An Algebraic Sparsified Nested Dissection Algorithm Using Low-Rank Approximations

Cambier, Léopold; Chen, Chao; Boman, Erik G.; Rajamanickam, Sivasankaran; Tuminaro, Raymond S.; Darve, Eric

doi:10.1137/19m123806x

An Algebraic Sparsified Nested Dissection Algorithm Using Low-Rank Approximations

Journal Article · Mon May 04 00:00:00 EDT 2020 · SIAM Journal on Matrix Analysis and Applications

DOI:https://doi.org/10.1137/19m123806x· OSTI ID:1697983

^[1]; ^[2]; Boman, Erik G. ^[3]; Rajamanickam, Sivasankaran ^[3]; Tuminaro, Raymond S. ^[3]; Darve, Eric ^[1]

Stanford Univ., CA (United States)
Univ. of Texas, Austin, TX (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Here, 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 fewer 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 $$\mathcal{O}$$(N), and the number of iterations grows slowly with N.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program

Grant/Contract Number:: AC04-94AL85000; NA0003525

OSTI ID:: 1697983

Report Number(s):: SAND--2019-0313J; 671477

Journal Information:: SIAM Journal on Matrix Analysis and Applications, Journal Name: SIAM Journal on Matrix Analysis and Applications Journal Issue: 2 Vol. 41; ISSN 0895-4798

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
hierarchical matrix
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nested dissection
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An Algebraic Sparsified Nested Dissection Algorithm Using Low-Rank Approximations

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