Abstract
A Modification of regularized Newton-type method for nonlinear ill-posed problems is considered. Using an a posteriori stopping rule proposed by Kaltenbacher, the convergence of the method is proved under certain conditions on the nonlinear operator. Optimal convergence rates are also shown under appropriate closeness and smoothness assumptions on the difference of the starting value and the solution. Some special cases of the method are also given. Numerical results confirm the corresponding theoretical statements.
Project supported by the National Natural Science Foundation of China (No.10871168) and supported by the Education of Zhejiang Province (No.Y200804144).
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References
Engl, H.W., Hanke, M., Neubauer, O.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)
Groetshch, C.W.: Inverse Problems in Mathematical Sciences. Vieweg, Braunschweig (1993)
Engl, H.W., Scherzer, O.: Convergence rates results for iterative methods for solving nonlinear ill-posed problems. In: Colton, D., et al. (eds.) Survey on Solution Methods for Inverse Problems, Wien, pp. 7–34. Springer, Heidelberg (2000)
Samanskii, V.: On a modification of the Newton method. Ukrain Mat. Z. 19, 133–138 (1967) (in Russian)
Otega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Acad. Press, New York (1970)
Kaltenbacher, B.: A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems. Numerische Mathematik 79, 501–528 (1998)
Zhao, Z., He, G.: Reconstruction of High Order Derivatives by a New Mollification Method. Applied Mathematics and Mechanics 29, 352–373 (2008)
Blaschke, B., Neubauer, A., Scherzer, O.: On convergence rates for the iteratively regularized Gauss-Newton method. IMA Journal of Numerical Analysis 17, 421–436 (1997)
Kaltenbacher, B.: Some Newton-type methods for the regularization of nonlinear ill-posed problems. Inverse Problems 13, 729–753 (1997)
Bauer, F., Hohage, T.: A Lepskij-type rule for regularized Newton methods. Inverse Problems 21, 1975–1991 (2005)
He, G., Liu, L.: A kind of implicit iterative methods for ill-posed operator equations. J. Comput. Math. 17, 275–284 (1999)
He, G., Meng, Z.: A Newton type iterative method for heat-conduction inverse problems. Applied Mathematics and Mechanics 28, 531–539 (2007)
Kaltenbacher, B., Neubauer, A.: Convergence of projected iterative regularized methods for nonlinear problems with smooth solutions. Inverse Problems 22, 1105–1119 (2006)
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Meng, Zh., Zhao, Zy., He, Gq. (2010). A Modification of Regularized Newton-Type Method for Nonlinear Ill-Posed Problems. In: Zhang, W., Chen, Z., Douglas, C.C., Tong, W. (eds) High Performance Computing and Applications. Lecture Notes in Computer Science, vol 5938. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11842-5_40
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DOI: https://doi.org/10.1007/978-3-642-11842-5_40
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