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), 499506
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.
 [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
(82h:65094), http://dx.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, BerlinNew York, 1980, pp. 161–214. MR 612324
(82j:65068)
 [3]
A.
R. Davies and R.
S. Anderssen, Optimization in the regularization of illposed
problems, J. Austral. Math. Soc. Ser. B 28 (1986),
no. 1, 114–133. MR 846786
(87j:45027), http://dx.doi.org/10.1017/S0334270000005221
 [4]
Joel
N. Franklin, On Tikhonov’s method for
illposed problems, Math. Comp. 28 (1974), 889–907. MR 0375817
(51 #12007), http://dx.doi.org/10.1090/S00255718197403758175
 [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
(85k:45020)
 [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 (81f:45001)
 [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
(57 #1877), http://dx.doi.org/10.1090/S00255718197404618951
 [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 WisconsinMadison, 1984.
 [10]
Alastair
Spence, Error bounds and estimates for eigenvalues of integral
equations, Numer. Math. 29 (1977/78), no. 2,
133–147. MR 0659480
(58 #31968)
 [11]
Andrey
N. Tikhonov and Vasiliy
Y. Arsenin, Solutions of illposed problems, V. H. Winston
& Sons, Washington, D.C.: John Wiley & Sons, New YorkToronto,
Ont.London, 1977. Translated from the Russian; Preface by translation
editor Fritz John; Scripta Series in Mathematics. MR 0455365
(56 #13604)
 [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 (87d:65148), http://dx.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
(57 #11036)
 [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. 488500. 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, SpringerVerlag, New York, 1980, pp. 161214. MR 612324 (82j:65068)
 [3]
 A. R. Davies & R. S. Anderssen, Optimization in the Regularization of IllPosed Problems, Centre for Mathematical Analysis Report 3085, Australian National University, Canberra, 1985. MR 846786 (87j:45027)
 [4]
 J. N. Franklin, "On Tikhonov's method for illposed problems," Math. Comp., v. 28, 1974, pp. 889907. 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. 152182. 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. 6980. 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 WisconsinMadison, 1984.
 [10]
 A. Spence, "Error bounds and estimates for eigenvalues of integral equations," Numer. Math., v. 29, 1978, pp. 133147. MR 0659480 (58:31968)
 [11]
 A. N. Tikhonov & V. Y. Arsenin, Solutions of Illposed 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. 109117. 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. 651667. MR 0471299 (57:11036)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198709061842
PII:
S 00255718(1987)09061842
Article copyright:
© Copyright 1987
American Mathematical Society
