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Asymptotic theory of filtering for linear operator equations with discrete noisy data

Authors: C. W. Groetsch and C. R. Vogel
Journal: Math. Comp. 49 (1987), 499-506
MSC: Primary 65J10; Secondary 49D15, 65R20
MathSciNet review: 906184
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Abstract: We consider Fredholm integral equations of the first kind with continuous kernels in which the data is discretely sampled and contaminated by white noise. A sufficient condition for the convergence of general filtering methods applied to such equations is derived. The condition in essence relates the decay rate of the singular values of the integral operator to the shape of the filter function used in the regularization method. Specific illustrations of the condition are given for Tikhonov regularization, the truncated singular value decomposition, and Landweber iteration.

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  • [1] R. S. Anderssen & P. M. Prenter, "A formal comparison of methods for the numerical solution of first kind integral equations," J. Austral. Math. Soc. Ser. B, v. 22, 1981, pp. 488-500. MR 626939 (82h:65094)
  • [2] M. Bertero, C. De Mol & G. A. Viano, "The stability of inverse problems," in Inverse Scattering Problems in Optics (H. P. Baltes, ed.), Topics in Current Physics, vol. 20, Springer-Verlag, New York, 1980, pp. 161-214. MR 612324 (82j:65068)
  • [3] A. R. Davies & R. S. Anderssen, Optimization in the Regularization of Ill-Posed Problems, Centre for Mathematical Analysis Report 30-85, Australian National University, Canberra, 1985. MR 846786 (87j:45027)
  • [4] J. N. Franklin, "On Tikhonov's method for ill-posed problems," Math. Comp., v. 28, 1974, pp. 889-907. MR 0375817 (51:12007)
  • [5] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, London, 1984. MR 742928 (85k:45020)
  • [6] M. A. Lukas, "Regularization," in The Application and Numerical Solution of Integral Equations (R. S. Anderssen et al., eds.), Sijthoff and Noordhoff, Alphen aan den Rijn, the Netherlands, 1980, pp. 152-182. MR 582981 (81f:45001)
  • [7] M. A. Lukas, Convergence Rates for Regularized Solutions, Report 5084, Colorado State University, 1984.
  • [8] M. Z. Nashed & G. Wahba, "Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind," Math. Comp., v. 28, 1974, pp. 69-80. MR 0461895 (57:1877)
  • [9] D. W. Nychka & D. D. Cox, Convergence Rates for Regularized Solutions of Integral Equations from Discrete Noisy Data, Technical Report No. 752, Department of Statistics, University of Wisconsin-Madison, 1984.
  • [10] A. Spence, "Error bounds and estimates for eigenvalues of integral equations," Numer. Math., v. 29, 1978, pp. 133-147. MR 0659480 (58:31968)
  • [11] A. N. Tikhonov & V. Y. Arsenin, Solutions of Ill-posed Problems (translated from the Russian), Wiley, New York, 1977. MR 0455365 (56:13604)
  • [12] C. R. Vogel, "Optimal choice of truncation level for the truncated SVD solution of linear first kind intregral equations when data are noisy," SIAM J. Numer. Anal., v. 23, 1986, pp. 109-117. MR 821908 (87d:65148)
  • [13] G. Wahba, "Practical approximate solutions to linear operator equations when the data are noisy," SIAM J. Numer. Anal., v. 14, 1977, pp. 651-667. MR 0471299 (57:11036)

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Article copyright: © Copyright 1987 American Mathematical Society

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