A preconditioner based on the shift-splitting method for generalized saddle point problems (original) (raw)
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SIAM Journal on Numerical Analysis, 2006
We examine block-diagonal preconditioners and efficient variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. We consider the general, nonsymmetric, nonsingular case. In particular, the (1,2) block need not equal the transposed (2,1) block. Our preconditioners arise from computationally efficient splittings of the (1,1) block. We provide analyses for the eigenvalue distributions and other properties of the preconditioned matrices. We extend the results of [de Sturler and Liesen 2003] to matrices with non-zero (2,2) block and to allow for the use of inexact Schur complements. To illustrate our eigenvalue bounds, we apply our analysis to a model Navier-Stokes problem, computing the bounds, comparing them to actual eigenvalue perturbations and examining the convergence behavior.
A note on constraint preconditioners for nonsymmetric saddle point problems
Numerical Linear Algebra with Applications, 2007
A class of constraint preconditioners for solving two-by-two block linear equations with the (1,2)-block being the transpose of the (2,1)-block and the (2,2)-block being zero was investigated in a recent paper of Cao (Numer. . In this short note, we extend his idea by allowing the (1,2)-block to be not equal to the transpose of the (2,1)-block. Results concerning the spectrum, the form of the eigenvectors and the convergence behaviour of a Krylov subspace method, such as GMRES are presented.
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arXiv (Cornell University), 2023
In this paper, we describe and analyze the spectral properties of a number of exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the FGMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center (1, 0) and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds.
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For the singular saddle-point problems with nonsymmetric positive definite (1, 1) block, we present a general constraint preconditioning (GCP) iteration method based on a singular constraint preconditioner. Using the properties of the Moore-Penrose inverse, the convergence properties of the GCP iteration method are studied. In particular, for each of the two different choices of the (1, 1) block of the singular constraint preconditioner, a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the GCP iteration method. Moreover, the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual (GMRES) and quasi-minimal residual (QMR) methods are also tested.
Block Preconditioners for Saddle Point Problems
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A general purpose block LU preconditioner for saddle point problems is presented. A major difference between the approach presented here and that of other studies is that an explicit, accurate approximation of the Schur complement matrix is efficiently computed. This is used to obtain a preconditioner to the Schur complement matrix which in turn defines a preconditioner for the global system. A number of variants are developed and results are reported for a few linear systems arising from CFD applications.
Block LU Preconditioners for Symmetric and Nonsymmetric Saddle Point Problems
SIAM Journal on Scientific Computing, 2003
In this paper, a block LU preconditioner for saddle point problems is presented. The main di erence between the approach presented here and that of other studies is that an explicit, accurate approximation of the Schur complement matrix is e ciently computed. This is used to compute a preconditioner to the Schur complement matrix that in turn de nes a preconditioner for a global iteration. The results indicate that this preconditioner is e ective on problems arising from CFD applications.
Mathematical Problems in Engineering, 2009
We establish two types of block triangular preconditioners applied to the linear saddle point problems with the singular 1,1 block. These preconditioners are based on the results presented in the paper of Rees and Greif 2007 . We study the spectral characteristics of the preconditioners and show that all eigenvalues of the preconditioned matrices are strongly clustered. The choice of the parameter is involved. Furthermore, we give the optimal parameter in practical. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.
Indefinite block triangular preconditioner for symmetric saddle point problems
Calcolo, 2013
In this paper, we consider an indefinite block triangular preconditioner for symmetric saddle point problems. The new eigenvalue distribution of the preconditioned matrix is derived and some corresponding results in Simoncini (Appl. Numer. Math. 49:63-80, 2004) and Wu et al. (Computing 84:183-208, 2009) are improved. Finally, numerical experiments of a model Stokes problem are reported.