Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2008, Article ID 192679, 19 pages
http://dx.doi.org/10.1155/2008/192679
Research Article

Minimization of Tikhonov Functionals in Banach Spaces

1Center for Industrial Mathematics, University of Bremen, Bremen 28334, Germany
2Fakultät für Maschinenbau, Helmut-Schmidt-Universität, Universität der Bundeswehr Hamburg, Holstenhofweg 85, Hamburg 22043, Germany

Received 3 July 2007; Accepted 31 October 2007

Academic Editor: Simeon Reich

Copyright © 2008 Thomas Bonesky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Citations to this Article [55 citations]

The following is the list of published articles that have cited the current article.

  • Markus Grasmair, Markus Haltmeier, and Otmar Scherzer, “ Sparse regularization with l q penalty term ,” Inverse Problems, vol. 24, no. 5, pp. 055020, 2008. View at Publisher · View at Google Scholar
  • F Schöpfer, T Schuster, and A K Louis, “An iterative regularization method for the solution of the split feasibility problem in Banach spaces,” Inverse Problems, vol. 24, no. 5, pp. 055008, 2008. View at Publisher · View at Google Scholar
  • D. A. Lorenz, “Convergence rates and source conditions for Tikhonov regularization with sparsity constraints,” Journal Of Inverse And Ill-Posed Problems, vol. 16, no. 5, pp. 463–478, 2008. View at Publisher · View at Google Scholar
  • F Schöpfer, and T Schuster, “Fast regularizing sequential subspace optimization in Banach spaces,” Inverse Problems, vol. 25, no. 1, pp. 015013, 2008. View at Publisher · View at Google Scholar
  • Kristian Bredies, “A forward–backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space,” Inverse Problems, vol. 25, no. 1, pp. 015005, 2008. View at Publisher · View at Google Scholar
  • D. A. Lorenz, and D. Trede, “Optimal convergence rates for Tikhonov regularization in Besov scales,” Inverse Problems, vol. 24, no. 5, 2008. View at Publisher · View at Google Scholar
  • Barbara Kaltenbacher, Frank Schoepfer, and Thomas Schuster, “Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems,” Inverse Problems, vol. 25, no. 6, 2009. View at Publisher · View at Google Scholar
  • T. Bonesky, and P. Maass, “Iterated soft shrinkage with adaptive operator evaluations,” Journal of Inverse and Ill-Posed Problems, vol. 17, no. 4, pp. 337–358, 2009. View at Publisher · View at Google Scholar
  • Andreas Neubauer, “On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces,” Inverse Problems, vol. 25, no. 6, pp. 065009, 2009. View at Publisher · View at Google Scholar
  • Torsten Hein, and Bernd Hofmann, “Approximate source conditions for nonlinear ill-posed problems-chances and limitations,” Inverse Problems, vol. 25, no. 3, 2009. View at Publisher · View at Google Scholar
  • K. Bredies, “An iterative thresholding-like algorithm for inverse problems with sparsity constraints in Banach space,” Journal Of Inverse And Ill-Posed Problems, vol. 17, no. 1, pp. 19–26, 2009. View at Publisher · View at Google Scholar
  • L Denis, D A Lorenz, and D Trede, “Greedy solution of ill-posed problems: error bounds and exact inversion,” Inverse Problems, vol. 25, no. 11, pp. 115017, 2009. View at Publisher · View at Google Scholar
  • Kristian Bredies, and Dirk A Lorenz, “Regularization with non-convex separable constraints,” Inverse Problems, vol. 25, no. 8, 2009. View at Publisher · View at Google Scholar
  • Andreas Neubauer, “Modified Tikhonov Regularization For Nonlinear Ill-Posed Problems In Banach Speces,” Journal Of Integral Equations And Applications, vol. 22, no. 2, pp. 341–351, 2010. View at Publisher · View at Google Scholar
  • Thomas Schuster, and Frank Schöpfer, “Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse,” Inverse Problems, vol. 26, no. 8, pp. 085006, 2010. View at Publisher · View at Google Scholar
  • Torsten Hein, and Kamil S. Kazimierski, “Accelerated Landweber iteration in Banach spaces,” Inverse Problems, vol. 26, no. 5, 2010. View at Publisher · View at Google Scholar
  • Barbara Kaltenbacher, and Bernd Hofmann, “Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces,” Inverse Problems, vol. 26, no. 3, pp. 035007, 2010. View at Publisher · View at Google Scholar
  • Radu Ioan Bot, and Bernd Hofmann, “An Extension Of The Variational Inequality Approach For Obtaining Convergence Rates In Regularization Of Nonlinear Ill-Posed Problems,” Journal Of Integral Equations And Applications, vol. 22, no. 3, pp. 369–392, 2010. View at Publisher · View at Google Scholar
  • Bangti Jin, and Dirk A. Lorenz, “Heuristic Parameter-Choice Rules For Convex Variational Regularization Based On Error Estimates,” Siam Journal On Numerical Analysis, vol. 48, no. 3, pp. 1208–1229, 2010. View at Publisher · View at Google Scholar
  • Bernd Hofmann, and Masahiro Yamamoto, “On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems,” Applicable Analysis, vol. 89, no. 11, pp. 1705–1727, 2010. View at Publisher · View at Google Scholar
  • Jens Flemming, and Bernd Hofmann, “A New Approach to Source Conditions in Regularization with General Residual Term,” Numerical Functional Analysis And Optimization, vol. 31, no. 3, pp. 254–284, 2010. View at Publisher · View at Google Scholar
  • Kazufumi Ito, and Bangti Jin, “A new approach to nonlinear constrained Tikhonov regularization,” Inverse Problems, vol. 27, no. 10, pp. 105005, 2011. View at Publisher · View at Google Scholar
  • Jens Flemming, and Bernd Hofmann, “Convergence rates in constrained Tikhonov regularization: equivalence of projected source conditions and variational inequalities,” Inverse Problems, vol. 27, no. 8, pp. 085001, 2011. View at Publisher · View at Google Scholar
  • Kamil S. Kazimierski, “Minimization of the Tikhonov functional in Banach spaces smooth and convex of power type by steepest descent in the dual,” Computational Optimization And Applications, vol. 48, no. 2, pp. 309–324, 2011. View at Publisher · View at Google Scholar
  • Markus Grasmair, Markus Haltmeier, and Otmar Scherzer, “The residual method for regularizing ill-posed problems,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2693–2710, 2011. View at Publisher · View at Google Scholar
  • Kazufumi Ito, Bangti Jin, and Tomoya Takeuchi, “A Regularization Parameter For Nonsmooth Tikhonov Regularization,” Siam Journal on Scientific Computing, vol. 33, no. 3, pp. 1415–1438, 2011. View at Publisher · View at Google Scholar
  • Bangti Jin, and Peter Maass, “Sparsity regularization for parameter identification problems,” Inverse Problems, vol. 28, no. 12, pp. 123001, 2012. View at Publisher · View at Google Scholar
  • Leitão, and M. Marques Alves, “On Landweber-Kaczmarz methods for regularizing systems of ill-posed equations in Banach spaces,” Inverse Problems, vol. 28, no. 10, 2012. View at Publisher · View at Google Scholar
  • K S Kazimierski, P Maass, and R Strehlow, “ Norm sensitivity of sparsity regularization with respect to p ,” Inverse Problems, vol. 28, no. 10, pp. 104009, 2012. View at Publisher · View at Google Scholar
  • Radu Ioan Bot, and Torsten Hein, “ Iterative regularization with a general penalty term—theory and application to L 1 and TV regularization ,” Inverse Problems, vol. 28, no. 10, pp. 104010, 2012. View at Publisher · View at Google Scholar
  • Li Chen, and Zhen Wu, “Dynamic programming principle for stochastic recursive optimal control problem with delayed systems,” ESAIM - Control, Optimisation and Calculus of Variations, vol. 18, no. 4, pp. 1005–1026, 2012. View at Publisher · View at Google Scholar
  • Armin Lechleiter, Kamil S Kazimierski, and Mirza Karamehmedovic, “ Tikhonov regularization in L p applied to inverse medium scattering ,” Inverse Problems, vol. 29, no. 7, pp. 075003, 2013. View at Publisher · View at Google Scholar
  • Markus Grasmair, “Variational inequalities and higher order convergence rates for Tikhonov regularisation on Banach spaces,” Journal of Inverse and Ill-Posed Problems, vol. 21, no. 3, pp. 379–394, 2013. View at Publisher · View at Google Scholar
  • Kristian Bredies, and Hanna Katriina Pikkarainen, “Inverse Problems In Spaces Of Measures,” Esaim-Control Optimisation and Calculus of Variations, vol. 19, no. 1, pp. 190–218, 2013. View at Publisher · View at Google Scholar
  • Martin Burger, Stanley Osher, Martin Burger, and Stanley Osher, “A Guide to the TV Zoo,” Level Set and Pde Based Reconstruction Methods in Imaging, pp. 1–70, 2013. View at Publisher · View at Google Scholar
  • Martin Burger, and Stanley Osher, “A guide to the TV zoo,” Lecture Notes in Mathematics, vol. 2090, pp. 1–70, 2013. View at Publisher · View at Google Scholar
  • Li-Wei Kuo, and D. R. Sahu, “Bregman Distance and Strong Convergence of Proximal-Type Algorithms,” Abstract and Applied Analysis, vol. 2013, pp. 1–12, 2013. View at Publisher · View at Google Scholar
  • Baohuai Sheng, and Peixin Ye, “The Learning Rates of Regularized Regression Based on Reproducing Kernel Banach Spaces,” Abstract and Applied Analysis, vol. 2013, pp. 1–10, 2013. View at Publisher · View at Google Scholar
  • R Strehlow, and K S Kazimierski, “Approximation of penalty terms in Tikhonov functionals—theory and applications in inverse problems,” Inverse Problems, vol. 30, no. 7, pp. 075005, 2014. View at Publisher · View at Google Scholar
  • Manabu Machida, and John C Schotland, “Inverse Born series for the radiative transport equation,” Inverse Problems, vol. 31, no. 9, pp. 095009, 2015. View at Publisher · View at Google Scholar
  • Fabio Margotti, and Andreas Rieder, “An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method,” Journal Of Inverse And Ill-Posed Problems, vol. 23, no. 4, pp. 373–392, 2015. View at Publisher · View at Google Scholar
  • Stephan W. Anzengruber, Bo Han, Wei Wang, and Ronny Ramlau, “A global minimization algorithm for Tikhonov functionals with sparsity constraints,” Applicable Analysis, vol. 94, no. 3, pp. 580–611, 2015. View at Publisher · View at Google Scholar
  • Jin Cheng, and Bernd Hofmannpp. 91–123, 2015. View at Publisher · View at Google Scholar
  • B Kaltenbacher, “A convergence rates result for an iteratively regularized Gauss–Newton–Halley method in Banach space,” Inverse Problems, vol. 31, no. 1, pp. 015007, 2015. View at Publisher · View at Google Scholar
  • Min Zhong, and Wei Wang, “A global minimization algorithm for Tikhonov functionals with $p-$convex ($p\,\geqslant \,2$) penalty terms in Banach spaces,” Inverse Problems, vol. 32, no. 10, pp. 104008, 2016. View at Publisher · View at Google Scholar
  • Maoguo Gong, Xiangming Jiang, and Hao Li, “Optimization methods for regularization-based ill-posed problems: a survey and a multi-objective framework,” Frontiers of Computer Science, vol. 11, no. 3, pp. 362–391, 2016. View at Publisher · View at Google Scholar
  • Lingling Du, Jing Li, and Jinping Wang, “The analysis study on nonlinear iterative methods for inverse problems,” Applicable Analysis, pp. 1–11, 2016. View at Publisher · View at Google Scholar
  • Huanxiang Liu, Baohuai Sheng, and Peixin Ye, “The improved learning rate for regularized regression with RKBSs,” International Journal of Machine Learning and Cybernetics, 2016. View at Publisher · View at Google Scholar
  • Uno Hämarik, Barbara Kaltenbacher, Urve Kangro, and Elena Resmerita, “Regularization by discretization in Banach spaces,” Inverse Problems, vol. 32, no. 3, pp. 035004, 2016. View at Publisher · View at Google Scholar
  • Fábio Margotti, “Mixed gradient-Tikhonov methods for solving nonlinear ill-posed problems in Banach spaces,” Inverse Problems, vol. 32, no. 12, pp. 125012, 2016. View at Publisher · View at Google Scholar
  • Claudio Estatico, Serge Gratton, Flavia Lenti, and David Titley-Peloquin, “A conjugate gradient like method for p-norm minimization in functional spaces,” Numerische Mathematik, 2017. View at Publisher · View at Google Scholar
  • Winnifried Wollner, and Roland Herzog, “A conjugate direction method for linear systems in Banach spaces,” Journal of Inverse and Ill-Posed Problems, vol. 25, no. 5, pp. 553–572, 2017. View at Publisher · View at Google Scholar
  • Martin Burger, Tapio Helin, and Hanne Kekkonen, “Large noise in variational regularization,” Transactions of Mathematics and Its Applications, 2018. View at Publisher · View at Google Scholar
  • Suthep Suantai, Yekini Shehu, and Prasit Cholamjiak, “Nonlinear iterative methods for solving the split common null point problem in Banach spaces,” Optimization Methods and Software, pp. 1–22, 2018. View at Publisher · View at Google Scholar
  • Suthep Suantai, Yekini Shehu, Prasit Cholamjiak, and Olaniyi S. Iyiola, “Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces,” Journal of Fixed Point Theory and Applications, vol. 20, no. 2, 2018. View at Publisher · View at Google Scholar