Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation

Bertaccini, Daniele; Golub, Gene H.; Serra‐Capizzano, Stefano; Possio, Cristina Tablino

doi:10.1007/s00211-004-0574-1

Cited by 75 publications

(91 citation statements)

References 37 publications

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“…Numerical examples confirm the theory and the effectiveness of the method [3,4,5]. The preconditioned HSS method was also applied to a discrete convection-diffusion equation and demonstrated its superiority with respect to the existing techniques [7].…”

Section: The Hermitian and Skew-hermitian Splliting (Hss) Methodssupporting

confidence: 54%

“…The first method is the filter algorithm [1] for the sparse eigenvalue problem, which consists of two algorithms: an Inexact Shift-Invert Lanczos method for efficiently obtaining good eigenvector approximations and an inexact Newton-type method to refine the eigenvector approximations to the desired accuracy. The second method is the Hermitian and Skew-Hermitian Splliting method [2][3][4][5][6][7][8] for solutions of linear systems.…”

Section: Methods and Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

Golub¹,

Ko²

2009

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Section: The Hermitian and Skew-hermitian Splliting (Hss) Methodssupporting

confidence: 54%

Section: Methods and Resultsmentioning

confidence: 99%

Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

Golub¹,

Ko²

2009

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“…This leads to the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method as follows. See also [2,3,4] and [11,12].…”

Section: The Preconditioned Hss Methodsmentioning

confidence: 99%

“…Hence, in a practical computation, it is a crucial but difficult problem to determine a good preconditioner P and choose an optimal iteration parameter α. For some discussions on this aspect, we refer the readers to [2,4,11,12].…”

Section: Consequently We Havementioning

confidence: 99%

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

2007

Self Cite

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Abstract. For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skewHermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

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“…The second approach is based on computing a preconditioner from the first matrix of a sequence and its further use in solving other systems (frozen preconditioner) [2]. The third approach consists in updating a preconditioner obtained from the matrix of one of the systems (seed preconditioner), and in repeating its updates when necessary [3,4]. The fourth approach is based on periodic recomputation of a preconditioner.…”

Section: Approaches To Solving the Sequence Of Linear Systemsmentioning

confidence: 99%

Choosing order of operations to accelerate strip structure analysis in parameter range

Kuksenko

Akhunov

Gazizov

2018

J. Phys.: Conf. Ser.

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Abstract. The paper considers the issue of using iteration methods in solving the sequence of linear algebraic systems obtained in quasistatic analysis of strip structures with the method of moments. Using the analysis of 4 strip structures, the authors have proved that additional acceleration (up to 2.21 times) of the iterative process can be obtained during the process of solving linear systems repeatedly by means of choosing a proper order of operations and a preconditioner. The obtained results can be used to accelerate the process of computer-aided design of various strip structures. The choice of the order of operations to accelerate the process is quite simple, universal and could be used not only for strip structure analysis but also for a wide range of computational problems. IntroductionDistributed circuits based on various strip structures are widely used in radioelectronic equipment, both as transmission lines that maintain proper characteristics for desired signals for a long time and as a basis for new protective devices. A strip structure consists of signal and ground conductors and a dielectric substrate. The separation between conductors, their thickness, other geometrical parameters and dielectric permittivity of a substrate can be repeatedly changed during simulation and optimization of elements and devices. These processes significantly increase computational costs. Generally, hardware accelerators (multicore workstations, clusters, graphical processing units) are used to decrease computational costs, while algorithmic methods are often ignored. A quasistatic approach, which is based on calculating electric capacitance with the method of moments [1], is widely used to decrease computational costs of strip structure analysis in contrast to the electrodynamic approach. The method of moments implies solving linear systems with dense matrix. This paper presents the research into application of iterative methods for solving dense linear systems repeatedly during multivariant analysis of strip structures using the method of moments with the optimal choice of order of operations.

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Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation

Cited by 75 publications

References 37 publications

Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

Solving large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

Choosing order of operations to accelerate strip structure analysis in parameter range

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