On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix

Duff, Iain S.; Koster, Jacko

doi:10.1137/s0895479899358443

Cited by 259 publications

(220 citation statements)

References 26 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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“…ILUTP Mem allows the user to set the values of 3 parameters that together control the memory required, and the accuracy provided, by the preconditioner; we used a range of values for lfil (0,1,2,3,4,5), droptol (0,.001,.01,.1), and pivtol (0,.1,1). In addition, before computing the ILU factorization, matrices were first permuted and scaled using MC64 [10,11] for stability, and then using COLAMD [12,13] for sparsity. Note that by using value-based ILU preconditioners, instead of the level-based ILU preconditioners used in [5], we are working with a much larger number of parameter values.…”

Section: Construction Of the Sample Spacementioning

confidence: 99%

Neural Networks for Predicting the Behavior of Preconditioned Iterative Solvers

Holloway

Chen

2007

Computational Science – ICCS 2007

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Abstract. We evaluate the effectiveness of neural networks as a tool for predicting whether a particular combination of preconditioner and iterative method will correctly solve a given sparse linear system Ax = b. We consider several scenarios corresponding to different assumptions about the relationship between the systems used to train the neural network and those for which the neural network is expected to predict behavior. Greater similarity between those two sets leads to better accuracy, but even when the two sets are very different prediction accuracy can be improved by using additional computation.

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“…A maximum weight perfect matching on a weighted graph is a perfect matching with maximum weight. Both the perfect matching problem and the maximum weight perfect matching problem are efficiently solvable [7,13]. We use mate(v), to denote the vertex matched to the vertex v in a matching M. That is if (r i , c j ) ∈ M, then mate(r i ) = c j and mate(c j ) = r i .…”

Section: Methods Descriptionmentioning

confidence: 99%

“…We use mate(v), to denote the vertex matched to the vertex v in a matching M. That is if (r i , c j ) ∈ M, then mate(r i ) = c j and mate(c j ) = r i . It is well known that perfect matchings in the bipartite graph G correspond to permutations which yield zero-free diagonals, see for example [7]. A matching edge (r i , c j ) is used to permute the column c j to the ith position, yielding a zero-free diagonal.…”

Section: Methods Descriptionmentioning

confidence: 99%

Heuristics for a Matrix Symmetrization Problem

Uçar

Parallel Processing and Applied Mathematics

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On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix

Cited by 259 publications

References 26 publications

Parallel sparse LU factorization on different message passing platforms

Parallel sparse LU factorization on different message passing platforms

Neural Networks for Predicting the Behavior of Preconditioned Iterative Solvers

Heuristics for a Matrix Symmetrization Problem

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