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Volume 5, Issue 1
Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

Alfio Borzì & Sergio González Andrade

DOI: 10.4208/nmtma.2011.m12si01

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 1-18.

Published online: 2012-05

[An open-access article; the PDF is free to any online user.]

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  • Abstract

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multigrid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical two-grid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

  • Keywords

Multigrid methods, Lavrentiev regularization, semismooth Newton methods, parabolic partial differential equations, optimal control theory.

  • AMS Subject Headings

35K10, 49K20, 49J20, 49M05, 65M55, 65C20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-1, author = {Alfio Borzì and Sergio González Andrade}, title = {Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {1}, pages = {1--18}, abstract = {

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multigrid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical two-grid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m12si01}, url = {http://global-sci.org/intro/article_detail/nmtma/5925.html} }
TY - JOUR T1 - Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem AU - Alfio Borzì & Sergio González Andrade JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 1 EP - 18 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2011.m12si01 UR - https://global-sci.org/intro/article_detail/nmtma/5925.html KW - Multigrid methods, Lavrentiev regularization, semismooth Newton methods, parabolic partial differential equations, optimal control theory. AB -

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multigrid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical two-grid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

Alfio Borzì and Sergio González Andrade. (2012). Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem. Numerical Mathematics: Theory, Methods and Applications. 5 (1). 1-18. doi:10.4208/nmtma.2011.m12si01
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