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

Groetsch, C. W.; Vogel, C. R.

doi:10.1090/S0025-5718-1987-0906184-2

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
DOI: https://doi.org/10.1090/S0025-5718-1987-0906184-2
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.

[1] R. S. Anderssen and P. M. Prenter, A formal comparison of methods proposed for the numerical solution of first kind integral equations, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 488–500. MR 626939, https://doi.org/10.1017/S0334270000002824
[2] M. Bertero, C. De Mol, and G. A. Viano, The stability of inverse problems, Inverse scattering problems in optics, Topics Current Phys., vol. 20, Springer, Berlin-New York, 1980, pp. 161–214. MR 612324
[3] A. R. Davies and R. S. Anderssen, Optimization in the regularization of ill-posed problems, J. Austral. Math. Soc. Ser. B 28 (1986), no. 1, 114–133. MR 846786, https://doi.org/10.1017/S0334270000005221
[4] Joel N. Franklin, On Tikhonov’s method for ill-posed problems, Math. Comp. 28 (1974), 889–907. MR 0375817, https://doi.org/10.1090/S0025-5718-1974-0375817-5
[5] C. W. Groetsch, The theory of Tikhonov regularization for Fredholm equations of the first kind, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 742928
[6] Robert S. Anderssen, Frank R. de Hoog, and Mark A. Lukas (eds.), The application and numerical solution of integral equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 6, Martinus Nijhoff Publishers, The Hague, 1980. MR 582981
[7] M. A. Lukas, Convergence Rates for Regularized Solutions, Report 5084, Colorado State University, 1984.
[8] M. Z. Nashed and Grace Wahba, Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind, Math. Comp. 28 (1974), 69–80. MR 0461895, https://doi.org/10.1090/S0025-5718-1974-0461895-1
[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] Alastair Spence, Error bounds and estimates for eigenvalues of integral equations, Numer. Math. 29 (1977/78), no. 2, 133–147. MR 0659480, https://doi.org/10.1007/BF01390333
[11] Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR 0455365
[12] C. R. Vogel, Optimal choice of a truncation level for the truncated SVD solution of linear first kind integral equations when data are noisy, SIAM J. Numer. Anal. 23 (1986), no. 1, 109–117. MR 821908, https://doi.org/10.1137/0723007
[13] Grace Wahba, Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal. 14 (1977), no. 4, 651–667. MR 0471299, https://doi.org/10.1137/0714044