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Volume 14, Issue 4-5
Overlapping Domain Decomposition Preconditioners for Unconstrained Elliptic Optimal Control Problems

Zhiyu Tan, Wei Gong & Ningning Yan

Int. J. Numer. Anal. Mod., 14 (2017), pp. 550-570.

Published online: 2017-08

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  • Abstract

In this paper, we propose several overlapping domain decomposition preconditioners for solving the unconstrained elliptic optimal control problem, based on the two level additive Schwarz algorithm. We consider the cases with controls on the whole domain and controls from a local subset. The latter case can be viewed as the subproblems when we solve the control-constrained control problem by using semi-smooth Newton method. When the controls act on the whole domain, we construct a symmetric and positive definite preconditioner which is proved to be robust combined with preconditioned MINRES method, and a symmetric and indefinite preconditioner which can be used in the preconditioned GMRES method and shows better numerical performance than the positive definite one. When the controls act on a local subset, we also construct a similar symmetric and indefinite preconditioner, the numerical experiments show its efficiency when combined with preconditioned GMRES method.

  • Keywords

Overlapping domain decomposition method, elliptic optimal control problem, preconditioned MINRES method, preconditioned GMRES method.

  • AMS Subject Headings

49K20, 65F10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-550, author = {}, title = {Overlapping Domain Decomposition Preconditioners for Unconstrained Elliptic Optimal Control Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {4-5}, pages = {550--570}, abstract = {

In this paper, we propose several overlapping domain decomposition preconditioners for solving the unconstrained elliptic optimal control problem, based on the two level additive Schwarz algorithm. We consider the cases with controls on the whole domain and controls from a local subset. The latter case can be viewed as the subproblems when we solve the control-constrained control problem by using semi-smooth Newton method. When the controls act on the whole domain, we construct a symmetric and positive definite preconditioner which is proved to be robust combined with preconditioned MINRES method, and a symmetric and indefinite preconditioner which can be used in the preconditioned GMRES method and shows better numerical performance than the positive definite one. When the controls act on a local subset, we also construct a similar symmetric and indefinite preconditioner, the numerical experiments show its efficiency when combined with preconditioned GMRES method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10049.html} }
TY - JOUR T1 - Overlapping Domain Decomposition Preconditioners for Unconstrained Elliptic Optimal Control Problems JO - International Journal of Numerical Analysis and Modeling VL - 4-5 SP - 550 EP - 570 PY - 2017 DA - 2017/08 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10049.html KW - Overlapping domain decomposition method, elliptic optimal control problem, preconditioned MINRES method, preconditioned GMRES method. AB -

In this paper, we propose several overlapping domain decomposition preconditioners for solving the unconstrained elliptic optimal control problem, based on the two level additive Schwarz algorithm. We consider the cases with controls on the whole domain and controls from a local subset. The latter case can be viewed as the subproblems when we solve the control-constrained control problem by using semi-smooth Newton method. When the controls act on the whole domain, we construct a symmetric and positive definite preconditioner which is proved to be robust combined with preconditioned MINRES method, and a symmetric and indefinite preconditioner which can be used in the preconditioned GMRES method and shows better numerical performance than the positive definite one. When the controls act on a local subset, we also construct a similar symmetric and indefinite preconditioner, the numerical experiments show its efficiency when combined with preconditioned GMRES method.

Zhiyu Tan, Wei Gong & Ningning Yan. (1970). Overlapping Domain Decomposition Preconditioners for Unconstrained Elliptic Optimal Control Problems. International Journal of Numerical Analysis and Modeling. 14 (4-5). 550-570. doi:
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