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A leapfrog semi-smooth Newton-multigrid method for semilinear parabolic optimal control problems

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Abstract

A new semi-smooth Newton multigrid algorithm is proposed for solving the discretized first order necessary optimality systems that characterizing the optimal solutions of a class of two dimensional semi-linear parabolic PDE optimal control problems with control constraints. A new computational scheme (leapfrog scheme) in time associated with the standard five-point stencil in space is established to achieve the second-order finite difference discretization. The convergence (or unconditional stability) of the proposed scheme is proved when assuming time-periodic solutions. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations and the optimal linear complexity of computing time.

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Acknowledgments

The authors would like to thank the two anonymous reviewers and Professor William W. Hager for their helpful comments and constructive suggestions that greatly contributed to improving the original version of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Statistical Sciences, Jackson State University, Jackson, MS, 39217, USA

    Jun Liu

  2. Department of Mathematics, Southern Illinois University, Carbondale, IL, 62901, USA

    Mingqing Xiao

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

Additional information

This work was supported in part by Dissertation Research Assistantship from Southern Illinois Univeristy and in part by the NSF1021203, NSF1419028 of the United States.

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Liu, J., Xiao, M. A leapfrog semi-smooth Newton-multigrid method for semilinear parabolic optimal control problems. Comput Optim Appl 63, 69–95 (2016). https://doi.org/10.1007/s10589-015-9759-z

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