Strong Convergence of Forward–Reflected–Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Control
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-valu...
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| Vydáno v: | Journal of scientific computing Ročník 94; číslo 3; s. 73 |
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| Jazyk: | angličtina |
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Springer US
01.03.2023
Springer Nature B.V |
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| ISSN: | 0885-7474, 1573-7691 |
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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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| AbstractList | 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. |
| ArticleNumber | 73 |
| Author | Izuchukwu, Chinedu Reich, Simeon Shehu, Yekini Taiwo, Adeolu |
| Author_xml | – sequence: 1 givenname: Chinedu orcidid: 0000-0002-8262-8605 surname: Izuchukwu fullname: Izuchukwu, Chinedu organization: School of Mathematics, University of the Witwatersrand – sequence: 2 givenname: Simeon orcidid: 0000-0003-0780-1559 surname: Reich fullname: Reich, Simeon organization: Department of Mathematics, The Technion – Israel Institute of Technology – sequence: 3 givenname: Yekini orcidid: 0000-0001-9224-7139 surname: Shehu fullname: Shehu, Yekini email: yekini.shehu@zjnu.edu.cn organization: Department of Applied Mathematics, Zhejiang Normal University – sequence: 4 givenname: Adeolu orcidid: 0000-0001-5939-935X surname: Taiwo fullname: Taiwo, Adeolu organization: Department of Mathematics, The Technion – Israel Institute of Technology |
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| Cites_doi | 10.1007/s40314-019-0855-z 10.1080/02331934.2020.1723586 10.1137/0716071 10.1080/02331934.2021.1981895 10.1137/18M1207260 10.1007/s10589-020-00217-8 10.1137/14097238X 10.1016/j.jmaa.2005.12.066 10.1007/978-3-642-65024-6 10.1080/10556788.2019.1619729 10.1080/00036811.2021.1892080 10.1007/s00025-022-01694-5 10.1007/s11075-018-0547-6 10.1007/s10092-018-0292-1 10.1007/s11075-019-00758-y 10.1007/s10957-010-9713-2 10.1080/01630563.2021.2001826 10.1007/s10288-020-00440-3 10.1080/02331934.2014.883510 10.1007/s11075-020-00999-2 10.1007/s10440-019-00297-7 10.1007/s10898-014-0150-x 10.1007/s10107-019-01416-w 10.1016/0022-247X(79)90234-8 10.1137/S1052623495290179 10.1007/s10898-021-01095-y 10.1137/S0363012998338806 10.1007/s11228-020-00542-4 10.1007/s11750-021-00620-2 10.1016/j.apnum.2021.12.006 10.1016/j.na.2011.09.005 10.1080/02331934.2017.1411485 10.1007/978-3-642-59073-3_11 |
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| Keywords | Viscosity iteration Monotone inclusion Strong convergence 47H10 49J20 Forward–reflected–backward method 49J40 47H09 Halpern’s iteration Inertial method |
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| SubjectTerms | Algorithms Approximation Computational Mathematics and Numerical Analysis Convergence Hilbert space Image restoration Inclusions Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Operators (mathematics) Optimal control Optimization Splitting Theoretical Viscosity |
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| Title | Strong Convergence of Forward–Reflected–Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Control |
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