The order of convergence for Landweber Scheme with \alpha,\beta-rule

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February 2012, 6(1): 133-146. doi: 10.3934/ipi.2012.6.133

The order of convergence for Landweber Scheme with $\alpha,\beta$-rule

Caifang Wang ^1, and Tie Zhou ^2,

1.	Department of Mathematics, Shanghai Maritime University, Shanghai 200135, China
2.	LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Received February 2011 Revised September 2011 Published February 2012

Abstract
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The Landweber scheme is widely used in various image reconstruction problems. In previous works, $\alpha,\beta$-rule is suggested to stop the Landweber iteration so as to get proper iteration results. The order of convergence of discrepancy principal (DP rule), which is a special case of $\alpha,\beta$-rule, with constant relaxation coefficient $\lambda$ satisfying $0<\lambda\sigma_1^2<1,~(\|A\|_{V,W}=\sigma_1>0)$ has been studied. A sufficient condition for convergence of Landweber scheme is that the value $\lambda_m\sigma_1^2$ should be lied in a closed interval, i.e. $0<\varepsilon\leq\lambda_m\sigma_1^2\leq2-\varepsilon$, $(0<\varepsilon<1)$. In this paper, we mainly investigate the order of convergence of the $\alpha,\beta$-rule with variable relaxation coefficient $\lambda_m$ satisfying $0 < \varepsilon\leq\lambda_m \sigma_1^2 \leq 2-\varepsilon$. According to the order of convergence, we can conclude that $\alpha,\beta$-rule is the optimal rule for the Landweber scheme.

Keywords: relaxation coefficient, stopping criterion, Landweber scheme, order of convergence..

Mathematics Subject Classification: Primary: 65F10; Secondary: 65G20, 65B9.

Citation: Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems and Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

References:

[1]	Y. Censor and T. Elfving, Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 40-58. doi: 10.1137/S089547980138705X.
[2]	M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction, IEEE Transactions on Medical Imaging, 22 (2003), 569-579. doi: 10.1109/TMI.2003.812253.
[3]	A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1016/0161-7346(84)90008-7.
[4]	G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, La Ricerca Scientifica, series II, Anno IX, XVI (1938), 326-333.
[5]	Y. Censor, D. Gordon and R. Gordon, Component avering: An efficient iterative parallel algorithm for large and sparse unstructured problems, Parallel Computing, 27 (2001), 777-808. doi: 10.1016/S0167-8191(00)00100-9.
[6]	L. Landweber, An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, 73 (1951), 615-624. doi: 10.2307/2372313.
[7]	R. J. Santos, Equivalence of regularization and truncated iteration for general ill-posed problems, Linear Algebra and its Applications, 236 (1996), 25-33. doi: 10.1016/0024-3795(94)00114-6.
[8]	F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
[9]	Y. Xiao, D. Michalski, Y. Censor and J. M. Galvin, Inherent smoothness of intensity patterns for intensity modulated radiation therapy generated by simultaneous projection algorithms, Physics in Medicine and Biology, 49 (2004), 3227-3245. doi: 10.1088/0031-9155/49/14/015.
[10]	P .D. Acton, S. R. Choi, K. Plossl and H. F. Kung, Quantification of dopamine transporters in the mouse brain using ultra-high resolution single-photon emission tomography, European Journal of Nuclear Medicine and Molecular Imaging, 29 (2002), 691-698. doi: 10.1007/s00259-002-0903-5.
[11]	W. Q. Yang and L. H. Peng, Image reconstruction algorithms for electrical capacitance tomography, Measurement Science & Technology, 14 (2003), R1-R13. doi: 10.1088/0957-0233/14/1/201.
[12]	J. Wang and Y. B. Zheng, On the convergence of generalized simultaneous iterative reconstruction algorithms, IEEE Transactions on Image Processing, 16 (2007), 1-6. doi: 10.1109/TIP.2006.887725.
[13]	G. Qu, C. Wang and M. Jiang, Necessary and sufficient convergence conditions for algebraic image reconstruction algorithms, IEEE Transactions on Image Processing, 18 (2009), 435-440. doi: 10.1109/TIP.2008.2008076.
[14]	G. Qu and M. Jiang, Landweber scheme for compact operator equation in Hilbert space and its applications, Communications in Numerical Methods in Engineering, 25 (2009), 771-786. doi: 10.1002/cnm.1196.
[15]	T. Zhang and B. Yu, Boosting with early stopping: Convergence and consistency, Annals of Statistics, 33 (2005), 1538-1579. doi: 10.1214/009053605000000255.
[16]	T. Elfving and T. Nikazad, Stopping rules for Landweber-type interation, Inverse Problems, 23 (2007), 1417-1432. doi: 10.1088/0266-5611/23/4/004.
[17]	A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996.
[18]	T. Hein and K. Kazimierski, Modifed Landweber iteration in Banach spaces-Convergence and convergence rates, Numerical Functional Analysis and Optimization, 31 (2010), 1158-1184.
[19]	M. Jiang, Iterative Algebraic Algorithms for Image Reconstruction, in " Medical Imaging Systems Technology," Vol. I, World Scientific, Singapore, (2005), 351-382.
[20]	C. W. Groetsch, "Inverse Problems in the Mathematical Sciences (Theory & Practice of Applied Geophysics)," Informatica International, Inc., 1993.
[21]	S. Aja-Fernández, R. Estépar, C. Alberola-López and C. Westin, Image quality assessment based on local variance, Proceedings IEEE Engineering in Medicine and Biology Society, 1 (2006), 4815-4818.

show all references

References:

[1]	Y. Censor and T. Elfving, Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 40-58. doi: 10.1137/S089547980138705X.
[2]	M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction, IEEE Transactions on Medical Imaging, 22 (2003), 569-579. doi: 10.1109/TMI.2003.812253.
[3]	A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1016/0161-7346(84)90008-7.
[4]	G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, La Ricerca Scientifica, series II, Anno IX, XVI (1938), 326-333.
[5]	Y. Censor, D. Gordon and R. Gordon, Component avering: An efficient iterative parallel algorithm for large and sparse unstructured problems, Parallel Computing, 27 (2001), 777-808. doi: 10.1016/S0167-8191(00)00100-9.
[6]	L. Landweber, An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, 73 (1951), 615-624. doi: 10.2307/2372313.
[7]	R. J. Santos, Equivalence of regularization and truncated iteration for general ill-posed problems, Linear Algebra and its Applications, 236 (1996), 25-33. doi: 10.1016/0024-3795(94)00114-6.
[8]	F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
[9]	Y. Xiao, D. Michalski, Y. Censor and J. M. Galvin, Inherent smoothness of intensity patterns for intensity modulated radiation therapy generated by simultaneous projection algorithms, Physics in Medicine and Biology, 49 (2004), 3227-3245. doi: 10.1088/0031-9155/49/14/015.
[10]	P .D. Acton, S. R. Choi, K. Plossl and H. F. Kung, Quantification of dopamine transporters in the mouse brain using ultra-high resolution single-photon emission tomography, European Journal of Nuclear Medicine and Molecular Imaging, 29 (2002), 691-698. doi: 10.1007/s00259-002-0903-5.
[11]	W. Q. Yang and L. H. Peng, Image reconstruction algorithms for electrical capacitance tomography, Measurement Science & Technology, 14 (2003), R1-R13. doi: 10.1088/0957-0233/14/1/201.
[12]	J. Wang and Y. B. Zheng, On the convergence of generalized simultaneous iterative reconstruction algorithms, IEEE Transactions on Image Processing, 16 (2007), 1-6. doi: 10.1109/TIP.2006.887725.
[13]	G. Qu, C. Wang and M. Jiang, Necessary and sufficient convergence conditions for algebraic image reconstruction algorithms, IEEE Transactions on Image Processing, 18 (2009), 435-440. doi: 10.1109/TIP.2008.2008076.
[14]	G. Qu and M. Jiang, Landweber scheme for compact operator equation in Hilbert space and its applications, Communications in Numerical Methods in Engineering, 25 (2009), 771-786. doi: 10.1002/cnm.1196.
[15]	T. Zhang and B. Yu, Boosting with early stopping: Convergence and consistency, Annals of Statistics, 33 (2005), 1538-1579. doi: 10.1214/009053605000000255.
[16]	T. Elfving and T. Nikazad, Stopping rules for Landweber-type interation, Inverse Problems, 23 (2007), 1417-1432. doi: 10.1088/0266-5611/23/4/004.
[17]	A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996.
[18]	T. Hein and K. Kazimierski, Modifed Landweber iteration in Banach spaces-Convergence and convergence rates, Numerical Functional Analysis and Optimization, 31 (2010), 1158-1184.
[19]	M. Jiang, Iterative Algebraic Algorithms for Image Reconstruction, in " Medical Imaging Systems Technology," Vol. I, World Scientific, Singapore, (2005), 351-382.
[20]	C. W. Groetsch, "Inverse Problems in the Mathematical Sciences (Theory & Practice of Applied Geophysics)," Informatica International, Inc., 1993.
[21]	S. Aja-Fernández, R. Estépar, C. Alberola-López and C. Westin, Image quality assessment based on local variance, Proceedings IEEE Engineering in Medicine and Biology Society, 1 (2006), 4815-4818.

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