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27
A sparse approximate inverse preconditioner for nonsymmetric linear systems
 SIAM J. SCI. COMPUT
, 1998
"... This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner f ..."
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Cited by 192 (22 self)
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This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient–type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell–Boeing collection and from Tim Davis’s collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.
Sparse Approximate Inverse Preconditioning For Dense Linear Systems Arising In Computational Electromagnetics
 Numerical Algorithms
, 1997
"... . We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising from industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pat ..."
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Cited by 59 (20 self)
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. We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising from industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pattern. Some strategies for determining the nonzero pattern of an approximate inverse are described. The results of numerical experiments suggest that sparse approximate inverse preconditioning is a viable approach for the solution of largescale dense linear systems on parallel computers. Key words. Dense linear systems, preconditioning, sparse approximate inverses, complex symmetric matrices, scattering calculations, Krylov subspace methods, parallel computing. AMS(MOS) subject classification. 65F10, 65F50, 65R20, 65N38, 7808, 78A50, 78A55. 1. Introduction. In the last decade, a significant amount of effort has been spent on the simulation of electromagnetic wave propagation phenomena to ad...
Wavelet Sparse Approximate Inverse Preconditioners
 BIT
, 1997
"... . There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle [21] and Chow and Saad [11] also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Ha ..."
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Cited by 36 (5 self)
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. There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle [21] and Chow and Saad [11] also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. HarwellBoeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices that come from elliptic PDE problems, this assumption may not necessarily hold. Our main idea is to look for a basis, other than the standard one, such that a sparse representation of the inverse is feasible. A crucial observation is that the kind of matrices we are interested in typically have a piecewise smooth inverse. We exploit this fact by applying wavelet techniques to construct a better sparse approximate inverse in the wavelet basis. We shall justify theoretically and numerically that our approach i...
Sparse Approximate Inverse Smoother for Multigrid
 SIAM J. Matrix Anal. Appl
, 1999
"... Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been ful ..."
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Cited by 31 (2 self)
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Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been fully exploited in multigrid algorithms. A careful examination of the iteration matrices of these approximate inverses indicates their superiority in smoothing the high frequency error in addition to their inherent parallelism. We propose a new class of sparse approximate inverse smoothers in this paper and present their analytic smoothing factors for constant coecient PDEs. Several distinctive features that make this technique special are: By adjusting the quality of the approximate inverse, the smoothing factor can be improved accordingly. For hard problems, this is useful.
Toward An Effective Sparse Approximate Inverse Preconditioner
 SIAM J. Matrix Anal. Appl
, 1999
"... . Sparse approximate inverse preconditioners have attracted much attention recently, because of their potential usefulness in a parallel environment. In this paper, we explore several performance issues related to e#ective sparse approximate inverse preconditioners (SAIPs) for the matrices derived f ..."
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Cited by 28 (3 self)
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. Sparse approximate inverse preconditioners have attracted much attention recently, because of their potential usefulness in a parallel environment. In this paper, we explore several performance issues related to e#ective sparse approximate inverse preconditioners (SAIPs) for the matrices derived from PDEs. Our refinements can significantly improve the quality of existing SAIPs and/or reduce the cost of computing them. For the test problems from the HarwellBoeing collection and some other applications, the performance of our preconditioners can be comparable or superior to incomplete LU (ILU) preconditioners with similar preconditioning cost. Key words. approximate inverse, globally coupled local inverse, ILU preconditioner, exponential decay AMS subject classifications. 15A09, 15A23, 65F10, 65F50, 65Y05 PII. S0895479897320071 1. Introduction. The use of preconditioned Krylov space methods has been proven to be a competitive solution technique for a wide range of large sparse matrix...
An MPI implementation of the SPAI preconditioner on the t3E
 INTL. J. HIGH PERF. COMPUT. APPL
, 1999
"... The authors describe and test spai_1.1, a parallel MPI implementation of the sparse approximate inverse (SPAI) preconditioner. They show that SPAI can be very effective for solving a set of very large and difficult problems on a Cray T3E. The results clearly show the value of SPAI (and approximate i ..."
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Cited by 27 (0 self)
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The authors describe and test spai_1.1, a parallel MPI implementation of the sparse approximate inverse (SPAI) preconditioner. They show that SPAI can be very effective for solving a set of very large and difficult problems on a Cray T3E. The results clearly show the value of SPAI (and approximate inverse methods in general) as the viable alternative to ILUtype methods when facing very large and difficult problems. The authors strengthen this conclusion by showing that spai_1.1 also has very good scaling behavior.
Numerical Experiments With Two Approximate Inverse Preconditioners
 BIT
, 1998
"... We present the results of numerical experiments aimed at comparing two recently proposed sparse approximate inverse preconditioners from the point of view of robustness, cost, and effectiveness. Results for a standard ILU preconditioner are also included. The numerical experiments were carried out o ..."
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Cited by 19 (7 self)
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We present the results of numerical experiments aimed at comparing two recently proposed sparse approximate inverse preconditioners from the point of view of robustness, cost, and effectiveness. Results for a standard ILU preconditioner are also included. The numerical experiments were carried out on a Cray C98 vector processor.
Sparse Numerical Linear Algebra: Direct Methods and Preconditioning
, 1996
"... Most of the current techniques for the direct solution of linear equations are based on supernodal or multifrontal approaches. An important feature of these methods is that arithmetic is performed on dense submatrices and Level 2 and Level 3 BLAS (matrixvector and matrixmatrix kernels) can be us ..."
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Cited by 19 (2 self)
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Most of the current techniques for the direct solution of linear equations are based on supernodal or multifrontal approaches. An important feature of these methods is that arithmetic is performed on dense submatrices and Level 2 and Level 3 BLAS (matrixvector and matrixmatrix kernels) can be used. Both sparse LU and QR factorizations can be implemented within this framework. Partitioning and ordering techniques have seen major activity in recent years. We discuss bisection and multisection techniques, extensions to orderings to block triangular form, and recent improvements and modifications to standard orderings such as minimum degree. We also study advances in the solution of indefinite systems and sparse leastsquares problems. The desire to exploit parallelism has been responsible for many of the developments in direct methods for sparse matrices over the last ten years. We examine this aspect in some detail, illustrating how current techniques have been developed or ...