A Lepskij-type stopping rule for regularized Newton methods

We investigate an a posteriori stopping rule of Lepskij-type for a class of regularized Newton methods and show that it leads to order optimal convergence rates for Holder and logarithmic source conditions without a priori knowledge of the smoothness of the solution. Numerical experiments show that this stopping rule yields results at least as good as, and in some situations significantly better than, Morozov's discrepancy principle.

[1]  O. Lepskii.  On a Problem of Adaptive Estimation in Gaussian White Noise , 1991 .

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

[3]  H. Engl,et al.  Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems , 1993 .

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

[5]  Thorsten Hohage,et al.  Logarithmic convergence rates of the iteratively regularized Gauss - Newton method for an inverse potential and an inverse scattering problem , 1997 .

[6]  B. Kaltenbacher.  Some Newton-type methods for the regularization of nonlinear ill-posed problems , 1997 .

[7]  Andreas Kirsch,et al.  Characterization of the shape of a scattering obstacle using the spectral data of the far field operator , 1998 .

[8]  B. Kaltenbacher.  A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinearill-posed problems , 1998 .

[9]  A. Goldenshluger,et al.  Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations , 2000 .

[10]  Qinian Jin,et al.  Tikhonov regularization and a posteriori rules for solving nonlinear ill posed problems , 2003 .

[11]  P. Mathé,et al.  Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789-803 , 2003 .

[12]  A. Bakushinsky,et al.  Iterative Methods for Approximate Solution of Inverse Problems , 2005 .

[13]  F. Bauer,et al.  Regularization without preliminary knowledge of smoothness and error behaviour , 2005, European Journal of Applied Mathematics.