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HIERARCHICAL ORTHOGONAL FACTORIZATION: SPARSE SQUARE MATRICES.

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Title: HIERARCHICAL ORTHOGONAL FACTORIZATION: SPARSE SQUARE MATRICES.
Authors: GNANASEKARAN, ABEYNAYA1 abeynaya@stanford.edu, DARVE, ERIC1 darve@stanford.edu
Source: SIAM Journal on Matrix Analysis & Applications. 2022, Vol. 43 Issue 1, p94-123. 30p.
Subject Terms: SPARSE matrices, FACTORIZATION, APPROXIMATION error, BENCHMARK problems (Computer science), LINEAR systems
Abstract: In this work, we develop a new fast algorithm, spaQR--sparsified QR--for solving large, sparse linear systems. The key to our approach lies in using low-rank approximations to sparsify the separators in a nested dissection based Householder QR factorization. First, a modified version of nested dissection is used to identify vertex separators and reorder the matrix. Then, classical Householder QR is used to factorize the separators, going from the leaves to the top of the elimination tree. After every level of separator factorization, we sparsify all the remaining separators by using low-rank approximations. This operation reduces the size of the separators without introducing any fill-in in the matrix. However, it introduces a small approximation error which can be controlled by the user. The resulting approximate factorization is stored as a sequence of sparse orthogonal and sparse upper-triangular factors. Hence, it can be applied efficiently to solve linear systems. We further improve the algorithm by using a block diagonal scaling. Then, we show a systematic analysis of the approximation error and effectiveness of the algorithm in solving linear systems. Finally, we perform numerical tests on benchmark unsymmetric problems to evaluate the performance of the algorithm. The factorization time scales as O(N log N) and the solve time scales as O(N). [ABSTRACT FROM AUTHOR]
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Abstract:In this work, we develop a new fast algorithm, spaQR--sparsified QR--for solving large, sparse linear systems. The key to our approach lies in using low-rank approximations to sparsify the separators in a nested dissection based Householder QR factorization. First, a modified version of nested dissection is used to identify vertex separators and reorder the matrix. Then, classical Householder QR is used to factorize the separators, going from the leaves to the top of the elimination tree. After every level of separator factorization, we sparsify all the remaining separators by using low-rank approximations. This operation reduces the size of the separators without introducing any fill-in in the matrix. However, it introduces a small approximation error which can be controlled by the user. The resulting approximate factorization is stored as a sequence of sparse orthogonal and sparse upper-triangular factors. Hence, it can be applied efficiently to solve linear systems. We further improve the algorithm by using a block diagonal scaling. Then, we show a systematic analysis of the approximation error and effectiveness of the algorithm in solving linear systems. Finally, we perform numerical tests on benchmark unsymmetric problems to evaluate the performance of the algorithm. The factorization time scales as O(N log N) and the solve time scales as O(N). [ABSTRACT FROM AUTHOR]
ISSN:08954798
DOI:10.1137/20M1373475