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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Citations
Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix
John Reid,Jennifer A. Scott +1 more
TL;DR: In this article, the authors proposed an unsymmetric variant of the reverse Cuthill-McKee algorithm to reduce the total bandwidth of a sparse symmetric matrix.
A parallel direct solver for large sparse highly unsymmetric linear systems
Iain S. Duff,Jennifer A. Scott +1 more
TL;DR: In this article, the problem is first subdivided into a number of loosely connected subproblems and a variant of sparse Gaussian elimination is then applied to each of the sub-problems in parallel.
•Journal Article
Stabilized bordered block diagonal forms for parallel sparse solvers
Iain S. Duff,Jennifer A. Scott +1 more
TL;DR: In this article, the duality between singly and doubly bordered block diagonal forms is considered and the idea of a stabilized doubly bounded block diagonal form is introduced, which is shown to be efficient both in computation time and the quality of the resulting block structure.
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Stabilized bordered block diagonal forms for parallel sparse solvers
Iain S. Duff,Jennifer A. Scott +1 more
- 01 Mar 2005
TL;DR: In this article, the duality between singly and doubly bordered block diagonal forms is considered and the idea of a stabilized doubly bounded block diagonal form is introduced, which is shown to be efficient both in computation time and the quality of the resulting block structure.
Solving intertemporal CGE models in parallel using a singly bordered block diagonal ordering technique
TL;DR: A direct ordering method that employs a special feature of an intertemporal Computable General Equilibrium (CGE) model to reorder its first-order partial derivative matrix into a Singly Bordered Block Diagonal (SBBD) form is introduced.
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References
Program Abstracts Sparse Matrix Symposium 1982 Held at Fairfield Glade, Tennessee, October 24-27, 1982,
Vivian A Jacobs
- 01 Jan 1982
TL;DR: This document contains abstracts of papers presented at the Sparse Matrix Symposium held in Fairfield Glade, Tennessee, Oct. 24-27, 1982.
1
On Algorithms for Obtaining a Maximum Transversal
TL;DR: A computer program for obtaining a permutation of a general sparse matrix that places a maxunum number of nonzero elements on its dmgonal is described and some imtml attempts at implementmg an algomthm with supermr asymptotic complexity are described.
•Journal Article
The design and use of algorithms for permuting large entries to the diagonal of sparse matrices
Iain S. Duff,Jacko Koster +1 more
TL;DR: In this article, the authors consider techniques for permuting a sparse matrix so that the diagonal of the permuted matrix has entries of large absolute value and discuss various criteria for this and consider their implementation as computer codes.
The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices
Iain S. Duff,Jacko Koster +1 more
TL;DR: This work considers techniques for permuting a sparse matrix so that the diagonal of the permuted matrix has entries of large absolute value and indicates several cases where such a permutation can be useful.
A column approximate minimum degree ordering algorithm
TL;DR: In this paper, a column preordering based on the nonzero pattern of a sparse matrix A is proposed to reduce fill-in in the Cholesky factorization of ATA.