Abstract
We consider the iterative solution of optimal control problems constrained by the time-harmonic parabolic equations. Due to the time-harmonic property of the control equations, a suitable discretization of the corresponding optimality systems leads to a large complex linear system with special two-by-two block matrix of saddle point form. For this algebraic system, an efficient preconditioner is constructed, which results in a fast Krylov subspace solver, that is robust with respect to the mesh size, frequency, and regularization parameters. Furthermore, the implementation is straightforward and the computational complexity is of optimal order, linear in the number of degrees of freedom. We show that the eigenvalue distribution of the corresponding preconditioned matrix leads to a condition number bounded above by 2. Numerical experiments confirming the theoretical derivations are presented, including comparisons with some other existing preconditioners.
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Acknowledgements
The authors are indebted to the reviewers for their careful reading and constructive advice on improving the structure of this article.
Funding
The work of Zhao-Zheng Liang is funded by the China Scholarship Council (File No. 201606180086) and by the National Natural Science Foundation of China (Grant No. 11771193). His work is performed during his visit at Uppsala University, Sweden. The work of Owe Axelsson was supported by the National Programme of Sustainability (NPU II) project IT4Innovations excellence in science–LQ1602 of the Ministry of Education, Youth and Sports of the Czech Republic.
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Liang, ZZ., Axelsson, O. & Neytcheva, M. A robust structured preconditioner for time-harmonic parabolic optimal control problems. Numer Algor 79, 575–596 (2018). https://doi.org/10.1007/s11075-017-0451-5
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DOI: https://doi.org/10.1007/s11075-017-0451-5