A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space

Inverse Problems

Volumn 25, Issue 1, 2009, Pages

  Bredies, Kristian ^a  

^a UNIVERSITY OF BREMEN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BANACH SPACES; DIFFERENTIAL EQUATIONS; IMAGE RECONSTRUCTION; INVERSE PROBLEMS; TOPOLOGY;

CONVERGENCE PROPERTIES; CONVEX FUNCTIONALS; DAUBECHIES; FORWARD-BACKWARD SPLITTING ALGORITHMS; FUNCTIONALS; HIGHER DIMENSIONS; IMAGE RESTORATIONS; ITERATIVE SOLUTIONS; LINEAR INVERSE PROBLEMS; NON SMOOTHES; RATES OF CONVERGENCES; SPLITTING METHODS; SUB-PROBLEMS; THRESHOLDING METHODS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 62649161771     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/25/1/015005     Document Type: Article

References (33)

1. Acar R and Vogel C 1994 Analysis of bounded variation penalty method for ill-posed problems Inverse Problems 10 1217-29
2. Alber Y I, Iusem A N and Solodov M V 1997 Minimization of nonsmooth convex functionals in Banach spaces J. Convex Anal. 4 235-55
3. Azé D and Penot J-P 1995 Uniformly convex and uniformly smooth convex functions Ann. Fac. Sci. Toulouse 4 705-30
4. Bredies K and Lorenz D A 2008 Iterated hard shrinkage for minimization problems with sparsity constraints SIAM J. Sci. Comput. 30 657-83
5. Bredies K and Lorenz D A 2008 Linear convergence of iterated soft-thresholding J. Fourier Anal. Appl. at press (DOI:10.1007/S00041-008-9041- 1)
6. Brezis H 1973 Opérateurs maximaux monotones et semi-groups de contractions dans les espaces de Hilbert North-Holland Mathematics Studies vol 5 (Amsterdam: North-Holland)
7. Butnariu D and Iusem A N 1997 Local moduli of convexity and their application to finding almost common fixed points of measurable families of operators Contemp. Math. 204 61-91
8. Butnariu D and Iusem A N 2000 Totally convex functions for fixed points computation and infinite dimensional optimization Applied Optimization vol 40 (Dordrecht: Kluwer)
9. Butnariu D, Iusem A N and Resmerita E 2000 Total convexity for powers of the norm in uniformly convex Banach spaces J. Convex Anal. 7 319-34
10. Chambolle A 2004 An algorithm for total variation minimization and applications J. Math. Imaging Vis. 20 89-97
11. Chen G H-G and Rockafellar R T 1997 Convergence rates in forward-backward splitting SIAM J. Optim. 7 421-44
12. Cioranescu I 1990 Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems (Dordrecht: Kluwer)
13. Combettes P L and Wajs V R 2005 Signal recovery by proximal forward-backward splitting Multiscale Model. Simul. 4 1168-200
14. Daubechies I, Defrise M and de Mol C 2004 An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Commun. Pure Appl. Math. 57 1413-57
15. Dunn J C 1981 Global and asymptotic convergence rate estimates for a class of projected gradient processes SIAM J. Control Optim. 19 368-400
16. Dvoretzky A 1961 Some results on convex bodies and Banach spaces Proc. Int. Symp. Linear Spaces (Jerusalem: Jerusalem Academic Press) pp 123-60
17. Ekeland I and Temam R 1976 Convex Analysis and Variational Problems (Amsterdam: North-Holland)
18. Feng G, Mellor R H, Bernstein M, Keller-Peck C, Nguyen Q T, Wallace M, Nerbonne J M, Lichtman J W and Sanes J R 2000 Imaging neuronal subsets in transgenic mice expressing multiple spectral variants of GFP Neurotechnique 28 41-51
19. p Ark. Mat. 3 239-44
20. Hintermüller M and Kunisch K 2004 Total bounded variation regularization as a bilaterally constrained optimization problem SIAM J. Appl. Math. 64 1311-33
21. Kohsaka F and Takahashi W 2004 Strong convergence of an iterative sequence for maximal monotone operators in a Banach space Abstr. Appl. Anal. 2004 239-49
22. Kunisch K and Ring W 1993 Regularization of nonlinear illposed problems with closed operators Numer. Funct. Anal. Optim. 14 389-404
23. Lindenstrauss J and Tzafriri L 1979 Classical Banach Spaces: II. Function Spaces (Berlin: Springer)
24. Lions P L and Mercier B 1979 Splitting algorithms for the sum of two nonlinear operators SIAM J. Numer. Anal. 16 964-79
25. p spaces, 1 < p ≤ 2 Illinois J. Math. 28 420-4
26. Meyer Y 1992 Wavelets and Operators (Cambridge Studies in Advanced Mathematics vol 37) (Cambridge: Cambridge University Press)
27. Ng M K, Qi L, Yang Y-F and Huang Y-M 2007 On semismooth Newton's methods for total variation minimization J. Math. Imaging Vis. 27 265-76
28. Schuster T, Maass P, Bonesky T, Kazimierski K S and Schöpfer F 2008 Minimization of Tikhonov functionals in Banach spaces Abstr. Appl. Anal. 2008 192679 (19 pp)
29. Showalter R E 1997 Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (Mathematical Surveys and Monographs vol 49) (Providence, RI: American Mathematical Society)
30. van Kempen G M P, van Vliet L J, Verveer P J and van der Voort H T M 1997 A quantitative comparison of image restoration methods for confocal microscopy J. Microsc. 185 354-65
31. Vese L A 2001 A study in the BV space of a denoising-deblurring variational problem Appl. Math. Optim. 44 131-61
32. Xu H-K 1991 Inequalities in Banach spaces with applications Nonlinear Anal. 16 1127-38
33. Xu Z B and Roach G F 1991 Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces J. Math. Anal. Appl. 157 189-210