Wavelet-based multilevel methods for linear ill-posed problems

BIT Numerical Mathematics

Volumn 51, Issue 3, 2011, Pages 669-694

  Klann, E ^a   Ramlau, R ^b   Reichel, L ^c  

^a JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^b JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^c KENT STATE UNIVERSITY   (United States)

Author keywords

Ill posed problem; Minimal residual method; Multilevel method; Wavelet

Indexed keywords

EID: 80052274973     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-011-0320-x     Document Type: Article

References (30)

1. Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996).
2. Bornemann, F. A., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Numer. Math. 75, 135-152 (1996).
3. Calvetti, D., Lewis, B., Reichel, L.: On the choice of subspace for iterative methods for linear discrete ill-posed problems. Int. J. Appl. Math. Comput. Sci. 11, 1069-1092 (2001).
4. Chen, Z., Xu, Y., Yang, H.: A multilevel augmentation method for solving ill-posed operator equations. Inverse Probl. 22, 155-174 (2006).
5. Cohen, A.: Numerical Analysis of Wavelet Methods. North-Holland, Amsterdam (2003).
6. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992).
7. Donatelli, M., Serra-Capizzano, S.: On the regularization power of multigrid-type algorithms. SIAM J. Sci. Comput. 27, 2053-2076 (2006).
8. Donatelli, M., Serra-Capizzano, S.: Filter factor analysis of an iterative multilevel regularizing method. Electron. Trans. Numer. Anal. 29, 163-177 (2008).
9. Engl, H. W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Dordrecht, Kluwer Academic (1996).
10. Español, M. I.: Multilevel methods for discrete ill-posed problems: application to deblurring, Ph. D. thesis, Department of Mathematics, Tufts University, Medford, MA (2009).
11. Groetsch, C. W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, London (1984).
12. Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Longman, Essex (1995).
13. Hanke, M., Vogel, C. R.: Two-level preconditioners for regularized inverse problems I: Theory. Numer. Math. 83, 385-402 (1999).
14. Huckle, T., Staudacher, J.: Multigrid preconditioning and Toeplitz matrices. Electron. Trans. Numer. Anal. 13, 81-105 (2002).
15. Jacobsen, M., Hansen, P. C., Saunders, M. A.: Subspace preconditioned LSQR for discrete ill-posed problems. BIT Numer. Math. 43, 975-989 (2003).
16. Justen, L., Ramlau, R.: A general framework for soft shrinkage with applications to blind deconvolution and wavelet denoising. Appl. Comput. Harmon. Anal. 26, 43-63 (2009).
17. King, J. T.: Multilevel algorithms for ill-posed problems. Numer. Math. 61, 311-334 (1992).
18. Louis, A. K., Maass, P., Rieder, A.: Wavelets: Theory and Applications. Wiley, Chichester (1997).
19. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1998).
20. Meyer, F. G.: Ondelettes sur l'Intervalle. Rev. Mat. Iberoam. 7, 115-133 (1992).
21. Morigi, S., Reichel, L., Sgallari, F.: Noise-reducing cascadic multilevel methods for linear discrete ill-posed problems. Numer. Algorithms 53, 1-22 (2010).
22. Morigi, S., Reichel, L., Sgallari, F.: Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal. Appl. Numer. Math. 60, 378-396 (2010).
23. Morigi, S., Reichel, L., Sgallari, F., Shyshkov, A.: Cascadic multiresolution methods for image deblurring. SIAM J. Imaging Sci. 1, 51-74 (2008).
24. Natterer, F.: The Mathematics of Computerized Tomography. Teubner, Stuttgart (1986).
25. Nemirovskii, A. S.: The regularization properties of the adjoint gradient method in ill-posed problems. U. S. S. R. Comput. Math. Math. Phys. 26(2), 7-16 (1986).
26. Plato, R.: The method of conjugate residuals for solving the Galerkin equations associated with symmetric positive semidefinite ill-posed problems. SIAM J. Numer. Anal. 35, 1621-1645 (1998).
27. Reichel, L., Shyshkov, A.: Cascadic multilevel methods for ill-posed problems. J. Comput. Appl. Math. 233, 1314-1325 (2010).
28. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003).
29. Scherzer, O.: An iterative multi level algorithm for solving nonlinear ill-posed problems. Numer. Math. 80, 579-600 (1998).
30. Strang, G., Nguyen, T.: Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley (1996).