• Original Articles •

### ON BLOCK PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS

Xiaoying Zhang, Yumei Huang

1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
• Received:2013-08-27 Revised:2014-01-15 Online:2014-05-15 Published:2014-05-22
• Supported by:

Yumei Huang is the corresponding author. The research of Huang is supported by NSFC Grant No. 11101195 and No. 11171371.

Xiaoying Zhang, Yumei Huang. ON BLOCK PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS[J]. Journal of Computational Mathematics, 2014, 32(3): 272-283.

Recently, Bai proposed a block-counter-diagonal and a block-counter-triangular preconditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in (Computing 91 (2011) 379-395). He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the block-counter-diagonal and the block-counter-triangular preconditioners. Experimental results show that the convergence analyses match well with the numerical results.

CLC Number:

 [1] Z.-Z. Bai, Construction and analysis of structured preconditioners for block two-by-two matrices, J Shanghai Univ (English Ed.), 8 (2004), 397-405.[2] Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), 791-815.[3] Z.-Z. Bai, Block preconditioners for elliptic PDE-constrained optimization problems, Computing, 91 (2011), 379-395.[4] Z.-Z. Bai and M.K. Ng, Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems, Numer. Math., 96 (2003), 197-220.[5] Z.-Z. Bai, M.K. Ng and Z.-Q.Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 31 (2009), 410-433.[6] M. Benzi, G.H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numerica, 14 (2005), 1-137.[7] H.C. Elman, D.J. Silvester and A.J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford and New York (2005).[8] C. Keller, N.I.M. Gould and A.J.Wathen, Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), 1300-1317.[9] O. Lass, M. Valleyjosand C.C. Douglas, Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems, Computing, 84 (2009), 27-48.[10] J.L. Lions, Optimal Control of Systems, Springer, Berlin and Heidelberg (1968).[11] M.F. Murphy, G.H. Golub and A.J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21 (2000), 1969-1972.[12] J.-Y. Pan, M.K. Ng and Z.-Z. Bai, New preconditioners for saddle point problems, Appl. Math. Comput., 172 (2006), 762-771.[13] T. Rees, H.S. Dollar and A.J. Wathen, Optimal solvers for PDE-constrained optimization, SIAM J. Sci. Comput., 32 (2010), 271-298.[14] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, PA, Second Ed., (2003).[15] R.S. Varga, Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1962.
 [1] Lu Zhang, Qifeng Zhang, Hai-wei Sun. A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(5): 708-732. [2] Yifen Ke, Changfeng Ma. MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS [J]. Journal of Computational Mathematics, 2021, 39(5): 733-754. [3] Daniele Boffi, Zhongjie Lu, Luca F. Pavarino. ITERATIVE ILU PRECONDITIONERS FOR LINEAR SYSTEMS AND EIGENPROBLEMS [J]. Journal of Computational Mathematics, 2021, 39(4): 633-654. [4] Kaibo Hu, Ragnar Winther. WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS [J]. Journal of Computational Mathematics, 2021, 39(3): 333-357. [5] Maohua Ran, Chengjian Zhang. A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS [J]. Journal of Computational Mathematics, 2020, 38(2): 239-253. [6] Davod Hezari, Vahid Edalatpour, Hadi Feyzollahzadeh, Davod Khojasteh Salkuyeh. ON THE GENERALIZED DETERIORATED POSITIVE SEMI-DEFINITE AND SKEW-HERMITIAN SPLITTING PRECONDITIONER [J]. Journal of Computational Mathematics, 2019, 37(1): 18-32. [7] Yang Cao, Zhiru Ren, Linquan Yao. IMPROVED RELAXED POSITIVE-DEFINITE AND SKEW-HERMITIAN SPLITTING PRECONDITIONERS FOR SADDLE POINT PROBLEMS [J]. Journal of Computational Mathematics, 2019, 37(1): 95-111. [8] Xinhui Shao, Chen Li, Tie Zhang, Changjun Li. A MODIFIED PRECONDITIONER FOR PARAMETERIZED INEXACT UZAWA METHOD FOR INDEFINITE SADDLE POINT PROBLEMS [J]. Journal of Computational Mathematics, 2018, 36(4): 579-590. [9] Davide Forti, Alfio Quarteroni, Simone Deparis. A PARALLEL ALGORITHM FOR THE SOLUTION OF LARGE-SCALE NONCONFORMING FLUID-STRUCTURE INTERACTION PROBLEMS IN HEMODYNAMICS [J]. Journal of Computational Mathematics, 2017, 35(3): 363-380. [10] Lingsheng Meng, Bing Zheng. STRUCTURED CONDITION NUMBERS FOR THE TIKHONOV REGULARIZATION OF DISCRETE ILL-POSED PROBLEMS [J]. Journal of Computational Mathematics, 2017, 35(2): 169-186. [11] Yonghong Ren, Fangfang Guo, Yang Li. NONLINEAR LAGRANGIANS FOR NONLINEAR PROGRAMMING BASED ON MODIFIED FISCHER-BURMEISTER NCP FUNCTIONS [J]. Journal of Computational Mathematics, 2015, 33(4): 396-414. [12] Xin He, Maya Neytcheva, Cornelis Vuik. ON PRECONDITIONING OF INCOMPRESSIBLE NON-NEWTONIAN FLOW PROBLEMS [J]. Journal of Computational Mathematics, 2015, 33(1): 33-58. [13] Minli Zeng, Guofeng Zhang. A NEW PRECONDITIONING STRATEGY FOR SOLVING A CLASS OF TIME-DEPENDENT PDE-CONSTRAINED OPTIMIZATION PROBLEMS [J]. Journal of Computational Mathematics, 2014, 32(3): 215-232. [14] Qiang Niu, Michael Ng. NUMERICAL STUDIES OF A CLASS OF COMPOSITE PRECONDITIONERS [J]. Journal of Computational Mathematics, 2014, 32(2): 136-151. [15] Guofeng Zhang, Zhong Zheng. BLOCK-SYMMETRIC AND BLOCK-LOWER-TRIANGULAR PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS [J]. Journal of Computational Mathematics, 2013, 31(4): 370-381.
Viewed
Full text

Abstract