Accelerated Landweber iteration in Banach spaces

We investigate a method of accelerated Landweber type for the iterative regularization of nonlinear ill-posed operator equations in Banach spaces. Based on an auxiliary algorithm with a simplified choice of the step-size parameter we present a convergence and stability analysis of the algorithm under consideration. We will close our discussion with the presentation of a numerical example.

[1]  E. Zeidler.  Nonlinear Functional Analysis and its Applications: III: Variational Methods and Optimization , 1984 .

[2]  Curtis R. Vogel,et al.  Well posedness and convergence of some regularisation methods for non-linear ill posed problems , 1989 .

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

[4]  K. Deimling.  Nonlinear functional analysis , 1985 .

[5]  Barbara Kaltenbacher,et al.  A Convergence Rates Result in Banach Spaces with Non-Smooth Operators , 2006 .

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

[7]  Bernd Hofmann,et al.  Approximate source conditions for nonlinear ill-posed problems—chances and limitations , 2009 .

[8]  P. Maass,et al.  Minimization of Tikhonov Functionals in Banach Spaces , 2008 .

[9]  Zongben Xu,et al.  Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces , 1991 .

[10]  B. Kaltenbacher,et al.  Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems , 2009 .

[11]  Tal Schuster,et al.  Nonlinear iterative methods for linear ill-posed problems in Banach spaces , 2006 .

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

[13]  Otmar Scherzer,et al.  A Convergence Rate Result for a Steepest Descent Method and a Minimal Error Method for the Solution of Nonlinear Ill-Posed Problems , 1995 .

[14]  Joram Lindenstrauss,et al.  Classical Banach spaces , 1973 .

[15]  Bernd Hofmann,et al.  On the nature of ill-posedness of an inverse problem arising in option pricing , 2003 .

[16]  I. Ciorǎnescu.  Geometry of banach spaces, duality mappings, and nonlinear problems , 1990 .

[17]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[18]  Bernd Hofmann,et al.  Analysis of Profile Functions for General Linear Regularization Methods , 2007, SIAM J. Numer. Anal..

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

[20]  O. Scherzer,et al.  A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators , 2007 .

[21]  Kamil S. Kazimierski,et al.  Modified Landweber Iteration in Banach Spaces—Convergence and Convergence Rates , 2010 .

[22]  Otmar Scherzer,et al.  Convergence Rates Results for Iterative Methods for Solving Nonlinear Ill-Posed Problems , 2000 .

[23]  Y. Kwok.  Mathematical models of financial derivatives , 2008 .