Class of Uzawa-NPHSS iteration method for solving nonsingular and singular saddle point problems

References

• Benzi M, Golub GH, Liesen J. Numerical solution of saddle point problems. Acta Numer. 1999;14(2):1–137.
   Google Scholar
• Betts JT, Kolmanovsky I. Practical methods for optimal control using nonlinear programming. Appl Mech Rev. 2002;55(4):B68.
   Google Scholar
• Perugia I, Simoncini V. Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer Linear Algebra Appl. 2000;7:585–616.
   Web of Science ®Google Scholar
• Elman HC, Silvester DJ, Wathen AJ. Performance and analysis of saddle point preconditioners for the discrete stead-state Navier-Stokes equations. Numer Math. 2002;90(4):665–688.
   Web of Science ®Google Scholar
• Yuan J-Y, Iusem AN. Preconditioned conjugate gradient method for generalized least squares problems. J Comput Appl Math. 1996;71(2):287–297.
   Web of Science ®Google Scholar
• Bramble JH, Pasciak JE, Vassilev AT. Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J Numer Anal. 1997;34(3):1072–1092.
   Web of Science ®Google Scholar
• Elman HC, Golub GH. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J Numer Anal. 1994;31(6):1645–1661.
   Web of Science ®Google Scholar
• Bramble JH, Pasciak JE, Vassilev AT. Uzawa type algorithms for nonsymmetric saddle point problems. Math Comput. 2000;69(230):667–689.
   Web of Science ®Google Scholar
• Bacuta C. A unified approach for Uzawa algorithms. SIAM J Numer Anal. 2006;44(6):2633–2649.
   Web of Science ®Google Scholar
• Bai Z-Z, Wang Z-Q. On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 2008;428(11-12):2900–2932.
   Web of Science ®Google Scholar
• Wang N-N, Li J-C, Li G, et al. Variants of the Uzawa method for three-order block saddle point problem. Appl Math Comput. 2017;305:188–202.
   Web of Science ®Google Scholar
• Li J-C, Wang N-N, Kong X. Uzawa-low method and preconditioned Uzawa-low method for three-order block saddle point problem. Appl Math Comput. 2015;269:626–636.
   Web of Science ®Google Scholar
• Bai Z-Z, Parlett BN, Wang Z-Q. On generalized successive overrelaxation methods for augmented linear systems. Numer Math. 2005;102(1):1–38.
   Web of Science ®Google Scholar
• Bai Z-Z, Golub GH, Ng MK. On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer Linear Algebra Appl. 2007;14(4):319–335.
   Web of Science ®Google Scholar
• Golub GH, Wu X, Yuan J-Y. SOR-like methods for augmented systems. BIT. 2001;41(1):71–85.
   Google Scholar
• Li J-C, Kong X. Optimal parameters of GSOR-like methods for solving the augmented linear systems. Appl Math Comput. 2008;204(1):150–161.
   Web of Science ®Google Scholar
• Bai Z-Z, Golub GH, Ng MK. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl. 2003;24(3):603–626.
   Web of Science ®Google Scholar
• Bai Z-Z, Golub GH, Pan J-Y. Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer Math. 2004;98(1):1–32.
   Web of Science ®Google Scholar
• Bai Z-Z, Golub GH. Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J Numer Anal. 2007;27:1–23.
   Web of Science ®Google Scholar
• Bai Z-Z. Optimal parameters in the HSS-like methods for saddle-point problems. Numer Linear Algebra Appl. 2009;16(6):447–479.
   Web of Science ®Google Scholar
• Rozloznik M, Simoncini V. Krylov subspace methods for saddle point problems with indefinite preconditioning. SIAM J Matrix Anal Appl. 2002;24(2):368–391.
   Web of Science ®Google Scholar
• Zheng B, Bai Z-Z, Yang X. On semi-convergence of parameterized Uzawa methods for singular saddle point problems. Linear Algebra Appl. 2009;431(5):808–817.
   Web of Science ®Google Scholar
• Zhang G-F, Wang S-S. A generalization of parameterized inexact Uzawa method for singular saddle point problems. Appl Math Comput. 2013;219(9):4225–4231.
   Web of Science ®Google Scholar
• Yang A-L, Dou Y, Wu Y-J. On generalized parameterized inexact Uzawa methods for singular saddle-point problems. Numer Algorithms. 2014;69(3):1–15.
   Web of Science ®Google Scholar
• Chen Y, Zhang N. A note on the generalization of parameterized inexact Uzawa method for singular saddle point problems. Appl Math Comput. 2014;235(4):318–322.
   Google Scholar
• Wu X, Silva BPB, Yuan J-Y. Conjugate gradient method for rank deficient saddle point problems. Numer Algorithms. 2004;35(2-4):139–154.
   Web of Science ®Google Scholar
• Yang A-L, Wu Y-J. The Uzawa-HSS method for saddle-point problems. Appl Math Lett. 2014;38:38–42.
   Web of Science ®Google Scholar
• Yang A-L, Li X, Wu Y-J. On semi-convergence of the Uzawa-HSS method for singular saddle-point problems. Appl Math Comput. 2015;252:88–98.
   Web of Science ®Google Scholar
• Huang Z-G, Wang L-G, Xu Z, et al. The generalized Uzawa-SHSS method for non-Hermitian saddle-point problems. Comput Appl Math. 2018;37(2):1213–1231.
   Google Scholar
• Zhang J, Shang J. A class of Uzawa-SOR methods for saddle point problems. Appl Math Comput. 2010;216(7):2163–2168.
   Web of Science ®Google Scholar
• Xiong J-S, Gao X-B. Semi-convergence analysis of Uzawa-AOR method for singular saddle point problems. Comput Appl Math. 2017;36(1):383–395.
   Google Scholar
• Liang Z-Z, Zhang G-F. PU-STS method for non-Hermitian saddle-point problems. Appl Math Lett. 2015;46:1–6.
   Web of Science ®Google Scholar
• Xiong J-S, Gao X-B. GSTS-Uzawa method for a class of complex singular saddle point problems. Comput Appl Math. 2017;36(1):11–12.
   Google Scholar
• Fan H-T, Wang X, Zheng B. A class of preconditioned generalized local PSS iteration methods for non-Hermitian saddle point problems. Comput Math Appl. 2016;72(4):1188–1204.
   Web of Science ®Google Scholar
• Li C-L, Ma C-F. The Uzawa-PPS iteration methods for nonsingular and singular non-Hermitian saddle point problems. Comput Math Appl. 2018;75(2):703–720.
   Web of Science ®Google Scholar
• Miao S-X. A new Uzawa-type method for saddle point problems. Appl Math Comput. 2017;300:95–102.
   Web of Science ®Google Scholar
• Wu Y-J, Li X, Yuan J-Y. A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems. Comput Appl Math. 2017;36(1):367–381.
   Google Scholar
• Saad Y. Iterative methods for sparse linear systems. 2nd ed. Philadelphia: SIAM; 2002.
   Google Scholar
• Berman A, Plemmons R. Nonnegative matrices in mathematical sciences. Philadelphia: SIAM; 1994.
   Google Scholar
• Bai Z-Z. Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl Math Comput. 2000;109:273–285.
   Web of Science ®Google Scholar
• Bai Z-Z. Motivations and realizations of Krylov subspace methods for large sparse linear systems. J Comput Appl Math. 2015;283:71–78.
   Web of Science ®Google Scholar
• Huang Y-M. A practical formula for computing optimal parameters in the HSS iteration methods. J Comput Appl Math. 2014;255:142–149.
   Web of Science ®Google Scholar
• Yang A-L. Scaled norm minimization method for computing the parameters of the HSS and the two-parameter HSS preconditioners. Numer Linear Algebra Appl. 2018;25:e2169.
   Web of Science ®Google Scholar