基于正则化的预处理子求解非对称广义鞍点问题
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摘要
推广了文献[33]的正则化技巧用于求解非对称的广义鞍点问题.证明了相应迭代法的无条件收敛性及相应的预处理矩阵的谱性质.基于该预处理子,提出了一种松弛的预处理形式,对其预处理后的系统的特征性质给出了相关结论.通过数值试验证明了所提出的预处理子的有效性.
The regularization technique of the literature [33] was further generalized to solve non-symmetric generalized saddle point problems. The unconditional convergence property of the corresponding iterative method and the spectral properties of the corresponding preconditioned matrix were proved. Based on the preconditioners, a relaxed preconditioning form was proposed and relevant conclusions about the eigenproperties of the preconditioned system were also arrive at. The effectiveness of the proposed new precondtioners was illustrated by numerical experiments.
The regularization technique of the literature [33] was further generalized to solve non-symmetric generalized saddle point problems. The unconditional convergence property of the corresponding iterative method and the spectral properties of the corresponding preconditioned matrix were proved. Based on the preconditioners, a relaxed preconditioning form was proposed and relevant conclusions about the eigenproperties of the preconditioned system were also arrive at. The effectiveness of the proposed new precondtioners was illustrated by numerical experiments.
引文
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[19] Cao Yang, Jiang Mei-qun, Yao Lin-quan. On parameterized block triangular preconditioners for generalized saddle point problems[J]. Applied Mathematics and Computation, 2010, 216:1777-1789.
[20] Cao Yang, Du Jun, Niu Qiang. Shift-splitting preconditioners for saddle point problems[J]. Journal of Computational and Applied Mathematics, 2014, 272:239-250.
[21] Cao Yang, Li Sen, Yao Lin-quan. A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems[J]. Applied Mathematics Letters, 2015,49:20-27.
[22] Cao Yang, Dong Jun-liang, Wang Yu-ming. A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier-Stokes equation[J]. Journal of Computational and Applied Mathematics, 2015, 273:41-60.
[23] Cao Yang, Ren Zhi-ru, Shi Quan. A simplified HSS preconditioner for generalized saddle point problems[J].BIT Numerical Mathematics, 2016, 56(2):423-439.
[24] Zhang Ju-li, Gu Chuan-qing. A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems[J]. BIT Numerical Mathematics, 2016, 56(2):587-604.
[25] Zhang Ju-li, Gu Chuan-qing, Zhang Ke. A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems[J]. Applied Mathematics and Computation, 2014, 249:468-479.
[26] Cao Yang, Wang An, Chen Yu-juan. A modified relaxed positive-semidefinite and skew-Hermitian splitting preconditioner for generalized saddle point problems[J].East Asian Journal on Applied Mathematics, 2017, 7(1):192-210.
[27] Xie Ya-jun, Ma Chang-feng. A modified positive-definite and skew-Hermitian splitting preconditioner for generalized saddle point problems from the NavierStokes equation[J]. Numerical Algorithms, 2016, 72(1):243-258.
[28] Benzi M, Golub G H. A preconditioner for generalized saddle point problems[J]. SIAM Journal Matrix Analysis and Application, 2004, 26(1):20-41.
[29] Bai Zhong-zhi, Golub G H, Lu Lin-zhang, et al. Block triangular and skew-Hermitian splitting methods for positive-definite linear systems[J]. SIAM Journal on Scientific Computing, 2005, 26(3):844-863.
[30] Shen Shu-qian. A note on PSS preconditioners for generalized saddle point problems[J]. Applied Mathematics and Computation, 2014, 237:723-729.
[31] Pestana J, Wathen A J. Natural preconditioning and iterative methods for saddle point systems[J]. SIAM Review, 2015, 57(1):71-91.
[32] Beik F P A, Benzi M. Iterative methods for double saddle point systems[J]. SIAM Journal on Matrix Analysis and Applications, 2018, 39(2):902-921.
[33] Bai Zhong-zhi, Benzi M. Regularized HSS iteration methods for saddle point linear systems[J]. BIT Numerical Mathematics, 2017, 57(2):287-311.
[34] Saad Y. Iteration methods for sparse linear systems[M].Philadelphia:SIAM, 2003.
[35] Cao Yang. Regularized DPSS preconditioners for nonHermitian saddle point problems[J]. Applied Mathematics Letters, 2018, 84:96-102.
[36] Elman H C, Ramage A, Silvester D J. Algorithm 866:IFISS, a Matlab toolbox for modelling incompressible flow[J]. ACM Transactions on Mathematical Software,2007, 33(2):14.
[2] Battermann A, Heinkenschloss M. Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems[M]//Control and Estimation of Distributed Parameter Systems. Basel:Birkh?user, 1998:15-32.
[3] Brezzi F, Fortin M. Mixed and hybrid finite element methods[M]. New York:Springer Science&Business Media, 2012.
[4] Elman H C, Silvester D J, Wathen A J. Finite elements and fast iterative solvers:with applications in incompressible fluid dynamics[M]. Oxford:Oxford University Press, 2014.
[5] Benzi M, Golub G H, Liesen J. Numerical solution of saddle point problems[J]. Acta Numerica, 2005(14):1-137.
[6]雍龙泉,刘三阳,史加荣,等.几类特殊矩阵及性质[J].兰州大学学报:自然科学版, 2016, 52(3):385-389.
[7]刘都超,王晓燕,姚景华.基于一类(p1(x), p2(x))方程的研究[J].兰州大学学报:自然科学版, 2016, 52(2):251-257.
[8]胡哲.一类含临界对流项的拟线性椭圆方程Neumann问题正解的先验估计与可解性[J].兰州大学学报:自然科学版, 2017, 53(4):534-540.
[9] Bai Zhong-zhi, Golub G H, Ng M K. Hermitian and skewHermitian splitting methods for non-Hermitian positive definite linear systems[J]. SIAM Journal of Matrix Analysis Application, 2003, 24(3):603-626.
[10] Bai Zhong-zhi, Golub G H. Accelerated Hermitian and skew-Hermitian splitting methods for saddle point problems[J]. IMA Journal of Numerical Analysis, 2007,27(1):1-23.
[11] Bai Zhong-zhi, Golub G H, Pan Jian-yu. Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems[J].Numerische Mathematik, 2004, 98(1):1-32.
[12] Bai Zhong-zhi, Wang Zeng-qi. On parameterized inexact Uzawa methods for generalized saddle point problems[J]. Linear Algebra Application, 2008, 428(11/12):2900-2932.
[13] Dai Li-fang, Liang Mao-lin, Fan Hong-tao. A new generalized parameterized inexact Uzawa method for solving saddle point problems[J]. Applied Mathematics and Computation, 2015, 265:414-430.
[14] Bai Zhong-zhi, Parlett B N, Wang Zeng-qi. On generalized successive overrelaxation methods for augmented linear systems[J]. Numerische Mathematik, 2005,102(1):1-38.
[15] Zhang Guo-feng, Yang Jian-lin, Wang Shan-shan. On generalized parameterized inexact Uzawa method for block two-by-two linear systems[J]. Journal of Computational and Applied Mathematics, 2014, 255:193-207.
[16] Saad Y, Schultz M H. GMRES:a generalized minimal residual algorithm for sloving nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1986, 7(3):856-869.
[17] Murphy M F, Golub G H, Wathen A J. A note on preconditioning for indefinite linear systems[J]. SIAM Journal on Scientific Computing, 2000, 21(6):1969-1972.
[18] Beik F P A, Benzi M, Chaparpordi S H A. On block diagonal and block triangular iterative schemes and preconditioners for stabilized saddle point problems[J].Journal of Computational and Applied Mathematics,2017, 326:15-30.
[19] Cao Yang, Jiang Mei-qun, Yao Lin-quan. On parameterized block triangular preconditioners for generalized saddle point problems[J]. Applied Mathematics and Computation, 2010, 216:1777-1789.
[20] Cao Yang, Du Jun, Niu Qiang. Shift-splitting preconditioners for saddle point problems[J]. Journal of Computational and Applied Mathematics, 2014, 272:239-250.
[21] Cao Yang, Li Sen, Yao Lin-quan. A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems[J]. Applied Mathematics Letters, 2015,49:20-27.
[22] Cao Yang, Dong Jun-liang, Wang Yu-ming. A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier-Stokes equation[J]. Journal of Computational and Applied Mathematics, 2015, 273:41-60.
[23] Cao Yang, Ren Zhi-ru, Shi Quan. A simplified HSS preconditioner for generalized saddle point problems[J].BIT Numerical Mathematics, 2016, 56(2):423-439.
[24] Zhang Ju-li, Gu Chuan-qing. A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems[J]. BIT Numerical Mathematics, 2016, 56(2):587-604.
[25] Zhang Ju-li, Gu Chuan-qing, Zhang Ke. A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems[J]. Applied Mathematics and Computation, 2014, 249:468-479.
[26] Cao Yang, Wang An, Chen Yu-juan. A modified relaxed positive-semidefinite and skew-Hermitian splitting preconditioner for generalized saddle point problems[J].East Asian Journal on Applied Mathematics, 2017, 7(1):192-210.
[27] Xie Ya-jun, Ma Chang-feng. A modified positive-definite and skew-Hermitian splitting preconditioner for generalized saddle point problems from the NavierStokes equation[J]. Numerical Algorithms, 2016, 72(1):243-258.
[28] Benzi M, Golub G H. A preconditioner for generalized saddle point problems[J]. SIAM Journal Matrix Analysis and Application, 2004, 26(1):20-41.
[29] Bai Zhong-zhi, Golub G H, Lu Lin-zhang, et al. Block triangular and skew-Hermitian splitting methods for positive-definite linear systems[J]. SIAM Journal on Scientific Computing, 2005, 26(3):844-863.
[30] Shen Shu-qian. A note on PSS preconditioners for generalized saddle point problems[J]. Applied Mathematics and Computation, 2014, 237:723-729.
[31] Pestana J, Wathen A J. Natural preconditioning and iterative methods for saddle point systems[J]. SIAM Review, 2015, 57(1):71-91.
[32] Beik F P A, Benzi M. Iterative methods for double saddle point systems[J]. SIAM Journal on Matrix Analysis and Applications, 2018, 39(2):902-921.
[33] Bai Zhong-zhi, Benzi M. Regularized HSS iteration methods for saddle point linear systems[J]. BIT Numerical Mathematics, 2017, 57(2):287-311.
[34] Saad Y. Iteration methods for sparse linear systems[M].Philadelphia:SIAM, 2003.
[35] Cao Yang. Regularized DPSS preconditioners for nonHermitian saddle point problems[J]. Applied Mathematics Letters, 2018, 84:96-102.
[36] Elman H C, Ramage A, Silvester D J. Algorithm 866:IFISS, a Matlab toolbox for modelling incompressible flow[J]. ACM Transactions on Mathematical Software,2007, 33(2):14.