Damped Jacobi Preconditioning and Coarse Grid Deflation for Conjugate Gradient Iteration on Parallel Computers

Mansfield, Lois

doi:10.1137/0912071

Cited by 39 publications

(26 citation statements)

References 6 publications

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“…Another preconditioning strategy that has proven successful when there are a few isolated extremal eigenvalues is deflation [20,16,17]. Let us define the projection P by…”

Section: Background: Preconditioning and Deflationmentioning

confidence: 99%

“…Nicolaides [20] chooses Z to be a piecewise constant interpolation from a set of m subdomains and points out that deflation might be effectively used with a conventional preconditioner. Mansfield [17] uses the same "subdomain deflation" in combination with damped Jacobi smoothing, obtaining a preconditioner which is related to the two-grid method.…”

Section: Background: Preconditioning and Deflationmentioning

confidence: 99%

See 1 more Smart Citation

On the Construction of Deflation-Based Preconditioners

Frank¹,

Vuik²

2001

SIAM J. Sci. Comput.

122

159

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Abstract. In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355-365], these theorems can provide direction in choosing a proper decomposition into subdomains. If grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication. Numerical experiments for a steady-state convection-diffusion problem are included.

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“…Another preconditioning strategy that has proven successful when there are a few isolated extremal eigenvalues is deflation [20,16,17]. Let us define the projection P by…”

Section: Background: Preconditioning and Deflationmentioning

confidence: 99%

Section: Background: Preconditioning and Deflationmentioning

confidence: 99%

On the Construction of Deflation-Based Preconditioners

Frank¹,

Vuik²

2001

SIAM J. Sci. Comput.

122

159

View full text Add to dashboard Cite

show abstract

“…Nicolaides [17] chooses Z to be a piecewise constant interpolation from a set of r subdomains and points out that deflation might be effectively used with a conventional preconditioner. Mansfield [13] uses the same "subdomain deflation" in combination with damped Jacobi smoothing, obtaining a preconditioner which is related to the two-grid method. In [11] Kolotilina uses a twofold deflation technique for simultaneously deflating the r largest and the r smallest eigenvalues using an appropriate deflating subspace of dimension r. Other authors have attempted to choose a subspace a priori that effectively represents the slowest modes.…”

Section: Introductionmentioning

confidence: 99%

A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow

Nabben¹,

Vuik²

2004

SIAM J. Numer. Anal.

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“…Note here that (11) can also be derived directly from the deflated equation (7) and (9). The augmented matrix in (11), denoted by hereafter, has been used for preconditioning of ill-conditioned system of linear equations [7].…”

Section: B Iec-sd Methodsmentioning

confidence: 99%

Deflation Techniques for Computational Electromagnetism: Theoretical Considerations

Igarashi

Watanabe

2011

IEEE Trans. Magn.

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Abstract-This paper describes deflation of finite element (FE) matrices for electromagnetic fields. The condition of the FE matrices is improved by the matrix deflation which replaces small eigenvalues with zeros. It is known that convergence of linear solvers for FE equations can be improved by using the AV method as well as explicit and implicit error correction (EC) methods which have been derived from the multigrid method. These numerical results are theorized on the basis of the matrix deflation. In particular, augmented matrices appeared in the AV and implicit EC methods are shown to have good conditioning after preconditioning. These results suggest that the above methods are based on a common mathematical principle.

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Damped Jacobi Preconditioning and Coarse Grid Deflation for Conjugate Gradient Iteration on Parallel Computers

Cited by 39 publications

References 6 publications

On the Construction of Deflation-Based Preconditioners

On the Construction of Deflation-Based Preconditioners

A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow

Deflation Techniques for Computational Electromagnetism: Theoretical Considerations

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