Convergence rates of the continuous regularized Gauss—Newton method

Abstract - In this paper a convergence proof is given for the continuous analog of the Gauss-Newton method for nonlinear ill-posed operator equations and convergence rates are obtained. Convergence for exact data is proved for nonmonotone operators under weaker source conditions than before. Moreover, nonlinear ill-posed problems with noisy data are considered and a priori and a posteriori stopping rules are proposed. These rules yield convergence of the regularized approximations to the exact solution as the noise level tends to zero. The convergence rates are optimal rates under the source conditions considered.

[1]  Barbara Kaltenbacher,et al.  On Broyden's method for the regularization of nonlinear ill-posed problems , 1998 .

[2]  A NUMERICAL METHOD FOR SOLVING NONLINEAR ILL-POSED PROBLEMS , 1999 .

[3]  Otmar Scherzer,et al.  A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions , 1998 .

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

[5]  Alexander G. Ramm,et al.  Stable solutions of some ill-posed problems , 1981 .

[6]  Continuous regularized Gauss-Newton-type algorithm for nonlinear ill-posed equations with simultaneous updates of inverse derivative , 2001, math-ph/0103042.

[7]  Alexander G. Ramm.  Linear Ill-Posed Problems and Dynamical Systems☆ , 2000 .

[8]  Continuous regularization of nonlinear ill-posed problems , 1999 .

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

[10]  Alexander G. Ramm,et al.  CONTINUOUS ANALOG OF THE GAUSS-NEWTON METHOD , 1999 .

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

[12]  Alexander G. Ramm,et al.  On stable numerical differentiation , 2001, Math. Comput..

[13]  Alexander G. Ramm,et al.  DYNAMICAL SYSTEMS AND DISCRETE METHODS FOR SOLVING NONLINEAR ILL-POSED PROBLEMS , 2000 .

[14]  Thorsten Hohage,et al.  Regularization of exponentially ill-posed problems , 2000 .

[15]  Alexander G. Ramm,et al.  Continuous modified Newton’s-type method for nonlinear operator equations , 2003 .

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