Abstract
Landweber scheme is a widely used method to get a stable solution of linear system. The iteration of the Landweber scheme is viewed as a solution of normal equation for a least-squares functional. However, in practice, regularized least-squares functional is considered so as to get a more suitable solution. In this paper, we consider a regularized optimization problem and study the regularized Landweber scheme. Using the eigenvalue decomposition and the result that two symmetric semi-positive definite matrices can be diagonalized simultaneously, we derive a presentation of the regularized Landweber scheme and then generate the convergence properties for the regularized Landweber iteration. Finally, we apply two-dimensional numerical examples to confirm the convergence conditions.
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Acknowledgments
The author thanks the referee for careful reading and the valuable comments. Caifang Wang was partially supported by National Natural Science Foundation of China (11401372) and a grant from Shanghai Municipal Commission for Science and Technology (13ZR1455500).
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Communicated by Ilio Galligani.
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Wang, C. The Convergence Properties for Regularized Landweber Method. J Optim Theory Appl 171, 262–275 (2016). https://doi.org/10.1007/s10957-016-0961-7
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DOI: https://doi.org/10.1007/s10957-016-0961-7