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Strong Convergence of Forward–Reflected–Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Control

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Abstract

In this paper, we propose and study several strongly convergent versions of the forward–reflected–backward splitting method of Malitsky and Tam for finding a zero of the sum of two monotone operators in a real Hilbert space. Our proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. We also develop inertial versions of our methods and strong convergence results are obtained for these methods when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous and monotone. Finally, we discuss some examples from image restorations and optimal control regarding the implementations of our methods in comparisons with known related methods in the literature.

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Not applicable.

Code Availability

The Matlab codes employed to run the numerical experiments are available upon request.

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Acknowledgements

The authors thank the anonymous referees for providing them with valuable comments and useful suggestions which helped them improve the earlier version of their paper.

Funding

The second author was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa

    Chinedu Izuchukwu

  2. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich & Adeolu Taiwo

  3. Department of Applied Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Yekini Shehu

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  1. Chinedu Izuchukwu
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  2. Simeon Reich
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  3. Yekini Shehu
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  4. Adeolu Taiwo
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Correspondence to Yekini Shehu.

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Izuchukwu, C., Reich, S., Shehu, Y. et al. Strong Convergence of Forward–Reflected–Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Control. J Sci Comput 94, 73 (2023). https://doi.org/10.1007/s10915-023-02132-6

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  • DOI: https://doi.org/10.1007/s10915-023-02132-6

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