American Institute of Mathematical Sciences

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August  2016, 10(3): 689-709. doi: 10.3934/ipi.2016017

An efficient projection method for nonlinear inverse problems with sparsity constraints

Deren Han 1, , Zehui Jia 1, , Yongzhong Song 1, and  David Z. W. Wang 2,

1. 

School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210023, China, China, China
2. 

School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798 Singapore

Received  July 2015 Revised  March 2016 Published  August 2016

In this paper, we propose a modification of the accelerated projective steepest descent method for solving nonlinear inverse problems with an $\ell_1$ constraint on the variable, which was recently proposed by Teschke and Borries (2010 Inverse Problems 26 025007). In their method, there are some parameters need to be estimated, which is a difficult task for many applications. We overcome this difficulty by introducing a self-adaptive strategy in choosing the parameters. Theoretically, the convergence of their algorithm was guaranteed under the assumption that the underlying mapping $F$ is twice Fréchet differentiable together with some other conditions, while we can prove weak and strong convergence of the proposed algorithm under the condition that $F$ is Fréchet differentiable, which is a relatively weak condition. We also report some preliminary computational results and compare our algorithm with that of Teschke and Borries, which indicate that our method is efficient.
Keywords: weak and strong convergence., sparsity, self-adaptive, nonlinear inverse problems, Projection method.
Mathematics Subject Classification: Primary: 90C33, 94A08, 97M5.
Citation: Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
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show all references

References:
[1]

Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[2]

Comput. Optim. Appl., 42 (2009), 173-193. doi: 10.1007/s10589-007-9083-3.  Google Scholar

[3]

Sci. China Math., 56 (2013), 2179-2186. doi: 10.1007/s11425-013-4683-0.  Google Scholar

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Commun. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.  Google Scholar

[5]

J. Fourier Anal. Appl., 14 (2008), 764-792. doi: 10.1007/s00041-008-9039-8.  Google Scholar

[6]

Inverse Problems Imag., 1 (2007), 29-46. doi: 10.3934/ipi.2007.1.29.  Google Scholar

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Adv. Imag. Electron Phys., 150 (2008), 1-51. doi: 10.1016/S1076-5670(07)00001-8.  Google Scholar

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IEEE Trans. Signal Process., 56 (2008), 5874-5890. doi: 10.1109/TSP.2008.929872.  Google Scholar

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Springer, Netherlands, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

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Springer-Verlag, Berlin, 2003.  Google Scholar

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Inverse Problems, 23 (2007), 2505-2526. doi: 10.1088/0266-5611/23/6/014.  Google Scholar

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IEEE J. Sel. Top. Signal Process., 1 (2007), 586-597. doi: 10.1109/JSTSP.2007.910281.  Google Scholar

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SIAM J. Numer. Anal., 46 (2008), 577-613. doi: 10.1137/0606668909.  Google Scholar

[14]

J. Ind. Man. Optim., 7 (2011), 1013-1026. doi: 10.3934/jimo.2011.7.1013.  Google Scholar

[15]

J. Optim. Theory Appl., 112 (2002), 129-143. doi: 10.1023/A:1013048729944.  Google Scholar

[16]

Inverse Problems, 26 (2010), 055002 (17pp). doi: 10.1088/0266-5611/26/5/055002.  Google Scholar

[17]

Inverse Problems, 25 (2009), 065003 (19pp). doi: 10.1088/0266-5611/25/6/065003.  Google Scholar

[18]

Inverse Problems, 21 (2005), 1571-1592. doi: 10.1088/0266-5611/21/5/005.  Google Scholar

[19]

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[20]

Appl. Comput. Harmon. Anal., 22 (2007), 43-60. doi: 10.1016/j.acha.2006.05.003.  Google Scholar

[21]

Inverse Problems, 26 (2010), 025007 (23pp). doi: 10.1088/0266-5611/26/2/025007.  Google Scholar

[22]

SIAM J. Sci. Comput., 31 (2008), 890-912. doi: 10.1137/080714488.  Google Scholar

[23]

Springer, New York, 1985. doi: 10.1007/978-1-4612-5020-3.  Google Scholar

[24]

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Mathematical Inequalities and Applications, 7 (2004), 453-456. doi: 10.7153/mia-07-45.  Google Scholar

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