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Journal of Electrical and Computer Engineering
Volume 2010 (2010), Article ID 930218, 16 pages
http://dx.doi.org/10.1155/2010/930218
Research Article

Smoothing and Regularization with Modified Sparse Approximate Inverses

Technische Universität München, Boltzmannstraße 3, 80748 Garching, Germany

Received 20 September 2010; Accepted 22 September 2010

Academic Editor: Owe Axelsson

Copyright © 2010 T. Huckle and M. Sedlacek. 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. S. Demko, W. F. Moss, and P. W. Smith, “On approximate-inverse preconditioners,” Mathematics of Computation, vol. 43, pp. 491–499, 1984. View at Google Scholar
  2. T. Huckle and A. Kallischko, “Frobenius norm minimization and probing for preconditioning,” International Journal of Computer Mathematics, vol. 84, no. 8, pp. 1225–1248, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  3. A. Kallischko, Modified sparse approximate inverses (MSPAI) for parallel preconditioning, Ph.D. thesis, Technische Universität München, 2008.
  4. M. W. Benson and P. O. Frederickson, “Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems,” Utilitas Mathematica, vol. 22, pp. 127–140, 1982. View at Google Scholar
  5. J. D. F. Cosgrove, J. C. Díaz, and A. Griewank, “Approximate inverse preconditionings for sparse linear systems,” International Journal of Computer Mathematics, vol. 44, pp. 91–110, 1992. View at Google Scholar
  6. M. J. Grote and T. Huckle, “Parallel preconditioning with sparse approximate inverses,” SIAM Journal of Scientific Computing, vol. 18, no. 3, pp. 838–853, 1997. View at Google Scholar
  7. R. M. Holland, A. J. Wathen, and G. J. Shaw, “Sparse approximate inverses and target matrices,” SIAM Journal of Scientific Computing, vol. 26, no. 3, pp. 1000–1011, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  8. O. Axelsson, Iterative Solution Methods, Cambridge University Press, New York, NY, USA, 1994.
  9. O. Axelsson and B. Polman, “A robust preconditioner based on algebraic substructuring and two-level grids,” in Robust Multigrid Methods, W. Hackbusch, Ed., vol. 23, pp. 1–26, Friedrich Vieweg & Sohn, 1988. View at Google Scholar
  10. T. F. C. Chan and T. P. Mathew, “The interface probing technique in domain decomposition,” SIAM Journal on Matrix Analysis and Applications, vol. 13, pp. 212–238, 1992. View at Google Scholar
  11. M. Benzi and M. Tůma, “Comparative study of sparse approximate inverse preconditioners,” Applied Numerical Mathematics, vol. 30, no. 2, pp. 305–340, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, Chelsea Publishing, NewYork, NY, USA, 1984.
  13. T. Huckle, “Compact fourier analysis for designing multigrid methods,” SIAM Journal on Scientific Computing, vol. 31, pp. 644–666, 2008. View at Google Scholar
  14. U. Trottenberg, C. W. Oosterlee, and A. Schüller, Multigrid, Academic Press, San Diego, Calif, USA, 2001.
  15. W.-P. Tang and W. L. Wan, “Sparse approximate inverse smoother for multigrid,” SIAM Journal on Matrix Analysis and Applications, vol. 21, no. 4, pp. 1236–1252, 2000. View at Google Scholar
  16. O. Bröker, M. J. Grote, C. Mayer, and A. Reusken, “Robust parallel smoothing for multigrid via sparse approximate inverses,” SIAM Journal of Scientific Computing, vol. 23, no. 4, pp. 1396–1417, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. N. Tikhonov, “Solution of incorrectly formulated problems and regularization method,” Soviet Mathematics. Doklady, vol. 4, pp. 1035–1038, 1963. View at Google Scholar
  18. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
  19. M. Hanke, “Iterative regularization techniques in image reconstruction,” in Proceedings of the Conference on Mathematical Methods in Inverse Problems for Partial Differential Equations, Mt. Holyoke, Mass, USA, 1998.
  20. M. Hanke, J. G. Nagy, and R. J. Plemmons, “Preconditioned iterative regularization for ill-posed problems,” in Numerical Linear Algebra and Scientific Computing, pp. 141–163, de Gruyter, Berlin, Germany, 1993. View at Google Scholar
  21. J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, “Iterative image restoration using approximate inverse preconditioning,” IEEE Transactions on Image Processing, vol. 5, no. 7, pp. 1151–1162, 1996. View at Google Scholar
  22. J. G. Nagy, “Accelerating convergence of iterative image restoration algorithms,” Tech. Rep. TR-2007-020, Emory University, Atlanta, Ga, USA, 2007, Proceedings of the Advanced Maui Opticaland Space Surveillance Technologies (AMOS) Conference. View at Google Scholar
  23. J. G. Nagy and D. P. O'Leary, “Restoring images degraded by spatially variant blur,” SIAM Journal of Scientific Computing, vol. 19, no. 4, pp. 1063–1082, 1998. View at Google Scholar
  24. P. C. Hansen, “Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems,” Numerical Algorithms, vol. 6, no. 1, pp. 1–35, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  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
  26. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM Review, vol. 34, pp. 561–580, 1992. View at Google Scholar