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Numerical approximation of a Tikhonov type regularizer by a discretized frozen steepest descent method. (English) Zbl 06802845
Summary: We present a frozen regularized steepest descent method and its finite dimensional realization for obtaining an approximate solution for the nonlinear ill-posed operator equation \(F(x)=y\). The proposed method is a modified form of the method considered by Argyros et al. (2014). The balancing principle considered by Pereverzev and Schock (2005) is used for choosing the regularization parameter. The error estimate is derived under a general source condition and is of optimal order. The provided numerical example proves the efficiency of the proposed method.
47A52 Linear operators and ill-posed problems, regularization
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
Full Text: DOI
[1] Engl, H. W.; Hanke, M.; Neubauer, A., (Regularization of Inverse Problems, Mathematics and its Applications, (2000), Kluwer Academic Publishers Dordrecht)
[2] Hettlich, F.; Rundell, W., A second degree method for nonlinear inverse problems, SIAM J. Numer. Anal., 37, 587-620, (2000) · Zbl 0946.35115
[3] Isakov, V., (Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, vol. 127, (2006), Springer New York) · Zbl 1092.35001
[4] Hanke, M.; Neubauer, A.; Scherzer, O., A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72, 21-37, (1995) · Zbl 0840.65049
[5] Ramlau, R., A modified Landweber-method for inverse problems, Numer. Funct. Anal. Optim., 20, 79-98, (1999) · Zbl 0970.65064
[6] Hanke, M., A regularizing Levenberg-Marquardt scheme, with application to inverse ground-water filtration problems, Inverse Problems, 13, 79-95, (1997) · Zbl 0873.65057
[7] Bakushinskii, A. W., The problem of the convergence of the iteratively regularized Gauss-Newton method, Comput. Math. Phys., 32, 1353-1359, (1992)
[8] Blaschke, B.; Neubauer, A.; Scherzer, O., On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal., 17, 421-436, (1997) · Zbl 0881.65050
[9] Hanke, M., Regularizing properties of a truncated Newton-cg algorithm for nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 18, 971-993, (1997) · Zbl 0899.65038
[10] George, S., On convergence of regularized modified newton’s method for nonlinear ill-posed problems, J. Inverse Ill-Posed Probl., 18, 133-146, (2010) · Zbl 1279.65069
[11] Kaltenbacher, B., Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13, 729-753, (1997) · Zbl 0880.65033
[12] Ramlau, R., TIGRA — an iterative algorithm for regularizing nonlinear ill-posed problems, Inverse Problems, 19, 433-465, (2003) · Zbl 1029.65059
[13] Scherzer, O., A convergence analysis of a method of steepest descent and two-step algorithm for nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 17, 197-214, (1996) · Zbl 0852.65048
[14] Neubauer, A.; Scherzer, O., A convergence rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems, J. Math. Anal. Appl., 14, 369-377, (1995) · Zbl 0826.65053
[15] Gilyazov, S. F., Iterative solution methods for inconsistent linear equations with nonself-adjoint operators, Moscow Univ. Comput. Math. Cybernet., 1, 8-13, (1997) · Zbl 0427.47007
[16] Vasin, V., Irregular nonlinear operator equations: tikhonov’s regularization and iterative approximation, J. Inverse Ill-Posed Probl., 21, 109-123, (2013) · Zbl 1276.65031
[17] I.K. Argyros, S. George, P. Jidesh, Inverse free iterative methods for nonlinear ill-posed operator equations, Int. J. Math. Math. Sci. , 2014, Article ID 754154, 8 pages, http://dx.doi.org/10.1155/2014/754154. · Zbl 1309.47065
[18] Argyros, I. K., Convergence and Applications of Newton-Type Iterations, (2008), Springer New York · Zbl 1153.65057
[19] Jin, Q., On a class of frozen regularized Gauss-Newton methods for nonlinear inverse problems, Math. Comp., 79, 2191-2211, (2010) · Zbl 1208.65073
[20] Groetsch, C. W.; Neubauer, A., Convergence of a general projection method for an operator equation of the first kind, Houston J. Math., 14, 201-208, (1988) · Zbl 0675.65051
[21] Kaltenbacher, B.; Neubauer, A.; Scherzer, O., Iterative Regularization Methods for Nonlinear Ill-Posed Problems, (2008), De Gruyter Berlin · Zbl 1145.65037
[22] Krisch, A., An Introduction To the Mathematical Theory of Inverse Problems, (1996), Springer New York
[23] Perverzev, S. V.; Probdorf, S., On the characterization of self-regularization properties of a fully discrete projection method for symms integral equation, J. Integral Equations Appl., 12, 113-130, (2000)
[24] Pereverzev, S.; Schock, E., On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal., 43, 2060-2076, (2005) · Zbl 1103.65058
[25] Semenova, E. V., Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators, Comput. Methods Appl. Math., 4, 444-454, (2010) · Zbl 1283.65102
[26] S. Lu, S.V. Pereverzev, Sparsity reconstruction by the standard Tikhonov method, RICAM-Report No. 2008-17, 2008.
[27] Vasin, V. V.; Prutkin, I. I.; Timerkhanova, L. Yu., Retrieval of a three-dimensional relief of geological boundary from gravity data, Izv. Phys. Solid Earth, 32, 58-62, (1996)
[28] V.V. Vasin, Modified processes of Newton type generating Fejer approximations of regularized solutions of nonlinear equations, in: Proceedings in Mathematics and Mechanics, URORAN, Tom 18, 3, 2012, (in Russian).
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