Results 1  10
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25
Approximate Inverse Techniques for BlockPartitioned Matrices
 SIAM J. Sci. Comput
, 1995
"... This paper proposes some preconditioning options when the system matrix is in blockpartitioned form. This form may arise naturally, for example from the incompressible NavierStokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to g ..."
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Cited by 43 (12 self)
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This paper proposes some preconditioning options when the system matrix is in blockpartitioned form. This form may arise naturally, for example from the incompressible NavierStokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner. The storage requirements for these preconditioners may be much less than for ILU preconditioners for tough, largescale CFD problems. The numerical experiments reported show that these preconditioners can help us solve difficult linear systems whose coefficient matrices are highly indefinite. 1 Introduction Consider the block partitioning of a matrix A, in the form A = ` B F E C ' (1) where the blocking naturally occurs due the ordering of the equations and the variables. Matrices of this form arise in many applications, such as in the incompressible NavierStokes equations, where the sc...
Distributed Schur Complement Techniques for General Sparse Linear Systems
 SIAM J. SCI. COMPUT
, 1997
"... This paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments. These techniques utilize the Schur complement system for deriving the preconditioning matrix in a number of ways. Two of these preconditioners consist of an approxima ..."
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Cited by 33 (13 self)
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This paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments. These techniques utilize the Schur complement system for deriving the preconditioning matrix in a number of ways. Two of these preconditioners consist of an approximate solution process for the global system, which exploit approximate LU factorizations for diagonal blocks of the Schur complement. Another preconditioner uses a sparse approximateinverse technique to obtain certain local approximations of the Schur complement. Comparisons are reported for systems of varying difficulty.
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 33 (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...
Block Preconditioners Based on Approximate Commutators
 SIAM J. SCI. COMPUT
, 2006
"... This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible NavierStokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [16]. ..."
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Cited by 24 (9 self)
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This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible NavierStokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [16]. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall e#cacy of this methodology. A drawback, however, is that it requires the construction of a convectiondiffusion operator (denoted Fp ) projected onto the discrete pressure space. This means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues, something often held only by the developers of the model. As an alternative, we consider automatic ways of computing Fp based on purely algebraic considerations. The new methods are closely related to the "BFBt preconditioner" of Elman [6]. We use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convectiondiffusion operators, together with methods for computing sparse approximate inverses, to generate the required matrix Fp automatically. We demonstrate that with this strategy, the favorable convergence properties of the preconditioning methodology are retained.
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 24 (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.
Harmonic Projection Methods for Large Nonsymmetric Eigenvalue Problems
 NUMER. LINEAR ALGEBRA APPL., 5, 33–55 (1998)
, 1998
"... The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other appro ..."
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Cited by 20 (9 self)
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The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to preconditioning methods, and to other methods for nonsymmetric eigenvalue problems that use the Rayleigh–Ritz procedure. The subject of estimating the boundary of the entire spectrum is briefly discussed, and the importance of preconditioning for interior eigenvalue problems is mentioned.
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 17 (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 ...
Preconditioned Krylov Subspace Methods for CFD Applications
 Proceedings of the International Workshop on Solution Techniques for LargeScale CFD Problems
, 1995
"... this paper we compare a number of standard preconditioning approaches to solve these problems. We test two accelerators, GMRES and DQGMRES, combined with a few threshold based preconditioners such as ILUT and approximate inverse techniques, on a number of linear systems arising from various models ..."
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Cited by 14 (4 self)
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this paper we compare a number of standard preconditioning approaches to solve these problems. We test two accelerators, GMRES and DQGMRES, combined with a few threshold based preconditioners such as ILUT and approximate inverse techniques, on a number of linear systems arising from various models
A New Approach to Parallel Preconditioning with Sparse Approximate Inverses
 Stanford University
, 1994
"... A new parallel preconditioner is presented for the solution of large, sparse, nonsymmetric linear systems of equations. A sparse approximate inverse is computed explicitly, and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and ..."
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Cited by 12 (0 self)
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A new parallel preconditioner is presented for the solution of large, sparse, nonsymmetric linear systems of equations. A sparse approximate inverse is computed explicitly, and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its application only requires a matrixvector product. The sparsity pattern of the approximate inverse is not imposed a priori but captured automatically. This keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum. Rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system, and the proximity of the approximate to the true inverse is estimated. An extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this new approach. 1 Introduction We consider the linear system of equations Ax = b; x; b 2 IR n : (1) Here A is a larg...
Parallel Implementation of a Sparse Approximate Inverse Preconditioner
 In SpringerVerlag, editor, Proceedings of Irregular'96
, 1996
"... . A parallel implementation of a sparse approximate inverse (spai) preconditioner for distributed memory parallel processors (dmpp) is presented. The fundamental spai algorithm is known to be a useful tool for improving the convergence of iterative solvers for illconditioned linear systems. The par ..."
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Cited by 10 (1 self)
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. A parallel implementation of a sparse approximate inverse (spai) preconditioner for distributed memory parallel processors (dmpp) is presented. The fundamental spai algorithm is known to be a useful tool for improving the convergence of iterative solvers for illconditioned linear systems. The parallel implementation (parspai) exploits the inherent parallelism in the spai algorithm and the data locality on the dmpps, to solve structurally symmetric (but nonsymmetric) matrices, which typically arise when solving partial differential equations (pdes). Some initial performance results are presented which suggest the usefulness of parspai for tackling such large size systems on present day dmpps in a reasonable time. The parspai preconditioner is implemented using the Message Passing Interface (mpi) and is embedded in the parallel library for unstructured mesh problems (plump). 1 Introduction We consider the linear system of equations Ax = b; x; b 2 IR n : (1) Here A is a large and...