Sobolev error estimates and a priori parameter selection for semi-discrete Tikhonov regularization

Journal of Inverse and Ill-Posed Problems

Volumn 17, Issue 9, 2009, Pages 845-869

  Krebs, J ^a   Louis, A K ^a   Wendland, H ^b  

^a SAARLAND UNIVERSITY   (Germany)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Generalized recovery; Ill posed problems; Learning theory; Noisy data; Regularization; Reproducing kernel Hilbert spaces

Indexed keywords

EID: 76249093421     PISSN: 09280219     EISSN: None     Source Type: Journal    
DOI: 10.1515/JIIP.2009.050     Document Type: Article

References (40)

1. R. A. Adams, Sobolev Spaces. Academic Press, New York, 1975.
2. R. Arcangéli, M. C. L. de Silanes, and J. J. Torrens, An extension of a bound for functions in Sobolev spaces, with applications to (m, s)-spline interpolation and smoothing. Numer. Math. 107 (2007), 181-211.
3. N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68 (1950), 337-404.
4. F. Bauer, S. Pereverzev, and L. Rosasco, On regularization algorithms in learning theory. J. Complexity 23 (2007), 52-72. (Pubitemid 46217137)
5. M. Bertero, C. De Mol, and E. Pike, Linear inverse problems with discrete data. I: General formulation and singular system analysis. Inverse Problems 1 (1985), 301-330.
6. -, Linear inverse problems with discrete data. II: Stability and regularization. Inverse Problems 4 (1988), 573-594.
7. S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York, 1994.
8. Z. Chen, Y. Xu, and H. Yang, Fast collocation methods for solving ill-posed integral equations of the first kind. Inverse Problems 24 (2008).
9. P. Craven and G. Wahba, Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31 (1979), 377-403.
10. F. Cucker and S. Smale, On the mathematical foundation of learning. Bull. Amer. Math. Soc. 39 (2002), 1-49.
11. E. De Vito, A. Caponnetto, and L. Rosasco, Discretization error analysis for Tikhonov regularization in learning theory. Analysis and Applications 4 (2006), 81-99.
12. E. De Vito, L. Rosasco, A. Caponnetto, U. D. Giovannini, and F. Odone, Learning from examples as an inverse problem. J. Machine Learning Research 6 (2005), 883-904.
13. G. E. Fasshauer, Solving partial differential equations by collocation with radial basis functions. In: Surface Fitting and Multiresolution Methods, A. L. Méhauté, C. Rabut, and L. L. Schumaker (Eds). Vanderbilt University Press, Nashville, 1997, 131-138.
14. C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions. Appl. Math. Comput. 93 (1998), 73-82. (Pubitemid 128682471)
15. P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems. SIAM J. Numer. Anal. 45 (2007), 1723-1741.
16. C. F. Lee, L. Ling, and R. Schaback, On convergent numerical algorithms for unsymmetric collocation. Advances in Comp. Math (to appear).
17. L. Ling, R. Opfer, and R. Schaback, Results on meshless collocation techniques. Engineering Analysis with Boundary Elements 30 (2006), 247-253.
18. A. K. Louis, Inverse und Schlecht Gestellte Probleme. B. G. Teubner, Stuttgart, 1989.
19. -, Approximate inverse for linear and some nonlinear problems. Inverse Problems 12 (1996), 175-190.
20. -, A unified approach to regularization methods for linear ill-posed problems. Inverse Problems 15 (1999), 489-498.
21. P. Mathe and S. V. Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data. Mathematics of Computation 75, 256 (2006), 1913-1929. (Pubitemid 44719738)
22. F. J. Narcowich, J. D. Ward, and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74 (2005), 643-763.
23. -, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx. 24 (2006), 175-186.
24. F. Natterer, The Mathematics of Computerized Tomography. SIAM, 2001.
25. R. Opfer, Multiscale kernels. Adv. Comput. Math. 25 (2006), 357-380. (Pubitemid 44394637)
26. -, Tight frame expansions of multiscale reproducing kernels in Sobolev spaces. Appl. Comput. Harmon. Anal. 20 (2006), 357-374.
27. S. V. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems. SIAM J. Numerical Analysis 43 (2005), 2060-2076. (Pubitemid 44598672)
28. R. Plato and G.Vainikko, On the regularization of projection methods for solving ill-posed problems. Numer. Math. 57 (1989), 63-79.
29. T. Poggio and S. Smale, The mathematics of learning: Dealing with data. Amer. Math. Soc. 50 (2003), 537-544.
30. M. Thamban Nair and S. V. Pereverzev, Regularized collocation method for Fredholm integral equations of the first kind. J. Complexity 23 (2007), 454-467. (Pubitemid 350122945)
31. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam, 1978.
32. V. Vapnik, Statistical Learning Theory. John Wiley and Sons, 1998.
33. G. Wahba, Smoothing noisy data by spline functions. Numer. Math. 24 (1975), 383-393.
34. -, Approximate solutions to linear operator equations when the data are noisy. SIAM J. Numer. Anal. 14 (1977), 651-667.
35. -, Spline Models for Observational Data. SIAM, Philadelphia, 1990.
36. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4 (1995), 389-396. (Pubitemid 126012128)
37. -, Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Theory 93 (1998), 258-272.
38. -, On the convergence of a general class of finite volume methods. SIAM J. Numer. Anal. 43 (2005), 987-1002.
39. -, Scattered Data Approximation. Cambridge University Press, Cambridge, 2005.
40. H. Wendland and C. Rieger, Approximate interpolation with applications to selecting smoothing parameters. Numer. Math. 101 (2005), 643-662.