A generalization of the positive-definite and skew-Hermitian splitting iteration

Previous Article
A new Bramble-Pasciak-like preconditioner for saddle point problems
NACO Home
This Issue
Next Article
On product-type generalized block AOR method for augmented linear systems

2012, 2(4): 811-821. doi: 10.3934/naco.2012.2.811

A generalization of the positive-definite and skew-Hermitian splitting iteration

Yang Cao ^1, , Wei- Wei Tan ^2, and Mei-Qun Jiang ^2,

1.	School of Transportation, Nantong University, Nantong, 226019, China
2.	School of Mathematical Sciences, Soochow University, Suzhou, 215006, China, China

Received December 2011 Revised September 2012 Published November 2012

Abstract
Related Papers

In this paper, a generalization of the positive-definite and skew-Hermitian splitting (GPSS) iteration is considered for solving non-Hermitian and positive definite systems of linear equations. Theoretical analysis shows that the GPSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrices. In some situations, this new scheme can outperform the Hermitian and skew-Hermitian splitting (HSS) method, the positive-definite and skew-Hermitian splitting (PSS) method, and the generalized HSS method (GHSS) and can be used as an efficient preconditioner. Numerical experiments using discretization of convection-diffusion-reaction equations demonstrate the effectiveness of the new method.

Keywords: preconditioning., PSS iteration, HSS iteration, Positive-definite matrix.

Mathematics Subject Classification: 65F10, 65N2.

Citation: Yang Cao, Wei- Wei Tan, Mei-Qun Jiang. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 811-821. doi: 10.3934/naco.2012.2.811

References:

[1]	Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle point problems, Numer. Linear Algebra Appl., 16 (2009), 447-479. doi: 10.1002/nla.626.
[2]	Z.-Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing, 89 (2010), 171-197. doi: 10.1007/s00607-010-0101-.
[3]	Z.-Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010), 93-111. doi: 10.1007/s00607-010-0077-0.
[4]	Z.-Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor., 56 (2011), 297-317. doi: 10.1007/s11075-010-9441-6.
[5]	Z.-Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), 1-23. doi: 10.1093/imanum/drl017.
[6]	Z.-Z. Bai, G. H. Golub and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28 (2006), 583-603. doi: 10.1137/050623644.
[7]	Z.-Z. Bai, G. H. Golub, L.-Z. Lu and J.-F. Yin, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 23 (2005), 844-863. doi: 10.1137/S1064827503428114.
[8]	Z.-Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626. doi: 10.1137/S0895479801395458.
[9]	Z.-Z. Bai, G. H. Golub and M. K. Ng, On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), 319-335. doi: 10.1002/nla.517.
[10]	Z.-Z. Bai, G. H. Golub and M. K. Ng, On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008), 413-440. doi: 10.1016/j.laa.2007.02.018.
[11]	Z.-Z. Bai, G. H. Golub and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1-32.
[12]	M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), 360-374. doi: 10.1137/080723181.
[13]	M. Benzi, M. J. Gander and G. H Golub, Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT, 43 (2003), 881-900. doi: 10.1023/B:BITN.0000014548.26616.65.
[14]	M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41. doi: 10.1137/S0895479802417106.
[15]	M. Benzi and X.-P. Guo, A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math., 61 (2011), 66-76. doi: 10.1016/j.apnum.2010.08.005.
[16]	L. C. Chan, M. K. Ng and N. K. Tsing, Spectral Analysis for HSS Preconditioners, Numer. Math. Theor. Meth. Appl., 1 (2008), 57-77.
[17]	M.-Q. Jiang and Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), 973-982. doi: 10.1016/j.cam.2009.05.012.
[18]	L. Li, T.-Z. Huang and X.-P. Liu, Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems, Comput. Math. Appl., 54 (2007), 147-159. doi: 10.1016/j.camwa.2006.12.024.
[19]	J.-Y. Pan, M. K. Ng and Z.-Z. Bai, New preconditioners for saddle point problems, Appl. Math. Comput., 172 (2006), 762-771. doi: 10.1016/j.amc.2004.11.016.
[20]	D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl.Math., 3 (1955), 28-41. doi: 10.1137/0103003.
[21]	Y. Saad, "Iterative Methods for Sparse Linear Systems," 2^nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[22]	A.-L Yang, J. An and Y.-J. Wu, A generalized preconditioned HSS method for non-Hermitian positive definite linear systems, Appl. Math. Comput., 216 (2010), 1715-1722. doi: 10.1016/j.amc.2009.12.032.

show all references

References:

[1]	Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle point problems, Numer. Linear Algebra Appl., 16 (2009), 447-479. doi: 10.1002/nla.626.
[2]	Z.-Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing, 89 (2010), 171-197. doi: 10.1007/s00607-010-0101-.
[3]	Z.-Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010), 93-111. doi: 10.1007/s00607-010-0077-0.
[4]	Z.-Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor., 56 (2011), 297-317. doi: 10.1007/s11075-010-9441-6.
[5]	Z.-Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), 1-23. doi: 10.1093/imanum/drl017.
[6]	Z.-Z. Bai, G. H. Golub and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28 (2006), 583-603. doi: 10.1137/050623644.
[7]	Z.-Z. Bai, G. H. Golub, L.-Z. Lu and J.-F. Yin, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 23 (2005), 844-863. doi: 10.1137/S1064827503428114.
[8]	Z.-Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626. doi: 10.1137/S0895479801395458.
[9]	Z.-Z. Bai, G. H. Golub and M. K. Ng, On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), 319-335. doi: 10.1002/nla.517.
[10]	Z.-Z. Bai, G. H. Golub and M. K. Ng, On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008), 413-440. doi: 10.1016/j.laa.2007.02.018.
[11]	Z.-Z. Bai, G. H. Golub and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1-32.
[12]	M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), 360-374. doi: 10.1137/080723181.
[13]	M. Benzi, M. J. Gander and G. H Golub, Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT, 43 (2003), 881-900. doi: 10.1023/B:BITN.0000014548.26616.65.
[14]	M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41. doi: 10.1137/S0895479802417106.
[15]	M. Benzi and X.-P. Guo, A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math., 61 (2011), 66-76. doi: 10.1016/j.apnum.2010.08.005.
[16]	L. C. Chan, M. K. Ng and N. K. Tsing, Spectral Analysis for HSS Preconditioners, Numer. Math. Theor. Meth. Appl., 1 (2008), 57-77.
[17]	M.-Q. Jiang and Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), 973-982. doi: 10.1016/j.cam.2009.05.012.
[18]	L. Li, T.-Z. Huang and X.-P. Liu, Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems, Comput. Math. Appl., 54 (2007), 147-159. doi: 10.1016/j.camwa.2006.12.024.
[19]	J.-Y. Pan, M. K. Ng and Z.-Z. Bai, New preconditioners for saddle point problems, Appl. Math. Comput., 172 (2006), 762-771. doi: 10.1016/j.amc.2004.11.016.
[20]	D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl.Math., 3 (1955), 28-41. doi: 10.1137/0103003.
[21]	Y. Saad, "Iterative Methods for Sparse Linear Systems," 2^nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[22]	A.-L Yang, J. An and Y.-J. Wu, A generalized preconditioned HSS method for non-Hermitian positive definite linear systems, Appl. Math. Comput., 216 (2010), 1715-1722. doi: 10.1016/j.amc.2009.12.032.

[1]	Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[2]	Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297
[3]	Erchuan Zhang, Lyle Noakes. Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices. Journal of Geometric Mechanics, 2019, 11 (2) : 277-299. doi: 10.3934/jgm.2019015
[4]	David Maxwell. Kozlov-Maz'ya iteration as a form of Landweber iteration. Inverse Problems and Imaging, 2014, 8 (2) : 537-560. doi: 10.3934/ipi.2014.8.537
[5]	Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[6]	Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61
[7]	Sihem Guerarra. Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA^=B. Numerical Algebra, Control and Optimization*, 2019, 9 (1) : 15-22. doi: 10.3934/naco.2019002
[8]	Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2253-2277. doi: 10.3934/dcdss.2019144
[9]	Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems and Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018
[10]	Óscar Vega-Amaya, Joaquín López-Borbón. A perturbation approach to a class of discounted approximate value iteration algorithms with borel spaces. Journal of Dynamics and Games, 2016, 3 (3) : 261-278. doi: 10.3934/jdg.2016014
[11]	Xijun Hu, Li Wu. Decomposition of spectral flow and Bott-type iteration formula. Electronic Research Archive, 2020, 28 (1) : 127-148. doi: 10.3934/era.2020008
[12]	Gemma Huguet, Rafael de la Llave, Yannick Sire. Fast iteration of cocycles over rotations and computation of hyperbolic bundles. Conference Publications, 2013, 2013 (special) : 323-333. doi: 10.3934/proc.2013.2013.323
[13]	Rich Stankewitz, Hiroki Sumi. Random backward iteration algorithm for Julia sets of rational semigroups. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2165-2175. doi: 10.3934/dcds.2015.35.2165
[14]	Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079
[15]	Xiaoman Liu, Jijun Liu. Image restoration from noisy incomplete frequency data by alternative iteration scheme. Inverse Problems and Imaging, 2020, 14 (4) : 583-606. doi: 10.3934/ipi.2020027
[16]	Jiulong Liu, Nanguang Chen, Hui Ji. Learnable Douglas-Rachford iteration and its applications in DOT imaging. Inverse Problems and Imaging, 2020, 14 (4) : 683-700. doi: 10.3934/ipi.2020031
[17]	Zhong-Zhi Bai. On convergence of the inner-outer iteration method for computing PageRank. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 855-862. doi: 10.3934/naco.2012.2.855
[18]	Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683
[19]	Scott Crass. New light on solving the sextic by iteration: An algorithm using reliable dynamics. Journal of Modern Dynamics, 2011, 5 (2) : 397-408. doi: 10.3934/jmd.2011.5.397
[20]	Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Yang Cao Wei- Wei Tan Mei-Qun Jiang
on Google Scholar
Yang Cao Wei- Wei Tan Mei-Qun Jiang

[Back to Top]