Kaczmarz methods for regularizing nonlinear ill-posed equations I: Convergence analysis

Inverse Problems and Imaging

Volumn 1, Issue 2, 2007, Pages 289-298

  Haltmeier, Markus ^a   Leitão, Antonio ^b   Scherzer, Otmar ^a,c  

^a UNIVERSITY OF INNSBRUCK   (Austria)

^b FEDERAL UNIVERSITY OF SANTA CATARINA   (Brazil)

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

Author keywords

Ill posed systems; Kaczmarz; Landweber; Regularization

Indexed keywords

EID: 85044958349     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2007.1.289     Document Type: Article

References (26)

1. A. B. Bakushinsky and M. Y. Kokurin, Iterative methods for approximate solution of inverse problems, Mathematics and Its Applications, vol. 577, Springer, Dordrecht, 2004.
2. L. Borcera, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136.
3. M. A. Breazeale, Schlieren photography in Physics, Proc. SPIE, 3581 (1998), 41–47.
4. M. Burger and B. Kaltenbacher, Regularizing newton-kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Anal., 44 (2006), 153–182.
5. P. Burgholzer, C. Hofer, G. Paltauf, M. Haltmeier, and O. Scherzer, Thermoacoustic tomography with integrating area and line detectors, IEEE Trans. Ultrason. Ferroelec. Freq. Contr., 52 (2005), 1577–1583.
6. H.W. Engl, M. Hanke, and A. Neubauer, “Regularization of Inverse Problems,” Kluwer Academic Publishers, Dordrecht, 1996.
7. C. L. Epstein, “Introduction to the Mathematics of Medical Imaging,” Pearson Prentice Hall, Upper Saddle River, NJ, 2003.
8. M. Haltmeier and T. Fidler, Mathematical challenges arising in thermoacoustic computed tomography with line detectors, arXiv:math.AP/0610155 (2006).
9. M. Hanke, Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim., 18 (1997), 971–993.
10. M. Hanke, A. Neubauer, and O. Scherzer, A convergence analysis of Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21–37.
11. S. Kaczmarz, Approximate solution of systems of linear equations, Internat. J. Control, 57 (1993), 1269–1271, Translated from the German.
12. B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative regularization methods for nonlinear ill–posed problems, 2006, submitted.
13. R. Kowar and O. Scherzer, Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems, Ill posed and inverse problems (book series), 23 (2002), 69–90.
14. R.A Kruger, W.L. Kiser, D.R. Reinecke, G.A. Kruger, and K.D. Miller, Thermoacoustic molecular imaging of small animals, Molecular Imaging, 2 (2003), 113–123.
15. L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math., 73 (1951), 615–624.
16. E. G. LeDet and C. I. Zanelli, A novel, rapid method to measure the effective aperture of array elements, IEEE Ultrasonics Symposium, (1999).
17. A. Leitao, P.A. Markowich, and J.P. Zubelli, Inverse problems for semiconductors: Models and methods, ch. in “Transport Phenomena and Kinetic Theory: Applications to Gases, Semiconductors, Photons, and Biological Systems” (Eds: C. Cercignani and E. Gabetta), Birkhäuser, Boston, 2006.
18. On inverse dopping profile problems for the stationary voltage-current map, Inv.Probl., 22 (2006), 1071–1088.
19. V.A. Morozov, “Regularization Methods for Ill–posed Problems,” CRC Press, Boca Raton, 1993.
20. F. Natterer, “The Mathematics of Computerized Tomography,” SIAM, Philadelphia, 2001.
21. Frank Natterer, Algorithms in tomography, State of the Art in Numerical Analysis, 63 (1997), 503–524.
22. Otmar Scherzer, A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 17 (1996), 197–214.
23. T.I. Seidman and C.R. Vogel, Well posedness and convergence of some regularisation methods for non–linear ill posed problems, Inverse Probl., 5 (1989), 227–238.
24. A. N. Tikhonov and V. Y. Arsenin, “Solutions of Ill-posed Problems,” John Wiley & Sons, Washington, D.C., 1977, Translation editor: Fritz John.
25. A.N. Tikhonov, Regularization of incorrectly posed problems, Soviet Math. Dokl., 4 (1963), 1624–1627.
26. M. Xu and L.V. Wang, Photoacoustic imaging in biomedicine, Review of Scientific Instruments, 77 (2006), 41–101.