Abstract
In this paper, a class of additive block triangular preconditioners are constructed for solving block two-by-two linear systems with symmetric positive (semi-)definite sub-matrices. Convergence analysis of the related splitting iteration method shows that it is almost unconditionally convergent and behaves problem independent with a convergence rate less than 0.5 under a practical parameter choice. Optimization of the preconditioned matrices, which have real and tight eigenvalue distributions, shows that it can result in an upper bound less than 2 for the condition number of the preconditioned matrices. Moreover, we also give a special consideration about the feasibility of the proposed preconditioner for solving more general problems with indefinite sub-matrices. Numerical experiments based on examples arising from complex symmetric linear systems and PDE-constrained optimization problems are presented to show the robustness and effectiveness of the proposed preconditioners compared with some other existing preconditioners.
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Axelsson, O., Boyanova, P., Kronbichler, M., Neytcheva, M., Wu, X.: Numerical and computational efficiency of solvers for two-phase problems. Comput. Math. Appl. 65(3), 301–314 (2013)
Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems Poisson and convection–diffusion control. Numer. Algorithm. 73(3), 631–663 (2016)
Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems Stokes control. Numer. Algorithm. 74(1), 19–37 (2017)
Axelsson, O., Farouq, S., Neytcheva, M.: A preconditioner for optimal control problems, constrained by Stokes equation with a time-harmonic control. J. Comput. Appl. Math. 310, 5–18 (2017)
Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7(4), 197–218 (2000)
Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithm. 66(4), 811–841 (2014)
Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011)
Bai, Z.-Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)
Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93(1), 41–60 (2015)
Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)
Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithm. 56, 297–317 (2011)
Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS Iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J. Numer. Anal. 33(1), 343–369 (2013)
Bai, Z.-Z., Chen, F., Wang, Z.-Q.: Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices. Numer. Algorithm. 62, 655–675 (2013)
Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24(3), 603–626 (2003)
Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28(3), 598–618 (2008)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)
Bosch, J., Stoll, M.: Preconditioning for vector-valued Cahn-Hilliard equations. SIAM J. Sci. Comput. 37(5), s216–s243 (2014)
Boyanova, P., Neytcheva, M.: Efficient numerical solution of discrete multi-component Cahn-Hilliard systems. Comput. Math. Appl. 67(1), 106–121 (2014)
Cao, S.-M., Feng, W., Wang, Z.-Q.: On a type of matrix splitting preconditioners for a class of block two-by-two linear systems. Appl. Math. Lett. 79, 205–210 (2018)
Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23(2), 480–498 (2001)
Elman, H.C., Ramage, A., Silvester, D.J.: Algorithm 866 IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33(2), 14 (2007)
Elman, H.C., Ramage, A., Silvester, D.J.: IFISS A computational laboratory for investigating incompressible flow problems. SIAM Rev. 56(2), 261–273 (2014)
Hezari, D., Edalatpour, V., Salkuyeh, D.K: Preconditioned GSOR Iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebera Appl. 22(4), 761–776 (2015)
Hezari, D., Salkuyeh, D.K., Edalatpour, V.: A new iterative method for solving a class of complex symmetric system of linear equations. Numer. Algorithm. 73(4), 927–955 (2016)
Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)
Krendl, W., Simoncini, V., Zulehner, W.: Stability estimates and structural spectral properties of saddle point problems. Numer. Math. 124(1), 183–213 (2013)
Lang, C., Ren, Z.-R.: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices. J. Eng. Math. 93(1), 87–98 (2015)
Liang, Z.-Z., Axelsson, O., Neytcheva, M.: A robust structured preconditioner for time-harmonic parabolic optimal control problems. Numer. Algorithm. 79(2), 575–596 (2017)
Liang, Z.-Z., Zhang, G.-F.: On SSOR iteration method for a class of block two-by-two linear systems. Numer. Algorithm. 71(3), 655–671 (2016)
Liao, L.-D., Zhang, G.-F., Li, R.-X.: Optimizing and improving of the C-to-R method for solving complex symmetric linear systems. Appl. Math. Lett. 336, 281–296 (2018)
Napov, A., Notay, Y.: An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), A1079–A1109 (2012)
Notay, Y: AGMG software and documentation; see http://agmg.eu/
Notay, Y.: Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput. 34(4), A2288–A2316 (2012)
Pearson, J.W., Wathen, A.J.: A new approximation of the schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 19(5), 816–829 (2012)
Praetorius, S., Voigt, A.: Development and analysis of a block-preconditioner for the phase-field crystal equation. SIAM J. Sci. Comput. 37(3), s425–s451 (2015)
Ren, Z.-R., Cao, Y., Zhang, L.-L.: On preconditioned MHSS real-valued iteration methods for a class of complex symmetric indefinite linear systems. East Asian J. Appl. Math. 6(2), 192–210 (2014)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Salkuyeh, D.K., Hezari, D., Edalatpour, V.: Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations. Int J. Comput. Math. 92(4), 802–815 (2015)
Sommerfeld, A.: Partial Differential Equations. Academic Press, New York (1949)
Van Rienen, U.: Numerical methods in computational electrodynamic: linear systems in practical applications. Springer, Berlin (2001)
Wang, T., Lu, L.-Z.: Alternating-directional PMHSS Iteration method for a class of two-by-two block linear systems. Appl. Math. Lett. 58, 159–164 (2016)
Xu, W.-W.: A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems. Appl. Math. Comput. 219, 10510–10517 (2013)
Yan, H.-Y., Huang, Y.-M.: Splitting-based block preconditioning methods for block two-by-two matrices of real square blocks. Appl. Math. Comput. 243, 825–837 (2014)
Zeng, M.-L., Zhang, G.-F.: Parameterized rotated block preconditioning techniques for block two-by-two systems with application to complex linear systems. Comput. Math. Appl. 70(12), 2946–2957 (2015)
Zheng, Q.-Q., Ma, C.-F.: Accelerated PMHSS Iteration methods for complex symmetric linear systems. Numer. Algorithm. 73(2), 501–516 (2016)
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We would like to express our sincere thanks to the unknown reviewer for his careful reading of the manuscript. His useful comments and valuable suggestions greatly improve the quality of the paper.
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This work was supported by the National Natural Science Foundation of China (Nos. 11801242 and 11771193) and the Fundamental Research Funds for the Central Universities (No. lzujbky-2018-31).
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Liang, ZZ., Zhang, GF. Robust additive block triangular preconditioners for block two-by-two linear systems. Numer Algor 82, 503–537 (2019). https://doi.org/10.1007/s11075-018-0611-2
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DOI: https://doi.org/10.1007/s11075-018-0611-2