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Volume 32, Issue 3
A New Preconditioning Strategy for Solving a Class of Time-Dependent PDE-Constrained Optimization Problems

Minli Zeng & Guofeng Zhang

DOI: 10.4208/jcm.1401-CR3

J. Comp. Math., 32 (2014), pp. 215-232.

Published online: 2014-06

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  • Abstract

In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distributed control problem involving the heat equation with two different functions. First a natural order-reduction is performed, and then the reduced-order linear system of equations is solved by the preconditioned MINRES algorithm with a new preconditioning techniques. The spectral properties of the preconditioned matrix are analyzed. Numerical results demonstrate that the preconditioning strategy for solving the large sparse systems discretized from the time-dependent problems is more effective for a wide range of mesh sizes and the value of the regularization parameter.

  • Keywords

PDE-constrained optimization, Reduced linear system of equations, Preconditioning, Saddle point problem, Krylov subspace methods.

  • AMS Subject Headings

65F08, 65F10, 65F50, 49M25, 49K20, 65N22.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-32-215, author = {Minli Zeng and Guofeng Zhang}, title = {A New Preconditioning Strategy for Solving a Class of Time-Dependent PDE-Constrained Optimization Problems}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {3}, pages = {215--232}, abstract = {

In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distributed control problem involving the heat equation with two different functions. First a natural order-reduction is performed, and then the reduced-order linear system of equations is solved by the preconditioned MINRES algorithm with a new preconditioning techniques. The spectral properties of the preconditioned matrix are analyzed. Numerical results demonstrate that the preconditioning strategy for solving the large sparse systems discretized from the time-dependent problems is more effective for a wide range of mesh sizes and the value of the regularization parameter.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1401-CR3}, url = {http://global-sci.org/intro/article_detail/jcm/9881.html} }
TY - JOUR T1 - A New Preconditioning Strategy for Solving a Class of Time-Dependent PDE-Constrained Optimization Problems AU - Minli Zeng & Guofeng Zhang JO - Journal of Computational Mathematics VL - 3 SP - 215 EP - 232 PY - 2014 DA - 2014/06 SN - 32 DO - http://doi.org/10.4208/jcm.1401-CR3 UR - https://global-sci.org/intro/article_detail/jcm/9881.html KW - PDE-constrained optimization, Reduced linear system of equations, Preconditioning, Saddle point problem, Krylov subspace methods. AB -

In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distributed control problem involving the heat equation with two different functions. First a natural order-reduction is performed, and then the reduced-order linear system of equations is solved by the preconditioned MINRES algorithm with a new preconditioning techniques. The spectral properties of the preconditioned matrix are analyzed. Numerical results demonstrate that the preconditioning strategy for solving the large sparse systems discretized from the time-dependent problems is more effective for a wide range of mesh sizes and the value of the regularization parameter.

Minli Zeng and Guofeng Zhang. (2014). A New Preconditioning Strategy for Solving a Class of Time-Dependent PDE-Constrained Optimization Problems. Journal of Computational Mathematics. 32 (3). 215-232. doi:10.4208/jcm.1401-CR3
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