A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization

Numerische Mathematik

Volumn 75, Issue 4, 1997, Pages 501-522

  Rieder, Andreas ^a  

^a SAARLAND UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031542819     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002110050250     Document Type: Article

References (27)

1. Beylkin, G., Coifman, R., Rokhlin, V. (1991): Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math., 44, 141-183
2. Chui, C.K. (1992): An Introduction to Wavelets. Wavelet Analysis and its Applications. Academic Press, Boston
3. Chui, C.K., Quak, E. (1992): Wavelets on a bounded interval. In D. Braess. Schumaker, L.L., editors, Numerical Methods in Approximation Theory, Vol. 9, International Series of Numerical Mathematics, Vol. 105, pages 53-75. Birkhäuser Verlag, Basel
4. Chui, C.K., Wang, J.Z. (1992): On compactly supported spline wavelets. Trans. Amer. Math. Soc., 330, 903-915
5. Ciarlet, P.G. (1978): The Finite Element Methods for Elliptic Problems. Sudies in Mathematics and its Applications. North-Holland, Amsterdam
6. Cohen, A., Daubechies, I., Vial, P. (1993): Wavelets on the interva and fast wavelet transforms. App. and Comp. Harm. Anal., 1, 54-81
7. Dahmen, W., Prößdorf, S., Schneider, R. (1993): Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution. Adv. in Comp. Math., 1, 259-335
8. Daubechies, I. (1988): Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math., 41, 906-966
9. Daubechies, I. (1992): Ten Lectures on Wavelets. CBMS-NSF Series in Applied Mathematics. SIAM Publications, Philadelphia
10. Davis, P.J., Rabinowitz, P. (1975): Methods of Numerical Integration. Computer Science and Applied Mathematics. Academic Press, New York
11. Deuflhard, P., Hohmann, A. (1994): Numerical Analysis: A First Course in Scientific Computation. de Gruyter Textbook, de Gruyter, Berlin, New York
12. Dicken, V., Maaß, P. (1996): Wavelet-Galerkin methods for ill-posed problems. J. Inv. Ill-Posed Problems, to appear
13. Donoho, D.L. (1995): Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. App. and Comp. Harm. Anal., 2, 101-126
14. Groetsch, C.W. (1984): The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston
15. Hackbusch, W. (1994): Iterative solution of large sparse systems of equations. Applied Mathematical Sciences. Springer-Verlag, New York
16. Harten, A., Yad-Shalom, I. (1994): Fast multiresolution algorithms for matrix-vector multiplication. SIAM J. Numer. Anal., 31, 1191-1218
17. King, J.T. (1992): Multilevel algorithms for ill-posed problems. Numer. Math., 61, 311-334
18. Louis, A.K. (1989): Inverse und schlecht gestellte Probleme. Studienbücher Mathematik. B. G. Teubner, Stuttgart, Germany (English translation in preparation)
19. Natterer, F. (1977): Regularisierung schlecht gestellter Probleme durch Projektionsverfahren. Numer. Math., 28, 329-341
20. Oswald, P. (1994): Multilevel Finite Element Approximation: Theory and Applications. Teubner Skripten zur Numerik. B. G. Teubner, Stuttgart, Germany
21. Plato, R., Vainikko, G.M. (1990): On the regularization of projection methods for solving ill-posed problems. Numer. Math., 57, 63-79
22. Rüde, U. (1993): Mathematical and Computational Techniques for Multilevel Adaptive Methods. Frontiers in Applied Mathematics. SIAM, Philadelphia
23. Strang, G. (1989): Wavelets and dilation equations: A brief introduction. SIAM Review, 31, 614-627
24. Strang, G., Fix, G. (1973): A Fourier analysis of the finite element variational method. In Constructive Aspects of Functional Analysis, Rome. Edizioni Cremonese
25. Sweldens, W., Piessens, R. (1994): Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions. SIAM J. Numer. Anal., 31, 1240-1264
26. Xia, X.-G., Nashed, M.Z. (1994): The Backus-Gilbert method for signals in reproducing Hubert spaces and wavelet subspaces. Inverse Problems, 10, 785-804
27. Yserentant, H. (1993): Old and new convergence proofs for multigrid methods. In Acta Numerica, pages 285-326. Cambridge University Press, New York