Iterative regularization with a general penalty term—theory and application to L1 and TV regularization

In this paper, we consider an iterative regularization scheme for linear ill-posed equations in Banach spaces. As opposed to other iterative approaches, we deal with a general penalty functional from Tikhonov regularization and take advantage of the properties of the regularized solutions which where supported by the choice of the specific penalty term. We present convergence and stability results for the presented algorithm. Additionally, we demonstrate how these theoretical results can be applied to L1- and TV-regularization approaches and close the paper with a short numerical example.

[1]  A. Baskakov,et al.  Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations , 2013 .

[2]  S. George,et al.  Iterative Methods for Nonlinear Ill-Posed Problems , 2012 .

[3]  Bin Dong,et al.  Fast Linearized Bregman Iteration for Compressive Sensing and Sparse Denoising , 2011, ArXiv.

[4]  Wotao Yin,et al.  Analysis and Generalizations of the Linearized Bregman Method , 2010, SIAM J. Imaging Sci..

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

[6]  Bernd Hofmann,et al.  An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems , 2010 .

[7]  Kamil S. Kazimierski,et al.  Accelerated Landweber iteration in Banach spaces , 2010 .

[8]  R. Boţ,et al.  Conjugate Duality in Convex Optimization , 2010 .

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

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

[11]  Otmar Scherzer,et al.  Variational Methods in Imaging , 2008, Applied mathematical sciences.

[12]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[13]  F. Schöpfer,et al.  Fast regularizing sequential subspace optimization in Banach spaces , 2008 .

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

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

[16]  Wotao Yin,et al.  Bregman Iterative Algorithms for (cid:2) 1 -Minimization with Applications to Compressed Sensing ∗ , 2008 .

[17]  Dirk A. Lorenz,et al.  Iterated Hard Shrinkage for Minimization Problems with Sparsity Constraints , 2008, SIAM J. Sci. Comput..

[18]  Alfred K. Louis,et al.  Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods , 2008 .

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

[20]  Y. Alber,et al.  Nonlinear Ill-posed Problems of Monotone Type , 2006 .

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

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

[23]  S. Osher,et al.  Convergence rates of convex variational regularization , 2004 .

[24]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .

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

[26]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[27]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

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

[29]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[30]  M. Hanke.  Regularization with differential operators : an iterative approach , 1992 .

[31]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .