Ordering techniques for singly bordered block diagonal forms for unsymmetric parallel sparse direct solvers
Yifan Hu,Jennifer A. Scott +1 more
TL;DR: New reordering algorithms that use only vertex separators and wide separators of this graph of AT + A (or BT + B) are proposed and can produce bordered forms that are competitive with those obtained using MONET.
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Abstract: The solution of large sparse linear systems of equations is one of the cornerstones of scientific computation. In many applications it is important to be able to solve these systems as rapidly as possible. One approach for very large problems is to reorder the system matrix to bordered block diagonal form and then to solve the block system using a coarse-grained parallel approach. In this paper, we consider the problem of efficiently ordering unsymmetric systems to singly bordered block diagonal form. Algorithms such as the MONET algorithm of Hu et al. (Comput. Chem. Eng. 23 (2000) 1631) that depend upon computing a representation of AAT can be prohibitively expensive when a single (or small number of) matrix factorizations are required. We therefore work with the graph of AT + A (or BT + B, where B is a row permutation of A) and propose new reordering algorithms that use only vertex separators and wide separators of this graph. Numerical experiments demonstrate that our methods are efficient and can produce bordered forms that are competitive with those obtained using MONET. Copyright © 2005 John Wiley & Sons, Ltd.
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TL;DR: Metis is copyrighted by the regents of the University of Minnesota as mentioned in this paper, and the content of which does not necessarily reflect the position or the policy of lhe government, and no official endorsement should be inferred.
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Coverings of Bipartite Graphs
A. L. Dulmage,N. S. Mendelsohn +1 more
TL;DR: In this paper, the concept of exterior coverings is introduced for decomposing bipartite graphs into two parts, an inadmissible part and a core, and then decomposing the core into irreducible parts and thus obtaining a canonical reduction of the graph.
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Graph theory and sparse matrix computation
Alan George,John R. Gilbert,Joseph W. H. Liu +2 more
- 01 Jan 1993
TL;DR: The articles in this volume are based on recent research on sparse matrix computations and examine graph theory as it connects to linear algebra, parallel computing, data structures, geometry and both numerical and discrete algorithms.
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Algorithm 575: Permutations for a Zero-Free Diagonal [F1]
TL;DR: This subroutine attempts to find a row permutation that makes the matrix have no zeros on its diagonal that is based on a depth first search with look-ahead technique.
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The design of a new frontal code for solving sparse, unsymmetric systems
Iain S. Duff,Jennifer A. Scott +1 more
TL;DR: The design, implementation, and performance of a frontal code for the solution of large, sparse, unsymmetric systems of linear equations, and the extensive use of higher-level BLAS kernels within MA42 are described.