DOI QR코드

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REGULARIZATION FOR THE PROBLEM OF FINDING A SOLUTION OF A SYSTEM OF NONLINEAR MONOTONE ILL-POSED EQUATIONS IN BANACH SPACES

  • Received : 2017.07.25
  • Accepted : 2018.02.21
  • Published : 2018.07.01

Abstract

The purpose of this paper is to present an operator method of regularization for the problem of finding a solution of a system of nonlinear ill-posed equations with a monotone hemicontinuous mapping and N inverse-strongly monotone mappings in Banach spaces. A regularization parameter choice is given and convergence rate of the regularized solutions is estimated. We also give the convergence and convergence rate for regularized solutions in connection with the finite-dimensional approximation. An iterative regularization method of zero order in a real Hilbert space and two examples of numerical expressions are also given to illustrate the effectiveness of the proposed methods.

Keywords

monotone mapping;hemicontinuous;strictly convex Banach space;$Fr{\acute{e}}chet$ differentiable;Browder-Tikhonov regularization

References

  1. Y. Alber and I. Ryazantseva, Nonlinear Ill-Posed Problems of Monotone Type, Springer, Dordrecht, 2006.
  2. P. K. Anh, N. Buong, and D. V. Hieu, Parallel methods for regularizing systems of equations involving accretive operators, Appl. Anal. 93 (2014), no. 10, 2136-2157. https://doi.org/10.1080/00036811.2013.872777
  3. P. K. Anh and C. Van Chung, Parallel iterative regularization methods for solving systems of ill-posed equations, Appl. Math. Comput. 212 (2009), no. 2, 542-550.
  4. P. K. Anh and V. T. Dung, Parallel iterative regularization algorithms for large overdetermined linear systems, Int. J. Comput. Methods 7 (2010), no. 4, 525-537. https://doi.org/10.1142/S0219876210002313
  5. I. K. Argyros and S. George, Expanding the applicability of Lavrentiev regularization methods for ill-posed equations under general source condition, Nonl. Funct. Anal. Appl. 19 (2014), no. 2, 177-192.
  6. J.-B. Baillon and G. Haddad, Quelques proprietes des operateurs angle-bornes et n-cycliquement monotones, Israel J. Math. 26 (1977), no. 2, 137-150. https://doi.org/10.1007/BF03007664
  7. A. B. Bakushinsky and M. Yu. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Mathematics and Its Applications (New York), 577, Springer, Dordrecht, 2004.
  8. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, translated from the Romanian, Editura Academiei Republicii Socialiste Romania, Bucharest, 1976.
  9. J. Baumeister, B. Kaltenbacher, and A. LeitLeitLeito, On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations, Inverse Probl. Imaging 4 (2010), no. 3, 335-350. https://doi.org/10.3934/ipi.2010.4.335
  10. N. Buong, Regularization for unconstrained vector optimization of convex functionals in Banach spaces, Comput. Math. Math. Phys. 46 (2006), no. 3, 354-360; translated from Zh. Vychisl. Mat. Mat. Fiz. 46 (2006), no. 3, 372-378. https://doi.org/10.1134/S096554250603002X
  11. N. Buong, Regularization extragradient method for Lipschitz continuous mappings and inverse strongly-monotone mappings in Hilbert spaces, Comput. Math. Math. Phys. 48 (2008), no. 11, 1927-1935. https://doi.org/10.1134/S096554250811002X
  12. N. Buong and N. D. Dung, A regularized parameter choice in regularization for a common solution of a finite system of ill-posed equations involving Lipschitz continuous and accretive mappings, Comput. Math. Math. Phys. 54 (2014), no. 3, 397-406. https://doi.org/10.1134/S0965542514030130
  13. N. Buong, N. T. T. Thuy, and T. T. Huong, A generalized quasi-residual principle in regularization for a solution of a finite system of ill-posed equations in Banach spaces, Nonl. Funct. Anal. Appl. 20 (2015), no. 2, 187-197.
  14. L. C. Ceng, C. F. Wen, and Y. Yao, Relaxed extragradient-like methods for systems of generalized equilibria with constraints of mixed equilibria, minimization and fixed point problems, J. Nonlinear Var. Anal. 1 (2017), 367-390.
  15. A. D. Cezaro, J. Baumeister, and A. Leitao, Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations, Inverse Probl. Imaging 5 (2011), no. 1, 1-17. https://doi.org/10.3934/ipi.2011.5.1
  16. A. D. Cezaro, M. Haltmeier, A. Leitao, and O. Scherzer, On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations, Appl. Math. Comput. 202 (2008), no. 2, 596-607.
  17. S. Y. Cho, Generalized mixed equilibrium and fixed point problems in a Banach space, J. Nonlinear Sci. Appl. 9 (2016), no. 3, 1083-1092. https://doi.org/10.22436/jnsa.009.03.36
  18. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and its Applications, 62, Kluwer Academic Publishers Group, Dordrecht, 1990.
  19. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
  20. M. Haltmeier, A. Leitao, and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations. I. Convergence analysis, Inverse Probl. Imaging 1 (2007), no. 2, 289-298. https://doi.org/10.3934/ipi.2007.1.289
  21. M. Hanke, A. Neubauer, and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math. 72 (1995), no. 1, 21-37. https://doi.org/10.1007/s002110050158
  22. Q. Jin and W. Wang, Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals, Inverse Problems 29 (2013), no. 8, 085011, 22 pp.
  23. S. Kaczmarz, Approximate solution of systems of linear equations, Internat. J. Control 57 (1993), no. 6, 1269-1271. https://doi.org/10.1080/00207179308934446
  24. B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
  25. B. Kaltenbacher, F. Schopfer, and T. Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems, Inverse Problems 25 (2009), no. 6, 065003, 19 pp.
  26. J. K. Kim and N. Buong, Regularization inertial proximal point algorithm for monotone hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces, J. Inequal. Appl. 2010 (2010), Art. ID 451916, 10 pp.
  27. J. K. Kim and N. Buong, Convergence rates in regularization for a system of nonlinear ill-posed equations with m-accretive operators, J. Inequal. Appl. 2014 (2014), 440, 9 pp. https://doi.org/10.1186/1029-242X-2014-9
  28. Y. Liu, A modified hybrid method for solving variational inequality problems in Banach spaces. J. Nonlinear Funct. Anal. 2017 (2017), Article ID 31.
  29. S. F. McCormick, The methods of Kaczmarz and row orthogonalization for solving linear equations and least squares problems in Hilbert space, Indiana Univ. Math. J. 26 (1977), no. 6, 1137-1150. https://doi.org/10.1512/iumj.1977.26.26090
  30. V. A. Morozov, Regularization Methods for Ill-Posed Problems, translated from the 1987 Russian original, CRC Press, Boca Raton, FL, 1993.
  31. F. Natterer, Algorithms in tomography, in The state of the art in numerical analysis (York, 1996), 503-523, Inst. Math. Appl. Conf. Ser. New Ser., 63, Oxford Univ. Press, New York, 1997.
  32. X. Qin and J.-C. Yao, Projection splitting algorithms for nonself operators, J. Nonlinear Convex Anal. 18 (2017), no. 5, 925-935.
  33. E. Resmerita, Regularization of ill-posed problems in Banach spaces: convergence rates, Inverse Problems 21 (2005), no. 4, 1303-1314. https://doi.org/10.1088/0266-5611/21/4/007
  34. N. T. T. Thuy, Regularization for a system of inverse-strongly monotone operator equations. Nonl. Funct. Anal. Appl. 17 (2012), no. 1, 71-87.
  35. N. T. T. Thuy, Regularization methods and iterative methods for variational inequality with accretive operator, Acta Math. Vietnam. 41 (2016), no. 1, 55-68. https://doi.org/10.1007/s40306-015-0123-2
  36. N. T. T. Thuy, P. T. Hieu, and J. J. Strodiot, Regularization methods for accretive variational inequalities over the set of common fixed points of nonexpansive semigroups, Optimization 65 (2016), no. 8, 1553-1567. https://doi.org/10.1080/02331934.2016.1166501
  37. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, translated from the Russian, preface by translation editor Fritz John, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977.
  38. M. M. Vainberg, Variational method and method of monotone operators in the theory of nonlinear equations, translated from the Russian by A. Libin, translation edited by D. Louvish, Halsted Press (A division of John Wiley & Sons), New York, 1973.

Acknowledgement

Supported by : National Research Foundation(NRF)