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.

Linked References

  1. A. K. Louis, Inverse und schlecht gestellte Probleme, Teubner Studienbücher Mathematik, B. G. Teubner, Stuttgart, Germany, 1989. View at Zentralblatt MATH · View at MathSciNet
  2. A. Rieder, No Problems with Inverse Problems, Vieweg & Sohn, Braunschweig, Germany, 2003. View at Zentralblatt MATH · View at MathSciNet
  3. H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, Dordrecht, The Netherlands, 2000.
  4. Y. I. Alber, “Iterative regularization in Banach spaces,” Soviet Mathematics, vol. 30, no. 4, pp. 1–8, 1986. View at Google Scholar · View at Zentralblatt MATH
  5. R. Plato, “On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations,” Numerische Mathematik, vol. 75, no. 1, pp. 99–120, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 460–489, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  7. D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 84919, p. 39, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  8. F. Schöpfer, A. K. Louis, and T. Schuster, “Nonlinear iterative methods for linear ill-posed problems in Banach spaces,” Inverse Problems, vol. 22, no. 1, pp. 311–329, 2006. View at Google Scholar · View at MathSciNet
  9. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Communications on Pure and Applied Mathematics, vol. 57, no. 11, pp. 1413–1457, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. K. Bredies, D. Lorenz, and P. Maass, “A generalized conditional gradient method and its connection to an iterative shrinkage method,” to appear in Computational Optimization and Applications.
  11. Y. I. Alber, A. N. Iusem, and M. V. Solodov, “Minimization of nonsmooth convex functionals in Banach spaces,” Journal of Convex Analysis, vol. 4, no. 2, pp. 235–255, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Reich and A. J. Zaslavski, “Generic convergence of descent methods in Banach spaces,” Mathematics of Operations Research, vol. 25, no. 2, pp. 231–242, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Reich and A. J. Zaslavski, “The set of divergent descent methods in a Banach space is σ-porous,” SIAM Journal on Optimization, vol. 11, no. 4, pp. 1003–1018, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, vol. 97 of Results in Mathematics and Related Areas, Springer, Berlin, Germany, 1979. View at Zentralblatt MATH · View at MathSciNet
  15. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. View at Zentralblatt MATH · View at MathSciNet
  16. R. Deville, G. Godefroy, and V. Zizler, Smoothness and Renormings in Banach Spaces, vol. 64 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK, 1993. View at Zentralblatt MATH · View at MathSciNet
  17. A. Dvoretzky, “Some results on convex bodies and Banach spaces,” in Proceedings of the International Symposium on Linear Spaces, pp. 123–160, Jerusalem Academic Press, Jerusalem, Israel, 1961. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. O. Hanner, “On the uniform convexity of Lp and lp,” Arkiv för Matematik, vol. 3, no. 3, pp. 239–244, 1956. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Z. B. Xu and G. F. Roach, “Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 157, no. 1, pp. 189–210, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Reich, “Review of I. Cioranescu “Geometry of Banach spaces, duality mappings and nonlinear problems”,” Bulletin of the American Mathematical Society, vol. 26, no. 2, pp. 367–370, 1992. View at Google Scholar
  21. L. M. Bregman, “The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, pp. 200–217, 1967. View at Google Scholar
  22. C. Byrne and Y. Censor, “Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization,” Annals of Operations Research, vol. 105, no. 1–4, pp. 77–98, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. I. Alber and D. Butnariu, “Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces,” Journal of Optimization Theory and Applications, vol. 92, no. 1, pp. 33–61, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Bregman monotone optimization algorithms,” SIAM Journal on Control and Optimization, vol. 42, no. 2, pp. 596–636, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. H. H. Bauschke and A. S. Lewis, “Dykstra's algorithm with Bregman projections: a convergence proof,” Optimization, vol. 48, no. 4, pp. 409–427, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  26. J. D. Lafferty, S. D. Pietra, and V. D. Pietra, “Statistical learning algorithms based on Bregman distances,” in Proceedings of the 5th Canadian Workshop on Information Theory, Toronto, Ontario, Canada, June 1997.
  27. I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, The Netherlands, 1976. View at MathSciNet
  28. R. R. Phelps, “Metric projections and the gradient projection method in Banach spaces,” SIAM Journal on Control and Optimization, vol. 23, no. 6, pp. 973–977, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. R. H. Byrd and R. A. Tapia, “An extension of Curry's theorem to steepest descent in normed linear spaces,” Mathematical Programming, vol. 9, no. 1, pp. 247–254, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. C. Canuto and K. Urban, “Adaptive optimization of convex functionals in Banach spaces,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 2043–2075, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. T. Figiel, “On the moduli of convexity and smoothness,” Studia Mathematica, vol. 56, no. 2, pp. 121–155, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet