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

References

• Bonesky T. Morozov’s discrepancy principle and Tikhonov-type functionals. Inverse Prob. 2009;25:015015.
   Web of Science ®Google Scholar
• Anzengruber SW, Ramlau R. Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators. Inverse Prob. 2010;26:025001.
   Web of Science ®Google Scholar
• Anzengruber SW. The discrepancy principle for Tikhonov regularization in Banach spaces: Regularization properties and rates of convergence. Saarbrücken: Südwestdeutscher Verlag für Hochschulschriften; 2012.
   Google Scholar
• Flemming J. Generalized Tikhonov regularization and modern convergence rate theory in Banach spaces. Aachen: Shaker Verlag; 2012.
   Google Scholar
• Schuster T, Kaltenbacher B, Hofmann B, Kazimierski KS. Regularization methods in Banach spaces. Berlin/Boston: Walter de Gruyter; 2012.
   Google Scholar
• Anzengruber SW, Ramlau R. Convergence rates for Morozov’s discrepancy principle using variational inequalities. Inverse Prob. 2011;27:105007.
   Web of Science ®Google Scholar
• Hofmann B, Mathé P. Parameter choice under variational inequalities. Inverse Prob. 2012;28:104006.
   Web of Science ®Google Scholar
• Hofmann B, Kaltenbacher B, Poeschl C, Scherzer O. A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Prob. 2007;23:987–1010.
   Web of Science ®Google Scholar
• Scherzer O, Grasmair M, Grossauer H, Haltmeier M, Lenzen F. Variational methods in imaging. New York: Springer-Verlag; 2009.
   Google Scholar
• Burger M, Flemming J, Hofmann B. Convergence rates in l1-regularization if the sparsity assumption fails. Inverse Prob. 2013;29:025013.
   Web of Science ®Google Scholar
• Boţ RI, Hofmann B. The impact of a curious type of smoothness conditions on convergence rates in l1-regularization. Eurasian J. Math. Comput. Appl. 2013;1:29–40.
   Google Scholar
• Burger M, Osher S. A guide to the TV Zoo. In: Level set and PDE based reconstruction methods in imaging (eds. Burger M, Osher S). Lecture Notes in Mathematics, Vol. 2090. Berlin: Springer; 2013.
   Google Scholar
• Burger M, Resmerita E, He L. Error estimation for Bregman iterations and inverse scale space methods in image restoration. Computing. 2007;81:109–135.
   Web of Science ®Google Scholar
• Grasmair M, Haltmeier M, Scherzer O. Sparse regularization with lq penalty term. Inverse Prob. 2008;24:1–13.
   Web of Science ®Google Scholar
• Grasmair M. Well-posedness and convergence rates for sparse regularization with sublinear lq penalty term. Inverse Probl. Imaging. 2009;3:383–387.
   Web of Science ®Google Scholar
• Lorenz DA. Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 2008;16:463–478.
   Web of Science ®Google Scholar
• Ramlau R, Zarzer CA. On the minimization of a Tikhonov functional with a non-convex sparsity constraint Electron. Trans. Numer. Anal. 2012;39:476–507.
   Web of Science ®Google Scholar
• Zarzer CA. On Tikhonov regularization with non-convex sparsity constraints. Inverse Prob. 2009;25:025006.
   Web of Science ®Google Scholar
• Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Vol. 375 of mathematics and its application. Dordrecht: Kluwer Academic Publishers; 1996.
   Google Scholar
• Burger M, Osher S. Convergence rates of convex variational regularization. Inverse Prob. 2004;20:1411–1421.
   Web of Science ®Google Scholar
• Bredies K, Lorenz DA. Regularization with non-convex separable constraints. Inverse Prob. 2009;25:085011.
   Web of Science ®Google Scholar
• Grasmair M. Non-convex sparse regularisation. J. Math. Anal. Appl. 2010;365:19–28.
   Web of Science ®Google Scholar
• Offtermatt J. A projection and variational regularization method for sparse inverse problems [PhD thesis]. Stuttgart: University of Stuttgart; 2012
   Google Scholar
• Tikhonov AN, Leonov AS, Yagola AG. Nonlinear ill-posed problems. London: Chapman & Hall; 1998.
   Google Scholar
• Grasmair M. Generalized Bregman distances and convergence rates for non-convex regularization methods. Inverse Prob. 2010;26:115014.
   Web of Science ®Google Scholar
• Hofmann B, Yamamoto M. On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems. Applicable Analy. 2010;89:1705–1727.
   Web of Science ®Google Scholar
• Grasmair M. An application of source inequalities for convergence rates of Tikhonov regularization with a non-differentiable operator (submitted). Preliminary version under arXiv:1209.2246v1. 2012.
   Google Scholar
• Clason C. l∞ fitting for inverse problems with uniform noise. Inverse Prob. 2012;28:104007.
   Web of Science ®Google Scholar