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A new semi-smooth Newton multigrid method for control-constrained semi-linear elliptic PDE problems

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Abstract

In this paper a new multigrid algorithm is proposed to accelerate the convergence of the semi-smooth Newton method that is applied to the first order necessary optimality systems arising from a class of semi-linear control-constrained elliptic optimal control problems. Under admissible assumptions on the nonlinearity, the discretized Jacobian matrix is proved to have an uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new numerical implementation that leads to a robust multigrid solver is employed to coarsen the grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the full-approximation-storage multigrid in current literature. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter.

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Acknowledgments

The authors would like to thank Professor David Gao of University of Ballarat for many helpful discussions during the preparation of this manuscript. The authors also want to thank the anonymous referees for their valuable suggestions that lead to the improvement of the paper. This publication was made possible by NPRP Grant #[4-451-2-168] from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.

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Authors and Affiliations

  1. Department of Mathematics, Southern Illinois University, Carbondale, Illinois,  62901, USA

    Jun Liu & Mingqing Xiao

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  1. Jun Liu
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  2. Mingqing Xiao
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Correspondence to Mingqing Xiao.

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This work was supported in part by the National Priority Research Project of Qatar under Grant No. 4-451-2-168, and in part by the NSF1021203 of the United States.

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Liu, J., Xiao, M. A new semi-smooth Newton multigrid method for control-constrained semi-linear elliptic PDE problems. J Glob Optim 64, 451–468 (2016). https://doi.org/10.1007/s10898-014-0206-y

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  • DOI: https://doi.org/10.1007/s10898-014-0206-y

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