We propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND -- sparsified Nested Dissection. It is based on nested dissection, sparsification and low-rank compression. After eliminating all interiors at a given level of the elimination tree, the algorithm sparsifies all separators corresponding to the interiors. This operation reduces the size of the separators by eliminating some degrees of freedom but without introducing any fill-in. This is done at the expense of a small and controllable approximation error. The result is an approximate factorization that can be used as an efficient preconditioner. We then perform several numerical experiments to evaluate this algorithm. We demonstrate that a version using orthogonal factorization and block-diagonal scaling takes less CG iterations to converge than previous similar algorithms on various kinds of problems. Furthermore, this algorithm is provably guaranteed to never break down and the matrix stays symmetric positive-definite throughout the process. We evaluate the algorithm on some large problems show it exhibits near-linear scaling. The factorization time is roughly O(N) and the number of iterations grows slowly with N.

中文翻译：

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使用低秩近似的代数稀疏嵌套剖析算法

我们提出了一种新算法，用于快速解决大型、稀疏、对称正定线性系统 spaND——稀疏嵌套剖析。它基于嵌套剖析、稀疏化和低秩压缩。在消除树的给定级别消除所有内部之后，该算法将与内部对应的所有分隔符进行稀疏化。此操作通过消除某些自由度但不引入任何填充来减小分隔符的大小。这是以小而可控的近似误差为代价的。结果是可以用作有效预处理器的近似因式分解。然后我们进行了几个数值实验来评估这个算法。我们证明，在各种问题上，使用正交分解和块对角线缩放的版本比以前的类似算法需要更少的 CG 迭代来收敛。此外，可以证明该算法永远不会崩溃，并且矩阵在整个过程中保持对称正定。我们在一些大问题上评估该算法表明它表现出接近线性的缩放。分解时间大约为 O(N)，迭代次数随着 N 缓慢增长。