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Volume 32, Issue 3
On Block Preconditioners for PDE-Constrained Optimization Problems

Xiaoying Zhang & Yumei Huang

DOI: 10.4208/jcm.1401-CR4

J. Comp. Math., 32 (2014), pp. 272-283.

Published online: 2014-06

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  • Abstract

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.

  • Keywords

PDE-constrained optimization, GMRES method, Preconditioner, Condition number, Asymptotic convergence factor.

  • AMS Subject Headings

65F08, 65F10.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-32-272, author = {Xiaoying Zhang and Yumei Huang}, title = {On Block Preconditioners for PDE-Constrained Optimization Problems}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {3}, pages = {272--283}, abstract = {

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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1401-CR4}, url = {http://global-sci.org/intro/article_detail/jcm/9885.html} }
TY - JOUR T1 - On Block Preconditioners for PDE-Constrained Optimization Problems AU - Xiaoying Zhang & Yumei Huang JO - Journal of Computational Mathematics VL - 3 SP - 272 EP - 283 PY - 2014 DA - 2014/06 SN - 32 DO - http://doi.org/10.4208/jcm.1401-CR4 UR - https://global-sci.org/intro/article_detail/jcm/9885.html KW - PDE-constrained optimization, GMRES method, Preconditioner, Condition number, Asymptotic convergence factor. AB -

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.

Xiaoying Zhang and Yumei Huang. (2014). On Block Preconditioners for PDE-Constrained Optimization Problems. Journal of Computational Mathematics. 32 (3). 272-283. doi:10.4208/jcm.1401-CR4
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