illustrate the robustness and efficiency of the proposed approach for solving large sparse symmetric and nonsymmetric linear systems.

中文翻译：

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通用稀疏线性系统的多色低阶预处理器

本文提出了一种用于求解一般大型稀疏线性方程组的多级并行预处理技术。通过对子域的邻接图进行多色处理，调用子域着色以对系数矩阵进行重新排序，从而形成两级块对角线结构。完整的二叉树结构$𝒯$然后构建用于便利预处理器的构建。利用的一个关键特性是观察到原始矩阵的逆与其块对角线近似之间的差异通常可以由低秩矩阵很好地近似。此属性和重新排序的矩阵的块对角线结构导致了多色低阶（MCLR）预调节器。MCLR预调节器的构建过程遵循自下而上遍历树的过程$𝒯$。所有不规则矩阵计算（例如ILU分解和相关的三角求解）都限于叶节点，在这些叶节点中，这些操作可以独立执行。非叶节点中的计算仅涉及易于优化的密集矩阵运算。为了进一步减少预处理Krylov子空间过程的迭代次数，我们将MCLR与一些经典的块松弛技术结合在一起。提出了各种测试问题的数值实验，以说明所提出的方法解决大型稀疏对称和非对称线性系统的鲁棒性和效率。