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An algorithm for the split feasible problem and image restoration. (English) Zbl 07299282

Summary: This paper proposes an accelerated algorithm for the split common fixed point problem, based on viscosity approximation methods and inertial effects. The main result will be applied to image restoration problems. This algorithm is constructed in such a way that its step sizes and the norm of a given linear operator are not related. Under some conditions, the strong convergence of the algorithm is obtained. Numerical investigations are carried out in order to illustrate high-performance of the present work, mainly using processing duration and the signal-to-noise ratio. It is also shown that this proposed algorithm is more efficient and effective than the published algorithm by Yao et al.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
65K10 Numerical optimization and variational techniques
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[1] Löfdahl, M. G., van Noort, M. J., Denker, C.: Solar image restoration. Modern solar facilities—advanced solar science. In: Proceedings of a workshop held at Gttingen September 27-29, 2006, 119-126,(2007)
[2] Kochher, R. , Oberoi , A., Goel, P.: Image restoration on mammography images. In: 2016 International Conference on Computing, Communication and Automation (ICCCA), Noida, 1170-1173 (2016)
[3] Eslahi, N.; Aghagolzadeh, A., Compressive sensing image restoration using adaptive curvelet thresholding and nonlocal Sparse Regularization, IEEE Trans. Image Process., 25, 7, 3126-3140 (2016) · Zbl 1408.94165 · doi:10.1109/TIP.2016.2562563
[4] Turci, A., The use of digital restoration within European film archives: a case study, Moving Image, 6, 1, 111-124 (2006)
[5] Chen, DQ; Zhang, H.; Cheng, LZ, A fast fixed point algorithm for total variation deblurring and segmentation, J. Math. Imaging Vis., 43, 3, 167-179 (2012) · Zbl 1255.68219 · doi:10.1007/s10851-011-0298-7
[6] Yang, H., Luo, X., Chen, L.: Solving adaptive image restoration problems via a modified projection algorithm, Math. Probl. Eng. 2016 (6132356) (2016) 11 pages · Zbl 1400.94038
[7] Sahragard, E.; Farsi, H.; Mohamadzadeh, S., Image restoration by variable splitting based on total variant regularizer, J. AI Data Min., 6, 13-33 (2018)
[8] Censor, Y.; Bortfeld, T.; Martin, B.; Trofimov, A., A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51, 2353-2365 (2006) · doi:10.1088/0031-9155/51/10/001
[9] Censor, Y.; Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8, 221-239 (1994) · Zbl 0828.65065 · doi:10.1007/BF02142692
[10] Anh, PK; Vinh, NT; Dung, VT, A new self-adaptive CQ algorithm with an application to the LASSO problem, J. Fixed Point Theory Appl., 20, 4, 142 (2018) · Zbl 1398.65149 · doi:10.1007/s11784-018-0620-8
[11] Suparatulatorn, R., Khemphet, A., Charoensawan, P., Suantai, S., Phudolsitthiphat, N.: Generalized self-adaptive algorithm for solving split common fixed point problem and its application to image restoration problem. Int. J. Comput. Math. 1-15, (2019)
[12] Suparatulatorn, R., Charoensawan, P., Poochinapan, K.: Inertial self-adaptive algorithm for solving split feasible problems with applications to image restoration. Math. Methods Appl. Sci (2019) · Zbl 07163596
[13] Censor, Y.; Segal, A., The split common fixed point problem for directed operators, J. Convex Anal., 16, 2, 587-600 (2009) · Zbl 1189.65111
[14] Moudafi, A., The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26, 5, 055007 (2010) · Zbl 1219.90185 · doi:10.1088/0266-5611/26/5/055007
[15] Maingé, PE, A viscosity method with no spectral radius requirements for the split common fixed point problem, Euro. J. Oper. Res., 235, 17-27 (2014) · Zbl 1305.65146 · doi:10.1016/j.ejor.2013.11.028
[16] Shehu, Y., New convergence theorems for split common fixed point problems in Hilbert spaces, J. Nonlinear Convex Anal., 16, 167-181 (2015) · Zbl 1317.47066
[17] Tang, Y-C; Peng, J-G; Liu, L-W, A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces, Math. Model. Anal., 17, 457-466 (2012) · Zbl 1267.47104 · doi:10.3846/13926292.2012.706236
[18] Shehu, Y.; Cholamjiak, P., Another look at the split common fixed point problem for demicontractive operators, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 110, 201-218 (2016) · Zbl 1338.47105 · doi:10.1007/s13398-015-0231-9
[19] Jailoka, P.; Suantai, S., The split common fixed point problem for multivalued demicontractive mappings and its applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113, 2, 689-706 (2019) · Zbl 07086841 · doi:10.1007/s13398-018-0496-x
[20] Yao, Y.; Leng, L.; Liou, YC, Strong convergence of an iteration for the split common fixed points of demicontractive operators, J. Nonlinear Convex Anal, 19, 197-205 (2018) · Zbl 1492.47102
[21] Suparatulatorn, R.; Suantai, S.; Phudolsitthiphat, N., Reckoning solution of split common fixed point problems by using inertial self-adaptive algorithms, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 113, 4, 3101-3114 (2019) · Zbl 07124357 · doi:10.1007/s13398-019-00676-7
[22] Padcharoen, A., Kumam, P., Cho, Y.J.: Split common fixed point problems for demicontractive operators. Numer. Algorithms 1-24 (2018) · Zbl 07101813
[23] Boikanyo, OA, A strongly convergent algorithm for the split common fixed point problem, Appl. Math. Comput., 265, 844-853 (2015) · Zbl 1410.65215 · doi:10.1016/j.amc.2015.05.130
[24] Cui, H.; Wang, F., Iterative methods for the split common fixed point problem in Hilbert spaces, Fixed Point Theory Appl., 2014, 78 (2014) · Zbl 1332.47041 · doi:10.1186/1687-1812-2014-78
[25] Yao, Y., Liou, Y.C., Postolache, M.: Self-adaptive algorithms for the split problem of the demicontractive operators. Optimization 1-10 (2017)
[26] Yao, Y.; Yao, JC; Liou, YC; Postolache, M., Iterative algorithms for split common fixed points of demicontractive operators without priori knowledge of operator norms, Carpathian J. Math, 34, 459-466 (2018) · Zbl 1449.47119
[27] Suparatulatorn, R., Cholamjiak, P., Suantai, S.: Self-adaptive algorithms with inertial effects for solving the split problem of the demicontractive operators, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114(1) (2020), Paper 40, pp. 16, 10.1007/s13398-019-00737-x · Zbl 1513.47132
[28] Nesterov, Y., A method of solving a convex programming problem with convergence rate \(O(1/k^2)\), Soviet Math. Doklady, 27, 2, 372-376 (1983) · Zbl 0535.90071
[29] Polyak, BT, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4, 5, 1-17 (1964) · Zbl 0147.35301 · doi:10.1016/0041-5553(64)90137-5
[30] Dong, QL; Cho, YJ; Zhong, LL; Rassias, TM, Inertial projection and contraction algorithms for variational inequalities, J. Global Opt., 70, 3, 687-704 (2018) · Zbl 1390.90568 · doi:10.1007/s10898-017-0506-0
[31] Lorenz, DA; Pock, T., An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51, 2, 311-325 (2015) · Zbl 1327.47063 · doi:10.1007/s10851-014-0523-2
[32] Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Ind. Manag. Opt. 3-11 (2018) · Zbl 1461.47035
[33] Maingé, PE, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219, 223-236 (2008) · Zbl 1156.65054 · doi:10.1016/j.cam.2007.07.021
[34] Cholamjiak, W., Khan, S.A., Yambangwai, D., Kazmi, K.R.: Strong convergence analysis of common variational inclusion problems involving an inertial parallelmonotone hybrid method for a novel application to image restoration, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114(2) (2020), Paper no 99, pp. 20, 10.1007/s13398-020-00827-1 · Zbl 1443.47055
[35] Majee, P.; Nahak, C., A modied iterative method for capturing a common solution of split generalized equilibrium problem and fixed point problem, RACSAM. Rev. R. Acad. Cienc. Exactas. Fís. Nat. Ser. A Mat. RACSAM, 112, 4, 1327-1348 (2018) · Zbl 1423.47045 · doi:10.1007/s13398-017-0428-1
[36] Moreau, JJ, Proprietes des applications prox, C. R. Acad. Sci. Paris Ser. A Math, 256, 1069-1071 (1963) · Zbl 0115.10802
[37] Moreau, JJ, Proximite et dualite dans un espace hilbertien, Bull. Soc. Math. France, 93, 272-299 (1965) · Zbl 0136.12101
[38] Maingé, PE, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325, 469-479 (2007) · Zbl 1111.47058 · doi:10.1016/j.jmaa.2005.12.066
[39] Xu, HK, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66, 1, 240-256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[40] Combettes, PL; Wajs, VR, Signal recovery by proximal forward-backward splitting, Multiscale Model Simul., 4, 1168-1200 (2005) · Zbl 1179.94031 · doi:10.1137/050626090
[41] Bauschke, HH; Combettes, PL, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2011), New York: CMS Books in Mathematics. Springer, New York · Zbl 1218.47001
[42] Iiduka, H.; Takahashi, W., Strong convergence theorems for nonexpansive nonself-mappings and inverse-strongly-monotone mappings, J. Convex Anal., 11, 69-79 (2004) · Zbl 1083.47054
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