Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

[1]  A. Tikhonov,et al.  Nonlinear Ill-Posed Problems , 1997 .

[2]  A. G. Yagola,et al.  Numerical solution of nonlinear ill-posed problems , 1998 .

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

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

[5]  Lin He,et al.  Error estimation for Bregman iterations and inverse scale space methods in image restoration , 2007, Computing.

[6]  O. Scherzer,et al.  Sparse regularization with lq penalty term , 2008, 0806.3222.

[7]  D. Lorenz,et al.  Convergence rates and source conditions for Tikhonov regularization with sparsity constraints , 2008, 0801.1774.

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

[9]  Markus Grasmair,et al.  Well-posedness and Convergence Rates for Sparse Regularization with Sublinear l q Penalty Term , 2009 .

[10]  Thomas Bonesky.  Morozov's discrepancy principle and Tikhonov-type functionals , 2008 .

[11]  K. Bredies,et al.  Regularization with non-convex separable constraints , 2009 .

[12]  Clemens Zarzer.  On Tikhonov regularization with non-convex sparsity constraints , 2009 .

[13]  R. Ramlau,et al.  ON THE MINIMIZATION OF A TIKHONOV FUNCTIONAL WITH A NON-CONVEX SPARSITY CONSTRAINT , 2009 .

[14]  R. Ramlau,et al.  Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators , 2010 .

[15]  Bernd Hofmann,et al.  On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems , 2010 .

[16]  Markus Grasmair,et al.  Non-convex sparse regularisation , 2010 .

[17]  M. Grasmair.  Generalized Bregman distances and convergence rates for non-convex regularization methods , 2010 .

[18]  Ronny Ramlau,et al.  Convergence rates for Morozov's discrepancy principle using variational inequalities , 2011 .

[19]  Bernd Hofmann,et al.  Parameter choice in Banach space regularization under variational inequalities , 2012 .

[20]  Christian Clason,et al.  L∞ fitting for inverse problems with uniform noise , 2012 .

[21]  M. Grasmair.  An Application of Source Inequalities for Convergence Rates of Tikhonov Regularization with a Non-differentiable Operator , 2012, 1209.2246.

[22]  Martin Burger,et al.  Convergence rates in $\mathbf{\ell^1}$-regularization if the sparsity assumption fails , 2012 .

[23]  Barbara Kaltenbacher,et al.  Regularization Methods in Banach Spaces , 2012, Radon Series on Computational and Applied Mathematics.

[24]  Jonas Offtermatt.  A projection and a variational regularization method for sparse inverse problems , 2012 .

[25]  Stanley Osher,et al.  A Guide to the TV Zoo , 2013 .

[26]  B. Hofmann,et al.  The impact of a curious type of smoothness conditions on convergence rates in l1-regularization , 2013 .

[27]  Lea Fleischer,et al.  Regularization of Inverse Problems , 1996 .