The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices

Duff, Iain S.; Koster, Jacko

doi:10.1137/s0895479897317661

Cited by 203 publications

(176 citation statements)

References 13 publications

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“…Duff and Koster investigated row and column permutations such that entries with large absolute values are moved to the diagonal of sparse matrices [24,25]. They suggest that putting large entries in diagonal ahead of the numerical factorization allows pivoting down the diagonal to be more stable.…”

Section: Large Diagonal Batch Pivotingmentioning

confidence: 99%

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Parallel sparse LU factorization on different message passing platforms

Shen

2006

Journal of Parallel and Distributed Computing

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Several message passing-based parallel solvers have been developed for general (nonsymmetric) sparse LU factorization with partial pivoting. Existing solvers were mostly deployed and evaluated on parallel computing platforms with high message passing performance (e.g., 1-10 µs in message latency and 100-1000 Mbytes/sec in message throughput) while little attention has been paid on slower platforms. This paper investigates techniques that are specifically beneficial for LU factorization on platforms with slow message passing. In the context of the S + distributed memory solver, we find that significant reduction in the application message passing overhead can be attained at the cost of extra computation and slightly weakened numerical stability. In particular, we propose batch pivoting to make pivot selections in groups through speculative factorization, and thus substantially decrease the interprocessor synchronization granularity. We experimented on three different message passing platforms with different communication speeds. While the proposed techniques provide no performance benefit and even slightly weaken numerical stability on an IBM Regatta multiprocessor with fast message passing, they improve the performance of our test matrices by 15-460% on an Ethernet-connected 16-node PC cluster. Given the different tradeoffs of communication-reduction techniques on different message passing platforms, we also propose a sampling-based runtime application adaptation approach that automatically determines whether these techniques should be employed for a given platform and input matrix.

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Section: Large Diagonal Batch Pivotingmentioning

confidence: 99%

“…Duff and Koster [24,25] and Li and Demmel [12] have explored permuting large entries to the diagonal as a way to reduce the need of pivoting during numerical factorization. Built on these results, our work is the first to quantitatively assess the effectiveness of these techniques on platforms with different message passing performance.…”

Section: Related Workmentioning

confidence: 99%

Parallel sparse LU factorization on different message passing platforms

Shen

2006

Journal of Parallel and Distributed Computing

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“…Therefore, in step (1) the diagonal matrices D r and D c are chosen in such a way that each row and each column of D r AD c have a largest entry equal to 1 in magnitude. The row permutation matrix P r is chosen to maximize the product of the diagonal entries in P r D r AD c with the MC64 code [10]. In step (2) any symmetric fill-reducing ordering can be computed based on the structure of A + A T , e.g.…”

Section: Supernode Pivotingmentioning

confidence: 99%

“…When partial pivoting is required to maintain numerical stability in direct methods for solving nonsymmetric linear systems, it is challenging to develop high performance parallel software because partial pivoting causes the computational task-dependency graph to change during runtime. It has been proposed recently that permuting the rows of the matrix prior to factorization to maximize the magnitude of its diagonal entries can be very effective in reducing the amount of pivoting during factorization [1,9,10,15]. The proposed technique, static pivoting, is an efficient alternative to partial pivoting for parallel sparse Gaussian elimination.…”

Section: Introductionmentioning

confidence: 99%

Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO

Schenk

Gärtner

2002

Lecture Notes in Computer Science

142

154

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Abstract. Supernode pivoting for unsymmetric matrices coupled with supernode partitioning and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to extend these concepts further and prepermutation of rows is used to place large matrix entries on the diagonal. Supernode pivoting allows dynamical interchanges of columns and rows during the factorization process. The BLAS-3 level efficiency is retained. An enhanced left-right looking scheduling scheme is uneffected and results in good speedup on SMP machines without increasing the operation count. These algorithms have been integrated into the recent unsymmetric version of the PARDISO solver. Experiments demonstrate that a wide set of unsymmetric linear systems can be solved and high performance is consistently achieved for large sparse unsymmetric matrices from real world applications.

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“…Here, P is chosen before factorization based solely on the values of the original A. A maximum weighted matching algorithm and the code MC64 [9] is currently employed. The algorithm chooses P to maximize the magnitude of the diagonal entries of P A.…”

Section: Introductionmentioning

confidence: 99%

Using Random Butterfly Transformations to Avoid Pivoting in Sparse Direct Methods

Baboulin¹,

Li²,

Rouet³

2015

Lecture Notes in Computer Science

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Abstract. We consider the solution of sparse linear systems using direct methods via LU factorization. Unless the matrix is positive definite, numerical pivoting is usually needed to ensure stability, which is costly to implement especially in the sparse case. The Random Butterfly Transformations (RBT) technique provides an alternative to pivoting and is easily parallelizable. The RBT transforms the original matrix into another one that can be factorized without pivoting with probability one. This approach has been successful for dense matrices; in this work, we investigate the sparse case. In particular, we address the issue of fill-in in the transformed system.

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The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices

Cited by 203 publications

References 13 publications

Parallel sparse LU factorization on different message passing platforms

Parallel sparse LU factorization on different message passing platforms

Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO

Using Random Butterfly Transformations to Avoid Pivoting in Sparse Direct Methods

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