Block-Symmetric and Block-Lower-Triangular Preconditioners for PDE-Constrained Optimization Problems
Year: 2013
Journal of Computational Mathematics, Vol. 31 (2013), Iss. 4 : pp. 370–381
Abstract
Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained optimization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices are derived. Numerical implementations show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1301-m4234
Journal of Computational Mathematics, Vol. 31 (2013), Iss. 4 : pp. 370–381
Published online: 2013-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: Saddle-point matrix Preconditioning PDE-constrained optimization Eigenvalue and eigenvector Regularization parameter.
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