Abstract
We further analyze the solution of a class of block two-by-two linear systems. Instead of using the preconditioned GMRES iteration methods, we propose a new approximation of the Schur complement based on the special structure of this kind of block two-by-two matrix, and construct a practical restrictive preconditioner accordingly. Subsequently, we propose a practical restrictively preconditioned conjugate gradient (RPCG) method to solve this class of linear systems. The convergence property of the practical RPCG method is similar to the RPCG method. Last, numerical experiments show that this method is more efficient than some classical preconditioned Krylov subspace iteration methods.
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References
Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)
Badalassi, V.E., Ceniceros, H.D., Banerjee, S.: Computation of multiphase systems with phase field models. J. Comput. Phys. 190, 371–397 (2003)
Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)
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., 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, 343–369 (2013)
Bai, Z.-Z., Li, G.-Q.: Restrictively preconditioned conjugate gradient methods for systems of linear equations. IMA J. Numer. Anal. 23, 561–580 (2003)
Bai, Z.-Z., Pan, J.-Y.: Matrix Analysis and Computations. SIAM, Philadelphia (2021)
Bai, Z.-Z., Wang, Z.-Q.: Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems. J. Comput. Appl. Math. 187, 202–226 (2006)
Bai, Z.-Z., Yin, J.-F., Su, Y.-F.: A shift-splitting preconditioner for non-Hermitian positive definite matrices. J. Comput. Math. 24, 40–46 (2006)
Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)
Bertaccini, D.: Efficient solvers for sequences of complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)
Berti, A., Bochicchio, I.: A mathematical model for phase separation: a generalized Cahn-Hilliard equation. Math. Methods Appl. Sci. 34, 1193–1201 (2011)
Cao, Y., Li, S.: Block triangular preconditioners based on symmetric-triangular decomposition for generalized saddle point problems. Appl. Math. Comput. 358, 262–277 (2019)
Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for complex symmetric linear systems. J. Comput. Appl. Math. 344, 691–700 (2018)
Day, D.D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23, 480–498 (2001)
Edalatpour, V., Hezari, D., Salkuyeh, D.K.: Accelerated generalized SOR method for a class of complex systems of linear equations. Math. Commun. 20, 37–52 (2015)
Freund, R.W.: Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Stat. Comput. 13, 425–448 (1992)
Golub, G.H., Yuan, J.-Y.: Symmetric-triangular decomposition and its applications. Part I: theorems and algorithms. BIT Numer. Math. 42, 814–822 (2002)
Hezari, D., Edalatpour, V., Salkuyeh, D.K.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebra Appl. 22, 761–776 (2015)
Lass, O., Vallejos, M., Borzì, A., Douglas, C.C.: Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems. Computing 84, 27–48 (2009)
Li, X.-A., Zhang, W.-H., Wu, Y.-J.: On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations. Appl. Math. Lett. 79, 131–137 (2018)
Liang, Z.-Z., Zhang, G.-F.: On SSOR iteration method for a class of block two-by-two linear systems. Numer. Algorithms 71, 1–17 (2016)
Peng, X.-F., Li, W.: On the restrictively preconditioned conjugate gradient method for solving saddle point problems. Int. J. Comput. Math. 93, 142–159 (2016)
Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32, 271–298 (2010)
Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)
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, 802–815 (2015)
Yan, T.-X., Ma, C.-F.: A modified generalized shift-splitting iteration method for complex symmetric linear systems. Appl. Math. Lett. 117, 107129 (2021)
Zhang, J.-H., Wang, Z.-W., Zhao, J.: Preconditioned symmetric block triangular splitting iteration method for a class of complex symmetric linear systems. Appl. Math. Lett. 86, 95–102 (2018)
Zhao, P.-P., Huang, Y.-M.: A restrictive preconditioner for the system arising in half-quadratic regularized image restoration. Appl. Math. Lett. 115, 106916 (2021)
Zheng, Z., Zeng, M.-L., Zhang, G.-F.: A variant of PMHSS iteration method for a class of complex symmetric indefinite linear systems. Numer. Algorithms 91, 283–300 (2022)
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Supported by the R&D Program of Beijing Municipal Education Commission, China (No. KM202011232019).
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Chen, F., He, SR. Practical Restrictively Preconditioned Conjugate Gradient Methods for a Class of Block Two-by-Two Linear Systems. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-023-00356-9
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DOI: https://doi.org/10.1007/s42967-023-00356-9