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77
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 58 (21 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...
A Lanczostype method for multiple starting vectors
 MATH. COMP
, 2000
"... Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczostype algorithm that ..."
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Cited by 54 (15 self)
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Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczostype algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left starting blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a builtin deflation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ lookahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczostype methods.
Lanczostype solvers for nonsymmetric linear systems of equations
 Acta Numer
, 1997
"... Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article ..."
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Cited by 41 (11 self)
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Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by lookahead are also discussed. www.DownloadPaper.ir
ML(k)BiCGSTAB: A BiCGSTAB variant based on multiple Lanczos starting vectors
 SIAM J. Sci. Comput
, 1999
"... Abstract. We present a variant of the popular BiCGSTAB method for solving nonsymmetric linear systems. The method, which we denote by ML(k)BiCGSTAB, is derived from a variant of the BiCG method based on a Lanczos process using multiple (k> 1) starting left Lanczos vectors. Compared with the origi ..."
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Cited by 27 (7 self)
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Abstract. We present a variant of the popular BiCGSTAB method for solving nonsymmetric linear systems. The method, which we denote by ML(k)BiCGSTAB, is derived from a variant of the BiCG method based on a Lanczos process using multiple (k> 1) starting left Lanczos vectors. Compared with the original BiCGSTAB method, our new method produces a residual polynomial which is of lower degree after the same number of steps, but which also requires fewer matrixvector products to generate, on average requiring only 1 + 1/k matvecs per step. Empirically, it also seems to be more stable and more quickly convergent. The new method can be implemented as a kterm recurrence and can be viewed as a bridge connecting the Arnoldibased FOM/GMRES methods and the Lanczosbased BiCGSTAB methods.
Recycling Subspace Information for Diffuse Optical Tomography
 SIAM J. Sci. Comput
, 2004
"... We discuss the efficient solution of a large sequence of slowly varying linear systems arising in computations for diffuse optical tomographic imaging. In particular, we analyze a number of strategies for recycling Krylov subspace information for the most efficient solution. We reconstruct threedim ..."
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Cited by 23 (4 self)
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We discuss the efficient solution of a large sequence of slowly varying linear systems arising in computations for diffuse optical tomographic imaging. In particular, we analyze a number of strategies for recycling Krylov subspace information for the most efficient solution. We reconstruct threedimensional...
QMRBased Projection Techniques for the Solution of NonHermitian Systems with Multiple RightHand Sides
, 2001
"... . In this work we consider the simultaneous solution of large linear systems of the form Ax (j) = b (j) ; j = 1; : : : ; K where A is sparse and nonHermitian. We describe singleseed and blockseed projection approaches to these multiple righthand side problems that are based on the QMR and bl ..."
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Cited by 21 (1 self)
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. In this work we consider the simultaneous solution of large linear systems of the form Ax (j) = b (j) ; j = 1; : : : ; K where A is sparse and nonHermitian. We describe singleseed and blockseed projection approaches to these multiple righthand side problems that are based on the QMR and block QMR algorithms, respectively. We use (block) QMR to solve the (block) seed system and generate the relevant biorthogonal subspaces. Approximate solutions to the nonseed systems are simultaneously generated by minimizing their appropriately projected (block) residuals. After the initial (block) seed has converged, the process is repeated by choosing a new (block) seed from among the remaining nonconverged systems and using the previously generated approximate solutions as initial guesses for the new seed and nonseed systems. We give theory for the singleseed case that helps explain the convergence behavior under certain conditions. Implementation details for both the singleseed and b...
Deflated iterative methods for linear equations with multiple righthand sides
, 2004
"... Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems ar ..."
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Cited by 20 (7 self)
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Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems are solved by combining restarted GMRES with a projection over the previously determined eigenvectors. This approach offers an alternative to block methods, and it can also be combined with a block method. It is useful when there are a limited number of small eigenvalues that slow the convergence. An example is given showing significant improvement for a problem from quantum chromodynamics. The second and subsequent righthand sides are solved much quicker than without the deflation. This new approach is relatively simple to implement and is very efficient compared to other deflation methods.