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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

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 & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
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show all references

References:
[1]

SIAM Journal on Matrix Analysis and Applications, 24 (2002), 40-58. doi: 10.1137/S089547980138705X.  Google Scholar

[2]

IEEE Transactions on Medical Imaging, 22 (2003), 569-579. doi: 10.1109/TMI.2003.812253.  Google Scholar

[3]

Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1016/0161-7346(84)90008-7.  Google Scholar

[4]

La Ricerca Scientifica, series II, Anno IX, XVI (1938), 326-333. Google Scholar

[5]

Parallel Computing, 27 (2001), 777-808. doi: 10.1016/S0167-8191(00)00100-9.  Google Scholar

[6]

American Journal of Mathematics, 73 (1951), 615-624. doi: 10.2307/2372313.  Google Scholar

[7]

Linear Algebra and its Applications, 236 (1996), 25-33. doi: 10.1016/0024-3795(94)00114-6.  Google Scholar

[8]

SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.  Google Scholar

[9]

Physics in Medicine and Biology, 49 (2004), 3227-3245. doi: 10.1088/0031-9155/49/14/015.  Google Scholar

[10]

European Journal of Nuclear Medicine and Molecular Imaging, 29 (2002), 691-698. doi: 10.1007/s00259-002-0903-5.  Google Scholar

[11]

Measurement Science & Technology, 14 (2003), R1-R13. doi: 10.1088/0957-0233/14/1/201.  Google Scholar

[12]

IEEE Transactions on Image Processing, 16 (2007), 1-6. doi: 10.1109/TIP.2006.887725.  Google Scholar

[13]

IEEE Transactions on Image Processing, 18 (2009), 435-440. doi: 10.1109/TIP.2008.2008076.  Google Scholar

[14]

Communications in Numerical Methods in Engineering, 25 (2009), 771-786. doi: 10.1002/cnm.1196.  Google Scholar

[15]

Annals of Statistics, 33 (2005), 1538-1579. doi: 10.1214/009053605000000255.  Google Scholar

[16]

Inverse Problems, 23 (2007), 1417-1432. doi: 10.1088/0266-5611/23/4/004.  Google Scholar

[17]

Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996.  Google Scholar

[18]

Numerical Functional Analysis and Optimization, 31 (2010), 1158-1184.  Google Scholar

[19]

World Scientific, Singapore, (2005), 351-382. Google Scholar

[20]

Informatica International, Inc., 1993. Google Scholar

[21]

Proceedings IEEE Engineering in Medicine and Biology Society, 1 (2006), 4815-4818. Google Scholar

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