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Preconditioning for PDE-constrained optimization with total variation regularization

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  • Li, Hongyi
  • Wang, Chaojie
  • Zhao, Di

Abstract

The efficient solution of PDE-constrained optimization problems is of great significance in many scientific and engineering applications. The key point of achieving this goal is to solve the linear systems arising from such problems more efficiently by preconditioning. In this paper, we consider preconditioning the linear systems arising from PDE-constrained optimization problems with total variation (TV) regularization. Based on the block structure of the coefficient matrix or its row and column transformed forms, we propose three kinds of new preconditioners by approximating the Schur complements in different ways. We analyze the eigenvalue properties of the preconditioned coefficient matrices and provide the eigenvalue bounds independent of mesh size. We also show the relations of eigenvalue bounds corresponding to different preconditioners. Numerical results illustrate the efficiency of the proposed preconditioners by comparison with other existing methods.

Suggested Citation

  • Li, Hongyi & Wang, Chaojie & Zhao, Di, 2020. "Preconditioning for PDE-constrained optimization with total variation regularization," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    • Handle: RePEc:eee:apmaco:v:386:y:2020:i:c:s009630032030429x
    DOI: 10.1016/j.amc.2020.125470
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    References listed on IDEAS

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    2. Ole Løseth Elvetun & Bjørn Fredrik Nielsen, 2016. "The split Bregman algorithm applied to PDE-constrained optimization problems with total variation regularization," Computational Optimization and Applications, Springer, vol. 64(3), pages 699-724, July.
    3. Wang, Chaojie & Li, Hongyi & Zhao, Di, 2019. "Improved block preconditioners for linear systems arising from half-quadratic image restoration," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    4. Xiaoliang Song & Bo Chen & Bo Yu, 2018. "An efficient duality-based approach for PDE-constrained sparse optimization," Computational Optimization and Applications, Springer, vol. 69(2), pages 461-500, March.
    5. Li, Hongyi & Qin, Xuyao & Zhao, Di & Chen, Jiaxin & Wang, Pidong, 2018. "An improved empirical mode decomposition method based on the cubic trigonometric B-spline interpolation algorithm," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 406-419.
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    Cited by:

    1. D. Hafemeyer & F. Mannel, 2022. "A path-following inexact Newton method for PDE-constrained optimal control in BV," Computational Optimization and Applications, Springer, vol. 82(3), pages 753-794, July.

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