The Parameterized Augmentation Block Preconditioner for Nonsymmetric Saddle Point Problems

doi:10.1007/s42967-024-00383-0

Abstract

Abstract: Based on the preconditioner presented by He and Huang (Comput Math Appl 62: 87–92, 2011), we introduce a parameterized augmentation block preconditioner for solving the nonsymmetric saddle point problems with the singular (1,1)-block. The theoretical analysis gives the eigenvalue and eigenvector properties of the corresponding preconditioned matrix, and numerical results confirm the effectiveness of the preconditioner for accelerating the convergence rate of the generalized minimal residual (GMRES) method when solving the large sparse nonsymmetric saddle point problems.

Key words: Augmentation, Preconditioner, Eigenvalue property, Generalized minimal residual (GMRES)

Bo Wu. The Parameterized Augmentation Block Preconditioner for Nonsymmetric Saddle Point Problems[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1704-1723.

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References

[1] Bai, Z.-J., Bai, Z.-Z.: On nonsingularity of block two-by-two matrices. Linear Algebra Appl. 439, 2388-2404 (2013)
[2] Bai, Z.-Z.: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput. 109, 273-285 (2000)
[3] Bai, Z.-Z.: Construction and analysis of structured preconditioners for block two-by-two matrices. J. of Shanghai Univ. 8, 397-405 (2004)
[4] Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791-815 (2006)
[5] Bai, Z.-Z.: Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks. J. Comput. Appl. Math. 237, 295-306 (2013)
[6] Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71-78 (2015)
[7] Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1-32 (2004)
[8] Bai, Z.-Z., Ng, M.K.: On inexact preconditioners for nonsymmetric matrices. SIAM J. Sci. Comput. 26, 1710-1724 (2005)
[9] Bai, Z.-Z., Pan, J.-Y.: Matrix Analysis and Computations. SIAM, Philadelphia (2021)
[10] Bai, Z.-Z., Tao, M.: On preconditioned and relaxed AVMM methods for quadratic programming problems with equality constraints. Linear Algebra Appl. 516, 264-285 (2017)
[11] Benzi, M., Liu, J.: Block preconditioning for saddle point systems with indefinite (1,1) block. Int. J. Comput. Math. 84, 1117-1129 (2007)
[12] Benzi, M., Wang, Z.: Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 33, 2761-2784 (2011)
[13] Benzi, M., Wathen, A.J.: Some preconditioning techniques for saddle point problems. Math. Ind. 13, 195-211 (2008)
[14] Cao, Z.-H.: Augmentation block preconditioners for saddle point-type matrices with singular (1,1) blocks. Numer. Linear Algebra Appl. 15, 515-533 (2008)
[15] Cao, Z.-H.: A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices. J. Comput. Appl. Math. 238, 109-115 (2013)
[16] Elman, H.C., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations. Numer. Math. 90, 665-688 (2002)
[17] Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, New York (2005)
[18] Greif, C., Schötzau, D.: Preconditioners for saddle point linear systems with highly singular (1,1) blocks. Electron. Trans. Numer. Anal. 22, 114-121 (2006)
[19] Greif, C., Schötzau, D.: Preconditioners for the discretized time-harmonic Maxwell equaitons in mixed form. Numer. Linear Algebra Appl. 14, 281-297 (2007)
[20] He, J., Huang, T.-Z.: Two augmentation preconditioners for nonsymmetric and indefinite saddle point linear systems with singular (1,1) blocks. Comput. Math. Appl. 62, 87-92 (2011)
[21] Huang, Z.-G., Wang, L.-G., Xu, Z., Cui, J.-J.: The improvements of the generalized shift-splitting preconditioners for non-singular and singular saddle point problems. Int. J. Comput. Math. 96, 797-820 (2019)
[22] Ipsen, I.C.F.: A note on preconditioning nonsymmetric matrices. SIAM J. Sci. Comput. 23, 1050-1051 (2001)
[23] Li, C.-X., Wu, S.-L.: Two block triangular preconditioners for nonsymmetric saddle point problems. Appl. Math. Comput. 269, 456-463 (2015)
[24] Martynova, T.S.: On augmented Lagrangian methods for saddle-point linear systems with singular or semidefinite (1,1) blocks. J. Comput. Math. 32, 297-305 (2014)
[25] Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21, 1969-1972 (2000)
[26] Perugia, I., Simoncini, V.: Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer. Linear Algebra Appl. 7, 585-616 (2000)
[27] Rees, T., Greif, C.: A preconditioner for linear systems arising from interior optimization methods. SIAM J. Sci. Comput. 29, 1992-2007 (2007)
[28] Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
[29] Shen, S.-Q., Huang, T.-Z., Zhang, J.-S.: Augmentation block triangular preconditioners for regularized saddle point problem. SIAM J. Matrix Appl. 33, 721-741 (2012)
[30] Wu, S.-L., Salkuyeh, D.K.: A shift-splitting preconditioner for asymmetric saddle point problems. Comput. Appl. Math. 39, 314 (2020)
[31] Zulenher, W.: Analysis of iterative methods for saddle point problems: a unified approach. Math. Comput. 71, 479-505 (2001)