Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-31T16:41:33.096Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimisation in the regularisation ill-posed problems

Published online by Cambridge University Press:  17 February 2009

A. R. Davies  and
R. S. Anderssen
A. R. Davies
Affiliation:
Department of Applied Mathematics, University College of Wales, Aberystyth SY23 3BZ, United Kingdom.
R. S. Anderssen
Affiliation:
Division of Mathematics and Statistics, C.S.I.R. O., P. O. Box 1965, Canberra City, A.C.T. 2601, Australia.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We survey the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations. Asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem. These methods include the discrepancy principle, cross-validation and maximum likelihood. The relationship between optimality and regularity is emphasized. New bounds on the constants appearing in asymptotic estimates are presented.

Type
Research Article
Information
The ANZIAM Journal , Volume 28 , Issue 1 , July 1986 , pp. 114 - 133
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Anderssen, R. S. and Bloomfield, P., “Numerical differentiation procedures for non-exact data”, Numer. Math. 22 (1974), 157–182.CrossRefGoogle Scholar
[2]Anderssen, R. S. and Prenter, P. M., “A formal comparison of methods proposed for the numerical solution of first kind integral equations”, J. Austral. Math. Soc. Ser. B 22 (1981), 488–500.CrossRefGoogle Scholar
[3]Baart, M. L., “Computation experience with the spectral smoothing method for differentiating noisy data”, J. Comput. Phys. 42 (1981), 141–151.CrossRefGoogle Scholar
[4]Craven, P. and Wahba, G., “Smoothing noisy data with spline functions”, Numer. Math. 31 (1979), 377–403.CrossRefGoogle Scholar
[5]Cullum, J., “The effective choice of smoothing norm in regularization”, Math. Comp. 33 (1979), 149–170.CrossRefGoogle Scholar
[6]Davies, A. R., “On the maximum likelihood regularization of Fredholm convolution equations of the first kind”, in Treatment of Integral Equations by Numerical Methods (eds. Baker, C. T. H. and Miller, G. F.) (Academic Press, London, 1982), 95–105.Google Scholar
[7]Davies, A. R. and Anderssen, R. S., “Improved estimates of statistical regularization parameters in Fourier differentiation and smoothing”, Research Report CMA-R01–85. Centre for Mathematical Analysis, Australian National University, 1985.Google Scholar
[8]Davies, A. R., Iqbal, M., Maleknejad, K. and Redshaw, T. C., “A comparison of statistical regularization and Fourier extrapolation methods for numerical convolution” in Numerical treatment of inverse problems in differential and integral equations (eds. Deuflhard, P. and Hairer, E.) (Birkhauser, Berlin, 1983), 320–334.CrossRefGoogle Scholar
[9]Gamber, H. A., “Choice of an optimal shape parameter when smoothing noisy data”, Comm. Statist. A-Theory Methods 8 (1979), 1425–1435.Google Scholar
[10]Groetsch, C. W., The theory of Tikhonov regularization for Fredholm equations of the first kind (Pitman, London, 1984).Google Scholar
[11]de Hoog, F. R., “Review of Fredholm equations of the first kind”, in The application and numerical solution of integral equations (eds. Anderssen, R. S., de Hoog, F. R. and Lukas, M. A.) (Sijthoff and Noordhoff, 1980), 119–134.CrossRefGoogle Scholar
[12]Klein, G., “On spline functions and statistical regularization of ill-posed problems”, J. Comput. Appl. Math. 5 (1979), 259–263.CrossRefGoogle Scholar
[13]Lukas, M. A., Regularization of linear operator equations, Ph.D. thesis, Australian National University, 1981.CrossRefGoogle Scholar
[14]Lukas, M. A., “Convergence rates for regularized solutions”, 1984. To be published.Google Scholar
[15]Morozov, V. A., “The error principle in the solution of operational equations by the regularization method”, USSR Comput. Math, and Math. Phys. 8 (1968), 63–87.CrossRefGoogle Scholar
[16]Tikhonov, A. N. and Arsenin, V. Y., Solutions of ill-posed problems (trans. from the Russian, Wiley, New York, 1977).Google Scholar
[17]Turchin, V. F., “Solution of the Fredholm equation of the first kind in a statistical ensemble of smooth functions”, USSR Comput. Math. and Math. Phys. 7 (1967), 79–96.CrossRefGoogle Scholar
[18]Turchin, V. F., “Selection of an ensemble of smooth functions for the solution of the inverse problem”, USSR Comput. Math. and Math. Phys. 8 (1968), 328–339.CrossRefGoogle Scholar
[19]Wahba, G., “Smoothing noisy data by spline functions”, Numer. Math. 24 (1975).CrossRefGoogle Scholar
[20]Wahba, G., “Practical approximate solutions to linear operator equations when the data are noisy”, SIAM J. Numer. Anal. 14 (1977), 651–667.CrossRefGoogle Scholar
[21]Wahba, G., “A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem”, Technical Report No. 712. Dept. of Statistics, University of Wisconsin-Madison, 1983.Google Scholar
[22]Whittle, P., “Some results in time series analysis”, Skand. Actuarietidskr. 35 (1952), 48–06.Google Scholar