We describe a second-order accurate approach to sparsifying the off-diagonal blocks in the hierarchical approximate factorizations of sparse symmetric positive definite matrices. The norm of the error made by the new approach depends quadratically, not linearly, on the error in the low-rank approximation of the given block. The analysis of the resulting two-level preconditioner shows that the preconditioner is second-order accurate as well. We incorporate the new approach into the recent Sparsified Nested Dissection algorithm [SIAM J. Matrix Anal. Appl., 41 (2020), pp. 715-746], and test it on a wide range of problems. The new approach halves the number of Conjugate Gradient iterations needed for convergence, with almost the same factorization complexity, improving the total runtimes of the algorithm. Our approach can be incorporated into other rank-structured methods for solving sparse linear systems.

中文翻译：

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稀疏SPD矩阵的二阶精确分层近似分解

我们描述了在稀疏对称正定矩阵的分层近似分解中稀疏非对角块的二阶精确方法。新方法产生的误差范数二次而非线性地取决于给定块的低秩近似中的误差。对生成的两级预处理器的分析表明，该预处理器也是二阶准确的。我们将新方法纳入最近的稀疏嵌套解剖算法 [SIAM J. Matrix Anal。Appl., 41 (2020), pp. 715-746]，并在广泛的问题上对其进行测试。新方法将收敛所需的共轭梯度迭代次数减半，分解复杂度几乎相同，从而提高了算法的总运行时间。