In this paper, we consider the matrices approximated in \({\mathscr{H}}^{2}\) format. The direct solution, as well as the preconditioning of systems with such matrices, is a challenging problem. We propose a non-extensive sparse factorization of the \({\mathscr{H}}^{2}\) matrix that allows to substitute direct \({\mathscr{H}}^{2}\) solution with the solution of the system with an equivalent sparse matrix of the same size. The sparse factorization is constructed of parameters of the \({\mathscr{H}}^{2}\) matrix. In the numerical experiments, we show the consistency of this approach in comparison with the other approximate block low-rank hierarchical solvers, such as HODLR [3], H2Lib [5], and IFMM [11].

中文翻译：

---

层次矩阵的简单非广义稀疏

在本文中，我们考虑以\（{\ mathscr {H}} ^ {2} \）格式近似的矩阵。直接解决方案以及使用此类矩阵对系统进行预处理是一个具有挑战性的问题。我们提出了\（{\ mathscr {H}} ^ {2} \）矩阵的非广义稀疏分解，该矩阵可以用解替换直接\（{\ mathscr {H}} ^ {2} \）解系统具有相同大小的等效稀疏矩阵。稀疏分解是由\（{\ mathscr {H}} ^ {2} \）矩阵的参数构成的。在数值实验中，我们证明了该方法与其他近似块低秩层次求解器（例如HODLR [3]，H2Lib [5]和IFMM [11]）的一致性。