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An algebraic sparsified nested dissection algorithm using low-rank approximations. (English) Zbl 1441.65048
The authors are interested in solving large symmetric positive-definite (SPD) sparse linear systems \(Ax=b\), \(A\in \mathbb{R}^{N\times N}\) and propose a new algorithm based on nested dissection, sparsification, and low-rank compression.
The approach is based on the idea of hierarchical interpolative factorization described by K. L. Ho and L. Ying [Commun. Pure Appl. Math. 69, No. 8, 1415–1451 (2016; Zbl 1353.35142)]. However, there are several differences, improvements, and novel capabilities. For example: the algorithm is completely general and can be applied to any symmetric positive definite matrix. The only required input is the sparse matrix itself. If some geometry information is available, it can be used to improve the quality of the ordering and clustering; inclusion of an additional step for scaling a diagonal block in the algorithm, greatly improving the accuracy of the preconditioner for only a small additional cost; the use of orthogonal (instead of interpolative) transformation, improving stability and ensuring that the preconditioner remains SPD when \(A\) is SPD. The authors 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 \(N\).

MSC:
65F50 Computational methods for sparse matrices
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65Y20 Complexity and performance of numerical algorithms
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