Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2017, Article ID 1375716, 8 pages
https://doi.org/10.1155/2017/1375716
Research Article

Vector Extrapolation Based Landweber Method for Discrete Ill-Posed Problems

1School of Mathematical Sciences/Research Center for Image and Vision Computing, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China
3Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, Netherlands

Correspondence should be addressed to Xian-Ming Gu; nc.evil@gnimnaixug

Received 10 June 2017; Revised 12 September 2017; Accepted 18 September 2017; Published 16 November 2017

Academic Editor: Qingling Zhang

Copyright © 2017 Xi-Le Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, CRC Press, Boca Raton, FL, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  2. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications, Springer Netherlands, Dordrecht, The Netherlands, 1996. View at MathSciNet
  3. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM Monographs on Mathematical Modeling and Computation, SIAM, Philadelphia, PA, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J-J. Mei, Y. Dong, T-Z. Huang, and W. Yin, “Cauchy Noise Removal by Nonconvex ADMM with Convergence Guarantees,” Journal of Scientific Computing, pp. 1–24, 2017. View at Publisher · View at Google Scholar
  5. X.-L. Zhao, F. Wang, T.-Z. Huang, M. K. Ng, and R. J. Plemmons, “Deblurring and sparse unmixing for hyperspectral images,” IEEE Transactions on Geoscience and Remote Sensing, vol. 51, no. 7, pp. 4045–4058, 2013. View at Publisher · View at Google Scholar · View at Scopus
  6. X.-L. Zhao, F. Wang, and M. K. Ng, “A new convex optimization model for multiplicative noise and blur removal,” SIAM Journal on Imaging Sciences, vol. 7, no. 1, pp. 456–475, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. P.-C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, PA, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Å. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  9. C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Transactions on Mathematical Software, vol. 8, no. 1, pp. 43–71, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  10. D. C. Fong and M. Saunders, “LSMR: an iterative algorithm for sparse least-squares problems,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2950–2971, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. P. Brianzi, P. Favati, O. Menchi, and F. Romani, “A framework for studying the regularizing properties of Krylov subspace methods,” Inverse Problems, vol. 22, no. 3, pp. 1007–1021, 2006. View at Publisher · View at Google Scholar · View at Scopus
  12. C. Vonesch and M. Unser, “A fast thresholded Landweber algorithm for wavelet-regularized multidimensional deconvolution,” IEEE Transactions on Image Processing, vol. 17, no. 4, pp. 539–549, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. H. Zhang and L. Z. Cheng, “Projected Landweber iteration for matrix completion,” Journal of Computational and Applied Mathematics, vol. 235, no. 3, pp. 593–601, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. L. Liang and Y. Xu, “Adaptive landweber method to deblur images,” IEEE Signal Processing Letters, vol. 10, no. 5, pp. 129–132, 2003. View at Publisher · View at Google Scholar · View at Scopus
  15. X.-M. Gu, T.-Z. Huang, H.-B. Li, S.-F. Wang, and L. Li, “Two CSCS-based iteration methods for solving absolute value equations,” Journal of Applied Analysis and Computation, vol. 7, no. 4, pp. 1336–1356, 2017. View at Google Scholar
  16. K. Jbilou and H. Sadok, “Vector extrapolation methods. Applications and numerical comparison,” Journal of Computational and Applied Mathematics, vol. 122, no. 1-2, pp. 149–165, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. A. Sidi, “Vector extrapolation methods with applications to solution of large systems of equations and to PageRank computations,” Computers & Mathematics with Applications, vol. 56, no. 1, pp. 1–24, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  18. D. A. Smith, W. F. Ford, and A. Sidi, “Extrapolation methods for vector sequences,” SIAM Review, vol. 29, no. 2, pp. 199–233, 1987. View at Publisher · View at Google Scholar · View at Scopus
  19. H.-F. Zhang, T.-Z. Huang, C. Wen, and Z.-L. Shen, “FOM accelerated by an extrapolation method for solving PageRank problems,” Journal of Computational and Applied Mathematics, vol. 296, pp. 397–409, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  20. K. Jbilou and H. Sadok, “LU implementation of the modified minimal polynomial extrapolation method for solving linear and nonlinear systems,” IMA Journal of Numerical Analysis, vol. 19, no. 4, pp. 549–561, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  21. A. Sidi, “Efficient implementation of minimal polynomial and reduced rank extrapolation methods,” Journal of Computational and Applied Mathematics, vol. 36, no. 3, pp. 305–337, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. A. Sidi, “Convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms,” SIAM Journal on Numerical Analysis, vol. 23, no. 1, pp. 197–209, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. M. Donatelli and S. Serra-Capizzano, “Anti-reflective boundary conditions and re-blurring,” Inverse Problems, vol. 21, no. 1, pp. 169–182, 2005. View at Publisher · View at Google Scholar · View at Scopus
  24. T. Elfving and T. Nikazad, “Stopping rules for Landweber-type iteration,” Inverse Problems, vol. 23, no. 4, article no. 004, pp. 1417–1432, 2007. View at Publisher · View at Google Scholar · View at Scopus
  25. J. G. Nagy, K. Palmer, and L. Perrone, “Iterative methods for image deblurring: a Matlab object-oriented approach,” Numerical Algorithms, vol. 36, no. 1, pp. 73–93, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus