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An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems. (English) Zbl 1410.65231

Summary: In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations \(F(x)=y\) where \(F:D(F)\subseteq X\to X\) is a nonlinear monotone operator or \(F'(x_0)\) is nonnegative selfadjoint operator defined on a real Hilbert space \(X\). We assume that only a noisy data \(y^\delta \in X\) with \(||y-y^{\delta}||\leqslant{\delta}\) are available. Further we assume that Fréchet derivative \(F'\) of \(F\) satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on \(x_0-\hat x\) the error \(||\hat x-x_{n,\alpha}^{\delta}||\) between the regularized approximation \(x_{n,\alpha}^{\delta}(x_{0,\alpha}^{\delta}:= x_0)\) and the solution \(\hat x\) is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem.

MSC:

65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47J05 Equations involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators
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[1] Argyros, I. K., Convergence and applications of Newton-type iterations, (2008), Springer · Zbl 1153.65057
[2] Bakushinsky, A. B., A regularizing algorithm based on the Newton-Kantorovich method for solving variational inequalities, Comput. Math. Math. Phys., 16, 16-23, (1976)
[3] Bakushinsky, A.; Goncharsky, A., Ill-posed problems: theory and applications, (1994), Kluwer Academic Publishers
[4] Bakushinsky, A.; Smirnova, A., On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 26, 1, 35-48, (2005) · Zbl 1068.47081
[5] Blaschke, B.; Neubauer, A.; Scherzer, O., On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal., 17, 3, 421-436, (1997) · Zbl 0881.65050
[6] Deuflhard, P.; Engl, H. W.; Scherzer, O., A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inverse Prob., 14, 5, 1081-1106, (1998) · Zbl 0915.65053
[7] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of inverse problems, Mathematics and Its Applications, vol. 375, (1996), Kluwer Academic Publishers Group Dordrecht, Boston, London, (viii+321pp) · Zbl 0859.65054
[8] George, S.; Elmahdy, A. I., An analysis of Lavrentiev regularization for nonlinear ill-posed problems using an iterative regularization method, Int. J. Comput. Appl. Math., 5, 3, 369-381, (2010)
[9] Hanke, M.; Neubauer, A.; Scherzer, O., A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72, 1, 21-37, (1995) · Zbl 0840.65049
[10] Qi-Nian, Jin., Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales, Inverse Prob., 16, 1, 187-197, (2000) · Zbl 0969.65048
[11] Qi-Nian, Jin., On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems, Math. Comput., 69, 232, 1603-1623, (2000) · Zbl 0962.65047
[12] Mahale, P.; Nair, M. T., Iterated Lavrentiev regularization for nonlinear ill-posed problems, ANZIAM J., 51, 191-217, (2009) · Zbl 1208.47058
[13] Nair, M. T.; Ravishankar, P., Regularized versions of continuous newton’s method and continuous modified newton’s method under general source conditions, Numer. Funct. Anal. Optim., 29, 9-10, 1140-1165, (2008) · Zbl 1155.65043
[14] Ramm, A. G., Inverse problems: mathematical and analytical techniques with applications to engineering, (2005), Springer · Zbl 1083.35002
[15] Tautanhahn, U., On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Prob., 18, 1, 191-207, (2002) · Zbl 1005.65058
[16] Vasin, V. V., Modified processes of Newton type generating fejer approximations for regularized solutions of nonlinear equations, Trudy Inst. Mat. Mekh. UrO RAN, 19, 2, 85-97, (2013), (in Russian)
[17] Vasin, V. V.; Prutkin, I. I.; Timerkhanova, L. Yu., Retrieval of a three-dimensional relief of geological boundary from gravity data, Izvestiya, Phys. Solid Earth, 32, 11, 58-62, (1996)
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