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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: 

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