A nonstandard approximation of pseudoinverse and a new stopping criterion for iterative regularization

Journal of Inverse and Ill-Posed Problems

Volumn 23, Issue 3, 2015, Pages 195-210

  Bakushinsky, Anatoly ^a   Smirnova, Alexandra ^b   Liu, Hui ^b  

^a INSTITUTE FOR SYSTEMS ANALYSIS   (Russian Federation)

^b GEORGIA STATE UNIVERSITY   (United States)

Author keywords

image restoration; inverse scattering problem; irregular operator equation; iterative regularization; stopping rule; TSVD

Indexed keywords

IMAGE RECONSTRUCTION; MATHEMATICAL OPERATORS;

INVERSE SCATTERING PROBLEMS; ITERATIVE REGULARIZATION; OPERATOR EQUATION; STOPPING RULE; TSVD;

INVERSE PROBLEMS;

EID: 84931068321     PISSN: 09280219     EISSN: None     Source Type: Journal    
DOI: 10.1515/jiip-2013-0086     Document Type: Article

References (40)

1. A. B. Bakushinsky, A general method of constructing regularizing algorithms for a linear incorrect equation in Hilbert space (in Russian), Zh. Vychisl. Mat. Mat. Fiz. 7 (1967), 672-677.
2. A. B. Bakushinsky, Iterative methods for nonlinear operator equations without regularity. New approach, Dokl. Russian Acad. Sci. 330 (1993), 282-284.
3. A. B. Bakushinsky, Iterative methods without saturation for solving degenerate nonlinear operator equations, Dokl. Russian Acad. Sci. 334 (1995), 7-8.
4. A. B. Bakushinsky, Iterative methods with fuzzy feedback for solving irregular operator equations, Dokl. Russian Acad. Sci. 428 (2009), no. 5, 1-3.
5. A. B. Bakushinsky and M. Y. Kokurin, Iterative Methods for Ill-Posed Operator Equations with Smooth Operators, Springer-Verlag, Dordrecht, 2004.
6. A. B. Bakushinsky, M. Y. Kokurin and A. Smirnova, Iterative Methods for Ill-Posed Problems. An Introduction, Inverse Ill-Posed Probl. Ser. 54, de Gruyter, Berlin, 2010.
7. A. B. Bakushinsky and A. Smirnova, On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems, Numer. Funct. Anal. Optim. 26 (2005), no. 1, 35-48.
8. A. B. Bakushinsky and A. Smirnova, Discrepancy principal for generalized GN iterations combined with the reverse connection control, J. Inverse Ill-Posed Probl. 18 (2010), no. 4, 421-432.
9. A. B. Bakushinsky and A. Smirnova, Irregular operator equations by iterative methods with undetermined reverse connection, J. Inverse Ill-Posed Probl. 18 (2010), no. 2, 147-165.
10. F. Bauer, T. Hohage and A. Munk, Iteratively regularized Gauss-Newton method for nonlinear inverse problems with random noise, SIAM J. Numer. Anal. 47 (2009), no. 3, 1827-1846.
11. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, London, 1998.
12. B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal. 17 (1997), 421-436.
13. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed., Appl. Math. Sci. 93, Springer-Verlag, New York, 2013.
14. H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, Dordecht, 1996.
15. O. P. Ferreira, M. L. N. Gonçalves and P. R. Oliveira, Local convergence analysis of the Gauss-Newton method under a majorant condition, J. Complexity 27 (2011), no. 1, 111-125.
16. J. Flemming and B. Hofmann, A new approach to source conditions in regularization with general residual term, Numer. Funct. Anal. Optim. 31 (2010), no. 1-3, 254-284.
17. J. Flemming and B. Hofmann, Convergence rates in constrained Tikhonov regularization: Equivalence of projected source conditions and variational inequalities, Inverse Problems 27 (2011), no. 8, 085001.
18. M. Hanke, Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim. 18 (1997), no. 9-10, 971-993.
19. T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and inverse scattering problem, Inverse Problems 13 (1997), 1279-1299.
20. T. Hohage and S. Langer, Acceleration techniques for regularized Newton methods applied to electromagnetic inverse medium scattering problems, Inverse Problems 26 (2010), no. 7, 074011.
21. Q. Jin, A convergence analysis of the iteratively regularized Gauss-Newton method under the Lipschitz condition, Inverse Problems 24 (2008), no. 4, 045002.
22. Q. Jin, On a class of frozen regularized Gauss-Newton methods for nonlinear inverse problems, Math. Comp. 79 (2010), no. 272, 2191-2211.
23. Q. Jin and U. Tautenhahn, On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems, Numer. Math. 111 (2009), no. 4, 509-558.
24. B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces, Inverse Problems 26 (2010), no. 3, 035007.
25. B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Ser. Comput. Appl. Math. 6, de Gruyter, Berlin, 2008
26. D. Langemann and M. Tasche, Phase reconstruction by a multilevel iteratively regularized Gauss-Newton method, Inverse Problems 24 (2008), no. 3, 035006.
27. S. Langer and T. Hohage, Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions, J. Inverse Ill-Posed Probl. 15 (2007), no. 3, 311-327.
28. K.-M. Lee, Inverse scattering via nonlinear integral equations for a Neumann crack, Inverse Problems 22 (2006), no. 6, 1989-2000.
29. P. Mahale and M. T. Nair, A simplified generalized Gauss-Newton method for nonlinear ill-posed problems, Math. Comp. 78 (2009), no. 265, 171-184.
30. J. G. Nagy, RestoreTools: An object oriented Matlab package for image restoration, 2002, http://www.mathcs.emory.edu/~nagy/RestoreTools.
31. J. G. Nagy, K. Palmer and L. Perrone, Iterative methods for image deblurring: A Matlab object-oriented approach, Numer. Algorithms 36 (2004), 73-93.
32. A. Neubauer, T. Hein, B. Hofmann, S. Kindermann and U. Tautenhahn, Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces, Appl. Anal. 89 (2010), no. 11, 1729-1743.
33. H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems 27 (2011), no. 3, 035005.
34. A. B. Smirnova, On convergence rates for iteratively regularized procedures with linear penalty terms, Inverse Problems 28 (2012), no. 8, 085005.
35. A. Smirnova and R. A. Renaut, A family of preconditioned iteratively regularized methods for nonlinear minimization, J. Inverse Ill-Posed Probl. 17 (2009), no. 4, 405-418.
36. A. B. Smirnova, R. A. Renaut and T. Khan, Convergence and application of a modified iteratively regularized Gauss-Newton algorithm, Inverse Problems 23 (2007), no. 4, 1547-1563.
37. V. V. Vasin and A. L. Ageev, Ill-Posed Problems with a Priori Information, Inverse Ill-Posed Probl. Ser., VNU, Utrecht, 1995.
38. C. R. Vogel, Numerical solution of a nonlinear ill-posed problem arizing in inverse scattering, Inverse Problems 1 (1985), 393-403.
39. Chinese tea supplier, Test Image 2, http://www.chinese-tea-supplier.com/organic-tie-guan-yin-wu-long-high-quality- 285.html.
40. Tamborine Mountain Produce Market, Test Image 1, http://www.tmbb.com.au/tamborine-mountain-produce-market/.