The residual method for regularizing ill-posed problems

Applied Mathematics and Computation

Volumn 218, Issue 6, 2011, Pages 2693-2710

  Grasmair, Markus ^a   Haltmeier, Markus ^b   Scherzer, Otmar ^a,c  

^a UNIVERSITY OF VIENNA   (Austria)

^b MAX PLANCK INSTITUTE FOR BIOPHYSICAL CHEMISTRY   (Germany)

^c JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

Convergence rates; Ill posed problems; Regularization; Residual method; Sparsity; Stability

Indexed keywords

CONVERGENCE RATES; ILL POSED PROBLEM; REGULARIZATION; RESIDUAL METHOD; SPARSITY;

BANACH SPACES; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL OPERATORS;

PROBLEM SOLVING;

EID: 80053233830     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2011.08.009     Document Type: Article

References (61)

1. R. Acar, and C.R. Vogel Analysis of bounded variation penalty methods for ill-posed problems Inverse Prob. 10 6 1994 1217 1229
2. T. Bonesky, K.S. Kazimierski, P. Maass, F. Schöpfer, and T. Schuster Minimization of Tikhonov functionals in Banach spaces Abstr. Appl. Anal. 19 2008 Art. ID 192679
3. R.I. Bo, and B. Hofmann An extension of the variational approach for obtaining convergence rates in regularization of nonlinear ill-posed problems J. Integral Equ. Appl. 22 2010 369 392
4. A. Braides Γ-convergence for beginners Oxford Lecture Series in Mathematics and its Applications vol. 22 2002 Oxford University Press Oxford
5. K. Bredies, and D. Lorenz Regularization with non-convex separable constraints Inverse Prob. 25 8 2009 085011 (14p)
6. M. Burger, and S. Osher Convergence rates of convex variational regularization Inverse Prob. 20 5 2004 1411 1421
7. E.J. Cands The restricted isometry property and its implications for compressed sensing C.R. Math. Acad. Sci. Paris 346 9-10 2008 589 592
8. E.J. Cands, J. Romberg, and T. Tao Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information IEEE Trans. Inf. Theory 52 2 2006 489 509 (Pubitemid 43174093)
9. E.J. Cands, J.K. Romberg, and T. Taom Stable signal recovery from incomplete and inaccurate measurements Commun. Pure Appl. Math. 59 8 2006 1207 1223 (Pubitemid 43988295)
10. E.J. Cands, and T. Tao Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52 12 2006
11. R. Chartrand Exact reconstructions of sparse signals via nonconvex minimization IEEE Signal Process. Lett. 14 2007 707 710 (Pubitemid 47434090)
12. C. Chaux, P.L. Combettes, J.-C. Pesquet, and V.R. Wajs A variational formulation for frame-based inverse problems Inverse Prob. 23 4 2007 1495 1518 (Pubitemid 47411222)
13. I. Cioranescu, and Geometry of Banach Spaces Duality mappings and nonlinear problems Mathematics and its Applications vol. 62 1990 Kluwer Dordrecht
14. I. Daubechies, M. Defrise, and C. De Mol An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Commun. Pure Appl. Math. 57 11 2004 1413 1457 (Pubitemid 39427442)
15. p sparse recovery can fail for 0 < p ≤ 1 IEEE Trans. Inf. Theory 55 5 2009 2203 2214
16. 1-norm near-solution approximates the sparsest near-solution Commun. Pure Appl. Math. 59 7 2006 907 934
17. 1-norm solution is also the sparsest solution Commun. Pure Appl. Math. 59 6 2006 797 829 (Pubitemid 43667226)
18. D.L. Donoho, M. Elad, and V.N. Temlyakov Stable recovery of sparse overcomplete representations in the presence of noise IEEE Trans. Inf. Theory 52 1 2006 6 18 (Pubitemid 43263116)
19. I. Ekeland, and R. Temam Convex Analysis and Variational Problems 1976 North-Holland Amsterdam
20. M. Elad Sparse and Redundant Representations 2010 Springer New York From theory to applications in signal and image processing, With a foreword by Alfred M. Bruckstein
21. H.W. Engl, M. Hanke, and A. Neubauer Regularization of inverse problems Mathematics and its Applications vol. 375 1996 Kluwer Academic Publishers Group Dordrecht
22. H.W. Engl, K. Kunisch, and A. Neubauer Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems Inverse Prob. 5 3 1989 523 540
23. J. Flemming, and B. Hofmann A new approach to source conditions in regularization with general residual term Numer. Funct. Anal. Optim. 31 3 2010 245 284
24. p spaces Springer Monographs in Mathematics 2007 Springer New York
25. q-minimization for 0 < q ≤ 1 Appl. Comput. Harmon. Anal. 26 3 2009 395 407
26. J.J. Fuchs Recovery of exact sparse representations in the presence of bounded noise IEEE Trans. Inf. Theory 51 10 2005 3601 3608 (Pubitemid 41448600)
27. C.R. Givens, and R.M. Shortt A class of Wasserstein metrics for probability distributions Michigan Math. J. 31 2 1984 231 240
28. q penalty term Inverse Prob. Imaging 3 3 2009 383 387
29. M. Grasmair Generalized Bregman distances and convergence rates for non-convex regularization methods Inverse Prob. 26 11 2010 115014
30. M. Grasmair Non-convex sparse regularisation J. Math. Anal. Appl. 365 1 2010 19 28
31. q penalty term Inverse Prob. 24 5 2008 055020
32. 1-regularization Commun. Pure Appl. Math. 64 2 2011 161 182
33. C.W. Groetsch The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind 1984 Pitman Boston
34. T. Hein Convergence rates for multi-parameter regularization in Banach spaces Int. J. Pure Appl. Math. 43 4 2008 593 614
35. T. Hein, and B. Hofmann Approximate source conditions for nonlinear ill-posed problems - chances and limitations Inverse Prob. 25 2009 035003
36. B. Hofmann, B. Kaltenbacher, C. Pöschl, and O. Scherzer A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators Inverse Prob. 23 3 2007 987 1010 (Pubitemid 46838837)
37. B. Hofmann, and M. Yamamoto Convergence rates for Tikhonov regularization based on range inclusions Inverse Prob. 21 3 2005 805 820 (Pubitemid 40666655)
38. V.K. Ivanov Ill-posed problems in topological spaces Sibirsk. Mat. Ž. 10 1969 1065 1074 In Russian
39. V.K. Ivanov, V.V. Vasin, and V.P. Tanana Theory of linear ill-posed problems and its applications second ed. Inverse and Ill-posed Problems Series 2002 VSP Utrecht Translated and revised from the 1978 Russian original
40. J.L. Kelley General Topology 1955 D. Van Nostrand Company Toronto-New York-London
41. K. Kuratowski Topology J. Jaworowski, Revised and Augmented New ed. Translated from the French vol. I 1966 Academic Press New York
42. J. Lindenstrauss, and L. Tzafriri Classical Banach spaces. II Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] vol. 97 1979 Springer Verlag Berlin Function spaces
43. D. Lorenz Convergence rates and source conditions for Tikhonov regularization with sparsity constraints J. Inverse Ill-Posed Prob. 16 5 2008 463 478
44. R.E. Megginson An Introduction to Banach Space Theory Graduate Texts in Mathematics vol. 183 1998 Springer Verlag New York
45. V.A. Morozov Regularization Methods for Ill-Posed Problems 1993 CRC Press Boca Raton
46. E. Resmerita Regularization of ill-posed problems in Banach spaces: convergence rates Inverse Prob. 21 4 2005 1303 1314 (Pubitemid 41090541)
47. E. Resmerita, and O. Scherzer Error estimates for non-quadratic regularization and the relation to enhancement Inverse Prob. 22 3 2006 801 814 (Pubitemid 43821177)
48. R. Saab, R. Chartrand, O. Yilmaz, Stable sparse approximations via nonconvex optimization, in: 33rd International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2008.
49. O. Scherzer, H.W. Engl, and K. Kunisch Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems SIAM J. Numer. Anal. 30 6 1993 1796 1838
50. O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen Variational methods in imaging Applied Mathematical Sciences vol. 167 2009 Springer New York
51. T.I. Seidman Convergent approximation methods for ill-posed problems. I. General theory Polish Academy of Sciences. Institute of Applied Cybernetics. Control Cybernet. 10 1-2 1981 31 49
52. T.I. Seidman, and C.R. Vogel Well posedness and convergence of some regularisation methods for non-linear ill posed problems Inverse Prob. 5 2 1989 227 238
53. I. Singer Abstract convex analysis Canadian Mathematical Society Series of Monographs and Advanced Texts 1997 John Wiley & Sons Inc., New York With a foreword by A.M. Rubinov, A Wiley-Interscience Publication
54. V.P. Tanana On a criterion for the convergence of the residual method Dokl. Akad. Nauk 343 1 1995 22 24
55. V.P. Tanana Methods for solution of nonlinear operator equations Inverse and Ill-posed Problems Series 1997 VSP Utrecht
56. A.N. Tikhonov, and V.Y. Arsenin Solutions of Ill-Posed Problems 1977 John Wiley & Sons Washington, DC
57. J.A. Tropp Just relax: convex programming methods for identifying sparse signals in noise IEEE Trans. Inf. Theory 52 3 2006 1030 1051 (Pubitemid 46444890)
58. V.V. Vasin Some tendencies in the Tikhonov regularization of ill-posed problems J. Inverse Ill-Posed Prob. 14 8 2006 813 840 (Pubitemid 44678803)
59. A. Visintin Strong convergence results related to strict convexity Comm. Partial Diff. Equ. 9 5 1984 439 466
60. Z.B. Xu, and G.F. Roach Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces J. Math. Anal. Appl. 157 1 1991 189 210
61. C.A. Zarzer On Tikhonov regularization with non-convex sparsity constraints Inverse Prob. 25 2009 025006