# American Institute of Mathematical Sciences

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

## An efficient projection method for nonlinear inverse problems with sparsity constraints

 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.
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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