Abstract
In this work, with respect to the regularization matrix of the new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for non-Hermitian saddle-point problems, we provide a class of Hermitian positive semidefinite matrices, which depend on certain parameters, for practical computations. A precise description about the eigenvalue distribution of the corresponding preconditioned matrix is given. The condition number of the eigenvector matrix, which partly determines the convergence rate of the related preconditioned Krylov subspace method, is also discussed in this work. Finally, some numerical experiments are carried out to identify the effectiveness of the presented special choices for the regularization matrix to solve the non-Hermitian saddle-point problems.
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References
Bai Z-Z (2000) Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl Math Comput 109:273–285
Bai Z-Z, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24:603–626
Bai Z-Z (2003) On the convergence of additive and multiplicative splitting iterations for systems of linear equations. J Comput Appl Math 154:195–214
Bai Z-Z, Golub GH, Pan J-Y (2004) Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer Math 98:1–32
Bai Z-Z, Golub GH, Lu L-Z, Yin J-F (2005) Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J Sci Comput 26:844–863
Bai Z-Z (2006) Structured preconditioners for nonsingular matrices of block two-by-two structures. Math Comput 75:791–815
Bai Z-Z, Golub GH (2007) Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J Numer Anal 27:1–23
Bai Z-Z (2009) Optimal parameters in the HSS-like methods for saddle-point problems. Numer Linear Algebra Appl 16:447–479
Bai Z-Z, Ng MK, Wang Z-Q (2009) Constraint preconditioners for symmetric indefinite matrices. SIAM J Matrix Anal Appl 31:410–433
Bai Z-Z, Benzi M, Chen F (2010) Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87:93–111
Bai Z-Z (2015) Motivations and realizations of Krylov subspace methods for large sparse linear systems. J Comput Appl Math 283:71–78
Bai Z-Z, Benzi M (2017) Regularized HSS iteration methods for saddle-point linear systems. BIT Numer Math 57:287–311
Bai Z-Z (2019) Regularized HSS iteration methods for stabilized saddle-point problems. IMA J Numer Anal 39:1888–1923
Bai Z-Z (2018) On spectral clustering of HSS preconditioner for generalized saddle-point matrices. Linear Algebra Appl 555:285–300
Benzi M, Golub GH (2004) A preconditioner for generalized saddle point problems. SIAM J Matrix Anal Appl 26:20–41
Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137
Benzi M, Guo X-P (2011) A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations. Appl Numer Math 61:66–76
Benzi M, Ng MK, Niu Q, Wang Z (2011) A relaxed dimensional factorization preconditioner for the incompressible Navier-Stokes equations. J Comput Phys 230:6185–6202
Benzi M, Deparis S, Grandperrin G, Quarteroni A (2016) Parameter estimates for the relaxed dimensional factorization preconditioner and application to hemodynamics. Comput Methods Appl Mech Eng 300:129–145
Cao Y, Yao L-Q, Jiang M-Q, Niu Q (2013) A relaxed HSS preconditioner for saddle point problems from meshfree discretization. J Comput Math 21:398–421
Cao Y, Jiang M-Q, Yao L-Q (2013) A modified dimensional split preconditioner for generalized saddle point problems. J Comput Appl Math 250:70–82
Cao Y, Dong J-L, Wang Y-M (2015) A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier-Stokes equation. J Comput Appl Math 273:41–60
Cao Y, Ren Z-R, Shi Q (2016) A simplified HSS preconditioner for generalized saddle point prob- lems. BIT Numer Math 56:423–439
Cao Y (2018) Regularized DPSS preconditioners for non-Hermitian saddle point problems. Appl Math Lett 84:96–102
Cao Z-H (2007) Positive stable block triangular preconditioners for symmetric saddle point problems. Appl Numer Math 57:899–910
Dollar HS (2007) Constraint-style preconditioners for regularized saddle-point problems. SIAM J Matrix Anal Appl 29:672–684
Elman HC, Ramage A, Silvester DJ (2014) IFISS: a computational laboratory for investigating incompressible flow problems. SIAM Rev. 56:261–273
Greenbaum A (1997) Iterative Methods for Solving Linear Systems. Society for Industrial and Applied Mathematics, SIAM, Philadelphia
Huang Y-M (2014) A practical formula for computing optimal parameters in the HSS iteration methods. J Comput Appl Math 255:142–149
Pan J-Y, Ng MK, Bai Z-Z (2006) New preconditioners for saddle point problems. Appl Math Comput 172:762–771
Saad Y (2003) Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia
Sturler ED, Liesen J (2005) Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. SIAM J Sci Comput 26:1598–1619
Zhang J-L (2018) An efficient variant of HSS preconditioner for generalized saddle point problems. Numer Linear Algebra Appl 25:e2166
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Communicated by Jinyun Yuan.
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The work is partially supported by National Natural Science Foundation of China (No. 11801362) and (No. 11601323).
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Zhang, JL. On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems. Comp. Appl. Math. 39, 191 (2020). https://doi.org/10.1007/s40314-020-01226-3
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DOI: https://doi.org/10.1007/s40314-020-01226-3
Keywords
- Non-Hermitian saddle-point problems
- New RDPSS preconditioner
- Regularization matrix
- Matrix similar transformation
- Spectral properties