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Robust preconditioning techniques for multiharmonic finite element method with application to time-periodic parabolic optimal control problems

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Abstract

We are concerned with efficient solutions of the time-periodic parabolic optimal control problems. By using the multiharmonic FEM, the linear algebraic equations characterizing the first-order optimality conditions can be decoupled into a series of parallel solvable block 4 × 4 linear systems with respect to the cosine and sine Fourier coefficients of the state and scaled control variables for different frequencies. Parameter robust preconditioners are proposed for solving these linear systems along with information on practical algorithm implementation and detailed spectral analysis. Problem independent eigenvalue bounds and upper bound approximations of the condition numbers of the eigenvector matrices are obtained for the preconditioned matrices. Such results ensure efficient Krylov subspace acceleration methods and a parameter-free Chebyshev acceleration method, which are both robust in view of all discretization and model parameters. Numerical experiments are presented to demonstrate the robustness and effectiveness of the proposed preconditioners within both Krylov subspace and Chebyshev accelerations compared with some already available preconditioned Krylov subspace methods.

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Acknowledgements

We would like to express our sincere thanks to the two anonymous reviewers for their insightful comments and valuable suggestions, which greatly improve the quality of the paper. We would also thank one of them for pointing us to [12, 13, 15] and references therein, which are unnoticed by us previously, about the parallelizable domain decomposition and multigrid methods for solving the time-dependent PDE-constrained optimal control problems.

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 11801242 and 11771193).

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  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Zhao-Zheng Liang & Guo-Feng Zhang

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  1. Zhao-Zheng Liang
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  2. Guo-Feng Zhang
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Correspondence to Zhao-Zheng Liang.

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Liang, ZZ., Zhang, GF. Robust preconditioning techniques for multiharmonic finite element method with application to time-periodic parabolic optimal control problems. Adv Comput Math 47, 67 (2021). https://doi.org/10.1007/s10444-021-09887-2

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