论文信息 - An iteratively regularized Gauss–Newton–Halley method for solving nonlinear ill-posed problems

An iteratively regularized Gauss–Newton–Halley method for solving nonlinear ill-posed problems

In this paper we analyze a second order method for regularizing nonlinear inverse problems, which comes from adapting Halley’s method to the ill-posed situation. The method is related to the iteratively regularized Gauss–Newton method and enhanced by the use of second derivatives of the forward operator. For parameter identification problems in PDEs, these second derivatives often require not much additional computational cost as compared to first order derivatives. We prove convergence and convergence rates under certain conditions on the forward operator in Hilbert spaces. Numerical tests for a problem of identifying a distributed coefficient in an elliptic PDE illustrate the performance of the method.

Barbara Kaltenbacher | B. Kaltenbacher

[1] Ioannis K. Argyros,et al. On the semi-local convergence of Halley's method under a center-Lipschitz condition on the second Fréchet derivative , 2012, Appl. Math. Comput..

[2] A. Neubauer,et al. On convergence rates for the Iteratively regularized Gauss-Newton method , 1997 .

[3] William Rundell,et al. A Second Degree Method for Nonlinear Inverse Problems , 1999, SIAM J. Numer. Anal..

[4] Bernd Hofmann,et al. How general are general source conditions? , 2008 .

[5] Otmar Scherzer,et al. Factors influencing the ill-posedness of nonlinear problems , 1994 .

[6] M. Hanke,et al. A convergence analysis of the Landweber iteration for nonlinear ill-posed problems , 1995 .

[7] Boro Döring,et al. Einige Sätze über das Verfahren der tangierenden Hyperbeln in Banach-Räumen , 1970 .

[8] Franz Schreier,et al. Iterative regularization methods for nonlinear problems , 2010 .

[9] George H. Brown,et al. On Halley's Variation of Newton's Method , 1977 .

[10] A. Bakushinskii. The problem of the convergence of the iteratively regularized Gauss-Newton method , 1992 .

[11] H. Engl,et al. Regularization of Inverse Problems , 1996 .

[12] H. Engl,et al. Convergence rates for Tikhonov regularisation of non-linear ill-posed problems , 1989 .

[13] Barbara Kaltenbacher,et al. Iterative Regularization Methods for Nonlinear Ill-Posed Problems , 2008, Radon Series on Computational and Applied Mathematics.