The Convergence Properties for Regularized Landweber Method
- 155 Downloads
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.
KeywordsRegularization Landweber Convergence properties Linear system
Mathematics Subject Classification65F22 65F10 47N10 49N45
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).
- 2.Byrne, C.L., Graham-Eagle, J.: A short proof of the convergence of the SART algorithm. Technical report, Department of Mathematical Sciences, University of Massachusetts, Lowell (2001)Google Scholar
- 9.Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications. Pure and Applied Mathematics. Wiley (1974). http://ci.nii.ac.jp/ncid/BA07787840
- 12.Cimmino, G.: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. La Ricerca Scientifica, Roma XVI, Series II, Anno IX 326–333 (1939)Google Scholar