Abstract
We introduce a regularized Newton method coupled with the parallel splitting-up technique for solving nonlinear ill-posed equations with smooth monotone operators. We analyze the convergence of the proposed method and carry out numerical experiments for nonlinear integral equations.
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Burger, M., Kaltenbacher, B.: Regularizing Newton–Kaczmarz methods for nonlinear ill-posed problems. SIAM Numer. Anal. 44, 153–182 (2006)
De Cezaro, A., Haltmeier, M., Leitão, A., Scherzer, O.: On steepest-descent-Kaczmarz method for regularizing systems of nonlinear ill-posed equations. Appl. Math. Comput. 202, 596–607 (2008)
Haltmeier, M., Leitão, A., Scherzer, O.: Kaczmarz methods for regularizing nonlinear ill-posed equations, I. Convergence analysis. Inverse Probl. Imaging 1(2), 289–298 (2007)
Haltmeier, M., Kowar, R., Leitão, A., Scherzer, O.: Kaczmarz methods for regularizing nonlinear ill-posed equations, II. Applications. Inverse Probl. Imaging 1(3), 507–523 (2007)
Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Walter de Gruyter, Berlin (2008)
Kowar, R., Scherzer, O.: Convergence analysis of a Landweber–Kaczmarz method for solving nonlinear ill-posed problems. In: Ill-Posed and Inverse Problems (Book Series), vol. 23, pp. 69–90 (2002)
Alber, Y., Ryazantseva, I.: Nonlinear Ill-Posed Problems of Monotone Type. Springer, New York (2006)
Hoang, N.S., Ramm, A.G.: An iterative scheme for solving nonlinear equations with monotone operators. BIT Numer. Math. 48(4), 725–741 (2008)
Hoang, N.S., Ramm, A.G.: Dynamical systems gradient methods for solving nonlinear equations with monotone operators. Acta Appl. Math. 106(3) (2009) 473–499
Hoang, N.S., Ramm, A.G.: Dynamical systems method for solving nonlinear equations with monotone operators. Math. Comput. 79(269) (2010) 239–258
Bakusinskii, A.B., Goncharskii, A.V.: Ill-Posed Problems. Numerical Methods and Applications. Moscow Univ. Press. In Russian (1989)
Janno, J.: Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation. Inverse Probl. 16, 133–148 (2000)
Tautenhahn, U.: On the method of Lavrentiev regularization for nonlinear ill-posed problems. Inverse Probl. 18, 191–207 (2002)
Anh, P.K., Chung, C.V.: Parallel iterative regularization methods for solving systems of ill-posed equations. Appl. Math. Comput. 212, 542–550 (2009)
Lü, T., Neittaanmäki, P., Tai, X.-C.: A parallel splitting up method for partial differential equations and its application to Navier–Stokes equations. RAIRO Math. Model. Numer. Anal. 26(6), 673–708 (1992)
Liu, F., Nashed, M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998)
Hein, T.: Convergence rates for multi-parameter regularization in Banach spaces. Int. J. Pure Appl. Math. 43(4), 593–614 (2008)
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Anh, P.K., Chung, C.V. Parallel regularized Newton method for nonlinear ill-posed equations. Numer Algor 58, 379–398 (2011). https://doi.org/10.1007/s11075-011-9460-y
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DOI: https://doi.org/10.1007/s11075-011-9460-y