Abstract
A distributed optimal control problem with the constraint of a parabolic partial differential equation is considered. Boundary value methods are used to solve the coupled initial/final value problems arising from the first order optimality conditions for this problem. We use a block triangular preconditioning strategy for solving the resulting two-by-two linear system. By making use of a matching strategy and a Kronecker product-based splitting technique we establish a Kronecker product-based approximation to the Schur complement. Since the Schur complement approximation is in a form of one Kronecker product structure, the preconditioner can be implemented efficiently. Numerical experiments are presented to illustrate the accuracy and computational efficiency of the proposed approach.
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Acknowledgements
The authors would like to thank the anonymous referee for valuable suggestions, which improved the original manuscript of the paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11301575, 11971085), the Natural Science Foundation Project of CQ CSTC (No. cstc2018jcyjAX0113),the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-M201800501), the Program of Chongqing Innovation Research Group Project in University (No. CXQT19018), and the Talent Project of Chongqing Normal University (Grant No. 02030307-0054).
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Chen, H., Huang, Q. Preconditioned iterative method for boundary value method discretizations of a parabolic optimal control problem. Calcolo 57, 5 (2020). https://doi.org/10.1007/s10092-019-0353-0
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DOI: https://doi.org/10.1007/s10092-019-0353-0
Keywords
- General linear multistep methods
- Boundary value methods
- Preconditioning
- Initial/final value problems
- PDE-constrained optimization