Controlling semi-convergence phenomenon in non-stationary simultaneous iterative methods

Document Type : Research Article


School of Mathematics, Iran University of Science and Technology, Tehran , Iran.


When applying the non-stationary simultaneous iterative methods for solving an ill-posed set of linear equations, the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence behavior of the non-stationary simultaneous iterative methods and obtain an upper bound for data error (noise error). Based on this bound, we propose new ways to specify the relaxation parameters to control the semi-convergence. The performance of our strategies is shown by examples taken from tomographic imaging.


1. Bertero, M. and Boccacci, P. Introduction to inverse problems in imaging. CRC press, 1998.
2. Brianzi, P., Benedetto, F.D. and Estatico, C. Improvement of spaceinvariant image deblurring by preconditioned landweber iterations. SIAM Journal on Scientific Computing, 30(3):1430-1458, 2008.
3. Censor, Y. and Elfving, T. Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem. SIAM Journal on Matrix Analysis and Applications, 24(1):40-58, 2002.
4. Censor, Y., Gordon, D. and Gordon, R. Component averaging: An effcient iterative parallel algorithm for large and sparse unstructured problems. Parallel computing, 27(6):777-808, 2001.
5. Cimmino, G. and Ricerche, C.N.D. Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. Istituto per le applicazioni del calcolo, 1938.
6. R. Davidi, G. T. Herman, and J. Klukowska. Snark09: A programming system for the reconstruction of 2d images from 1d projections. The CUNY Institute for Software Design and Development, New York, 2009.
7. Tommy Elfving, Per Christian Hansen, and Touraj Nikazad. Semiconvergence and relaxation parameters for projected SIRT algorithms. SIAM Journal on Scientific Computing, 34(4):A2000-A2017, 2012.
8. Elfving, T., Nikazad, T. and Hansen, P.C. Semiconvergence and relaxation parameters for a class of SIRT algorithms, Electronic Transactions on Numerical Analysis, 37:321-336, 2010.
9. Engl, H.W., Hanke, M. and Neubauer, A. Regularization of inverse problems, volume 375. Springer Science & Business Media, 1996.
10. Guy C. and Ffytche, D. An introduction to the principles of medical imaging, World Scientific, 2005.
11. Hansen, P.C. Rank-de_cient and discrete ill-posed problems: numerical aspects of linear inversion, volume 4. Siam, 1998.
12. Hansen, P.C. and Hansen, M.S.AIR tools-a MATLAB package of algebraic iterative reconstruction methods, Journal of Computational and Applied Mathematics, 236(8):2167-2178, 2012.
13. Herman, G.T. Fundamentals of computerized tomography: image reconstruction from projections, Springer Science & Business Media, 2009.
14. Jiang, M. and Wang, G. Convergence studies on iterative algorithms for image reconstruction, Medical Imaging, IEEE Transactions on, 22(5):569-579, 2003.
15. Kaczmarz, S. Angenherte ausung von systemen linearer gleichungen. Bulletin International de lAcademie Polonaise des Sciences et des Lettres, 35:355-357, 1937.
16. Kak, A.C. and Slaney, M. Principles of computerized tomographic imaging, volume 33. Siam, 1988.
17. Landweber, L. An iteration formula for Fredholm integral equations of the first kind, American journal of mathematics, pages 615-624, 1951.
18. Natterer, F. The mathematics of computerized tomography, John Wiley, New York, 1986.
19. Piana, M. and Bertero, M. Projected landweber method and preconditioning, Inverse Problems, 13(2):441, 1997.
20. Radon, J. ber die bestimmung von funktionen durch ihre integralwerte lngs gewisser mannigfaltigkeiten, Classic papers in modern diagnostic radiology, 5, 2005.
21. Sluis, A. and Vorst, H. Sirt-and cg-type methods for the iterative solution of sparse linear least-squares problems, Linear Algebra and its Applications, 130:257-303, 1990.
22. Webb, S. From the watching of shadows: the origins of radiological tomography, CRC Press, 1990.