A new iterative method for solving the systems arisen from finite element discretization of a time-harmonic parabolic optimal control problems. (English) Zbl 1540.65356
Summary: In this paper, we focus on solving a class of two-by-two block complex system of linear equations arising from finite element discretization of a distributed optimal control constrained by a time-harmonic parabolic equations. We propose a new iterative method for solving the obtained system. In each iteration of the method a two-by-two block system of linear equations with real coefficient matrix should be solved. We solve this system inexactly using the generalized minimal residual (GMRES) and the Chebyshev acceleration methods in conjunction with the real-valued preconditioned square block (PRESB) preconditioner. The convergence and spectral properties of the method are discussed. Numerical results in 2-dimensional case are presented to demonstrate the efficiency of the method.
MSC:
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 65F08 | Preconditioners for iterative methods |
| 35K25 | Higher-order parabolic equations |
| 49M25 | Discrete approximations in optimal control |
| 49M41 | PDE constrained optimization (numerical aspects) |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
preconditioner; finite element; PDE-constrained; optimization; GMRES; PRESB; Chebyshev accelerationSoftware:
MinResReferences:
| [1] | Axelsson, O.; Boyanova, P.; Kronbichler, M.; Neytcheva, M.; Wu, X., Numerical and computational efficiency of solvers for two-phase problems, Comput. Math. Appl., 65, 301-314 (2013) · Zbl 1319.76025 |
| [2] | Axelsson, O.; Lukas, D., Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems, J. Numer. Math., 27, 1-21 (2019) · Zbl 1428.65040 |
| [3] | Axelsson, O.; Neytcheva, M.; Ahmad, B., A comparison of iterative methods to solve complex valued linear algebraic systems, Numer. Algorithms, 66, 811-841 (2014) · Zbl 1307.65034 |
| [4] | Axelsson, O.; Neytcheva, M.; Liang, Z.-Z., Parallel solution methods and preconditioners for evolution equations, Math. Model. Anal., 23, 287-308 (2018) · Zbl 1488.65064 |
| [5] | Axelsson, O.; Salkuyeh, D. K., A new version of a preconditioning method for certain two-by-two block matrices with square blocks, BIT, 59, 321-342 (2018) · Zbl 07074115 |
| [6] | Bai, Z.-Z., Block alternating splitting implicit iterative methods for saddle-point problems from time-harmonic eddy current models, Numer. Linear Algebra Appl., 19, 914-936 (2012) · Zbl 1289.65048 |
| [7] | Bai, Z.-Z.; Benzi, M.; Chen, F., Modified HSS iterative methods for a class of complex symmetric linear systems, Computing, 87, 93-111 (2010) · Zbl 1210.65074 |
| [8] | Bai, Z.-Z.; Benzi, M.; Chen, F., On preconditioned MHSS iterative methods for complex symmetric linear systems, Numer. Algorithms, 56, 297-317 (2011) · Zbl 1209.65037 |
| [9] | Bai, Z.-Z.; Benzi, M.; Chen, F.; Wang, Z.-Q., Preconditioned MHSS iterative 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) · Zbl 1271.65100 |
| [10] | 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, 603-626 (2003) · Zbl 1036.65032 |
| [11] | Bai, Z.-Z.; Golub, G. H.; Ng, M. K., On successive overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14, 319-335 (2007) · Zbl 1199.65097 |
| [12] | Bai, Z.-Z.; Golub, G. H.; Ng, M. K., On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428, 413-440 (2008) · Zbl 1135.65016 |
| [13] | Gunzburger, M.; Trenchea, C., Optimal control of the time-periodic MHD equations, Nonlinear Anal., 63, 1687-1699 (2005) · Zbl 1224.49053 |
| [14] | Jiang, M.-Q.; Cao, Y., On local Hermitian and skew-Hermitian splitting iterative methods for generalized saddle point problems, J. Comput. Appl. Math., 231, 973-982 (2009) · Zbl 1221.65091 |
| [15] | Kollmann, M.; Kolmbauer, M., A preconditioned minres solver for time-periodic parabolic optimal control problems, Numer. Linear Algebra Appl., 20, 761-784 (2012) · Zbl 1313.49035 |
| [16] | Kolmbauer, M.; Langer, U., A robust preconditioned MINRES solver for distributed time-periodic eddy current optimal control problems, SIAM J. Sci. Comput., 34, 785-809 (2012) · Zbl 1258.49059 |
| [17] | Krendl, W.; Simoncini, V.; Zulehner, W., Stability estimates and structural spectral properties of saddle point problems, Numer. Math., 124, 183-213 (2013) · Zbl 1269.65032 |
| [18] | Liang, Z.-Z.; Axelsson, O.; Neytcheva, M., A robust structured preconditioner for time-harmonic parabolic optimal control problems, Numer. Algorithms, 79, 575-596 (2018) · Zbl 1400.49041 |
| [19] | Rees, T.; Dollar, H. S.; Wathen, A. J., Optimal solvers for PDE-constrained optimization, SIAM J. Sci. Comput., 32, 271-298 (2010) · Zbl 1208.49035 |
| [20] | Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469 (1993) · Zbl 0780.65022 |
| [21] | Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042 |
| [22] | 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) · Zbl 0599.65018 |
| [23] | Zheng, Z.; Zhang, G.-F.; Zhu, M.-Z., A block alternating splitting iterative method for a class of block two-by-two complex linear systems, Comput. Math. Appl., 288, 203-214 (2015) · Zbl 1328.65078 |
| [24] | Zhu, M.; Zhang, G.-F.; Zheng, Z.; Liang, Z.-Z., On HSS-based sequential two-stage method for non-Hermitian saddle point problems, Appl. Math. Comput., 242, 907-916 (2014) · Zbl 1336.65051 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
