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Inverse free iterative methods for nonlinear ill-posed operator equations. (English) Zbl 1309.47065
Summary: We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equation \(F(x)=y\). The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by R. Ramlau [Inverse Probl. 19, No. 2, 433–465 (2003; Zbl 1029.65059)]. The regularization parameter is chosen according to the balancing principle considered by S. Pereverzev and E. Schock [SIAM J. Numer. Anal. 43, No. 5, 2060–2076 (2005; Zbl 1103.65058)]. The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.

MSC:
47J06 Nonlinear ill-posed problems
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
Software:
TIGRA
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References:
[1] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, The Netherlands, 1996. · Zbl 0859.65054
[2] Q.-N. Jin and Z.-Y. Hou, “On an a posteriori parameter choice strategy for tikhonov regularization of nonlinear ill-posed problems,” Numerische Mathematik, vol. 83, no. 1, pp. 139-159, 1999. · Zbl 0932.65065 · doi:10.1007/s002110050442
[3] U. Tautenhahn and Q.-N. Jin, “Tikhonov regularization and a posteriori rules for solving nonlinear ill posed problems,” Inverse Problems, vol. 19, no. 1, pp. 1-21, 2003. · Zbl 1030.65061 · doi:10.1088/0266-5611/19/1/301
[4] H. W. Engl, K. Kunisch, and A. Neubauer, “Convergence rates for Tikhonov regularisation of non-linear ill-posed problems,” Inverse Problems, vol. 5, no. 4, article 007, pp. 523-540, 1989. · Zbl 0695.65037 · doi:10.1088/0266-5611/5/4/007
[5] A. Neubauer, “Tikhonov regularisation for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation,” Inverse Problems, vol. 5, no. 4, article 008, pp. 541-557, 1989. · Zbl 0695.65038 · doi:10.1088/0266-5611/5/4/008
[6] R. Ramlau, “TIGRA-an iterative algorithm for regularizing nonlinear ill-posed problems,” Inverse Problems, vol. 19, no. 2, pp. 433-465, 2003. · Zbl 1029.65059 · doi:10.1088/0266-5611/19/2/312
[7] M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the Landweber iteration for nonlinear ill-posed problems,” Numerische Mathematik, vol. 72, pp. 21-37, 1995. · Zbl 0840.65049 · doi:10.1007/s002110050158
[8] R. Ramlau, “Modified Landweber method for inverse problems,” Numerical Functional Analysis and Optimization, vol. 20, no. 1, pp. 79-98, 1999. · Zbl 0970.65064 · doi:10.1080/01630569908816882
[9] M. Hanke, “A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems,” Inverse Problems, vol. 13, no. 1, pp. 79-95, 1997. · Zbl 0873.65057 · doi:10.1088/0266-5611/13/1/007
[10] A. B. Bakushinskii, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Computational Mathematics and Mathematical Physics, vol. 32, no. 9, pp. 1353-1359, 1992.
[11] B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA Journal of Numerical Analysis, vol. 17, no. 3, pp. 421-436, 1997. · Zbl 0881.65050 · doi:10.1093/imanum/17.3.421
[12] M. Hanke, “Regularizing properties of a truncated Newton-cg algorithm for nonlinear inverse problems,” Numerical Functional Analysis and Optimization, vol. 18, no. 9-10, pp. 971-993, 1997. · Zbl 0899.65038 · doi:10.1080/01630569708816804
[13] S. George, “On convergence of regularized modified Newton/s method for nonlinear ill-posed problems,” Journal of Inverse and Ill-Posed Problems, vol. 18, no. 2, pp. 133-146, 2010. · Zbl 1279.65069 · doi:10.1515/jiip.2010.004
[14] B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inverse Problems, vol. 13, no. 3, pp. 729-753, 1997. · Zbl 0880.65033 · doi:10.1088/0266-5611/13/3/012
[15] O. Scherzer, “A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems,” Numerical Functional Analysis and Optimization, vol. 17, no. 1-2, pp. 197-214, 1996. · Zbl 0852.65048 · doi:10.1080/01630569608816691
[16] S. Pereverzev and E. Schock, “On the adaptive selection of the parameter in regularization of ill-posed problems,” SIAM Journal on Numerical Analysis, vol. 43, no. 5, pp. 2060-2076, 2005. · Zbl 1103.65058 · doi:10.1137/S0036142903433819
[17] I. K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer, New York, NY, USA, 2008. · Zbl 1153.65057 · doi:10.1007/978-0-387-72743-1
[18] E. V. Semenova, “Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators,” Computational Methods in Applied Mathematics, vol. 10, no. 4, pp. 444-454, 2010. · Zbl 1283.65102 · doi:10.2478/cmam-2010-0026
[19] S. Lu and S. V. Pereverzev, “Sparsity reconstruction by the standard Tikhonov method,” RICAM-Report, 2008.
[20] V. V. Vasin, I. I. Prutkin, M. Timerkhanova, and L. Yu, “Retrieval of a three-dimensional relief of geological boundary from gravity data,” Izvestiya, Physics of the Solid Earth, vol. 32, no. 11, pp. 58-62, 1996.
[21] V. V. Vasin, “Modified processes of Newton type generating Fejer approximations of regularized solutions of nonlinear equations,” Proceedings in Mathematics and Machanics, vol. 19, no. 2, pp. 85-97, 2013 (Russian).
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