Wavelet multilevel solvers for linear Ill-posed problems stabilized by Tikhonov regularization

Wavelet Analysis and Its Applications

Volumn 6, Issue C, 1997, Pages 347-380

  Rieder, Andreas ^a  

^a SAARLAND UNIVERSITY   (Germany)

Author keywords

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Indexed keywords

EID: 77957173388     PISSN: 1874608X     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-608X(97)80010-6     Document Type: Chapter

References (35)

1. Beylkin G., Coifman R., and Rokhlin V. Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math. 44 (1991) 141-183
2. Chui C.K. An Introduction to Wavelets (1992), Academic Press, Oxford
3. Chui C.K., and Quak E. Wavelets on a bounded interval. In: Braess D., and Schumaker L.L. (Eds). Numerical Methods in Approximation Theory vol. 9 (1992), Birkhäuser, Boston, MA 53-75
4. Chui C.K., and Wang J.Z. On compactly supported spline wavelets. Trans. Amer. Math. Soc. 330 (1992) 903-915
5. Ciarlet P.G. The Finite Element Method for Elliptic Problems (1978), North-Holland, Basel
6. Cohen A., Daubechies I., and Vial P. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 (1993) 54-81
7. Dahmen W., Prößdorf S., and Schneider R. Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution. Adv. Comput. Math. 1 (1993) 259-335
8. Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 906-966
9. Daubechies I. Ten Lectures on Wavelets (1992), SIAM, Amsterdam
10. Davis P.J., and Rabinowitz P. Methods of Numerical Integration (1975), Academic Press, Philadelphia, PA
11. Deuflhard P., and Hohmann A. Numerical Analysis: A First Course in 1cientific Computation (1994), de Gruyter, New York
12. to appear. Dicken V., and Maaß P. Wavelet-Galerkin methods for ill-posed problems. J. Inverse Ill-Posed Probl. (1996)
13. Donoho D.L. Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (1995) 101-126
14. Engl H.W. Regularization methods for the stable solution of inverse problems. Surveys Math. Indust. 3 (1993) 71-143
15. Griebel M., and Oswald P. On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70 (1995) 163-180
16. Groetsch C.W. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (1984), Pitman, New York
17. Hackbusch W. Iterative Solution of Large Sparse Systems of Equations (1994), Springer, Boston
18. Harten A., and Yad-Shalom I. Fast multiresolution algorithms for matrix-vector multiplication. SIAM J. Numer. Anal. 31 (1994) 1191-1218
19. King J.T. Multilevel algorithms for ill-posed problems. Numer. Math. 61 (1992) 311-334
20. King J.T., and Neubauer A. A variant of finite-dimensional Tikhonov regularization with a- posteriori parameter choice. Computing 40 (1988) 91-109
21. Kirsch A., Kress R., Monk P., and Zinn A. Two methods for solving the inverse acoustic scattering problem. Inverse Problems 4 (1988) 749-770
22. Kremer J., and Louis A.K. On the mathematical foundations of hyperthermia therapy. Math. Methods Appl. Sci. 13 (1990) 467-479
23. Liu J. A multiresolution method for distributed parameter estimation. SIAM J. Sci. Comput. 14 (1993) 389-405
24. English translation in preparation. Louis A.K. Inverse und schlecht gestellte Probleme (1989), Teubner, New York
25. English version: Wiley, Chichester 1997, to appear. Louis A.K., Maaß P., and Rieder A. Wavelets: Theorie und Anwendungen (1994), Teubner, Stuttgart
26. Oswald P. Multilevel Finite Element Approximation: Theory and Applications. Teubner Skr. Numer. (1994), Teubner, Stuttgart
27. Plato R., and Vainikko G. On the regularization of projection methods for solving ill-posed problems. Numer. Math. 57 (1990) 63-79
28. Quak E., and Weyrich N. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval. Appl. Comput. Harmon. Anal. 1 (1994) 217-231
29. to appear. Rieder A. A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization. Numer. Math. (1996)
30. Schumaker L.L. Spline Functions: Basic Theory (1981), Wiley, Stuttgart
31. Strang G., and Fix G.J. An Analysis of the Finite Element Method (1973), Prentice-Hall, New York
32. Sweldens W., and Piessens R. Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions. SIAM J. Numer. Anal. 31 (1994) 1240-1264
33. Xia X.G., and Nashed M.Z. The Backus-Gilbert method for signals in reproducing Hilbert spaces and wavelet subspaces. Inverse Problems 10 (1994) 785-804
34. Xu J. Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581-613
35. Yserentant H. Old and new convergence proofs for multigrid methods. Acta Numerica (1993), Cambridge Univ. Press, Englewood Cliffs 285-326