TY - JOUR
T1 - On Block Preconditioners for PDE-Constrained Optimization Problems
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 - <p style="text-align: justify;">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.</p>