The effective choice of the smoothing norm in regularization
Author:
Jane Cullum
Journal:
Math. Comp. 33 (1979), 149170
MSC:
Primary 65R20; Secondary 41A25, 65D25
MathSciNet review:
514816
Fulltext PDF Free Access
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Abstract: We consider illposed problems of the form where g and K are given, and we must compute f. The Tikhonov regularization procedure replaces (1) by a oneparameter family of minimization problemsMinimize where is a smoothing norm chosen by the user. We demonstrate by example that the choice of is not simply a matter of convenience. We then show how this choice affects the convergence rate, and the condition of the problems generated by the regularization. An appropriate choice for depends upon the character of the compactness of K and upon the smoothness of the desired solution.
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 A. N. TIKHONOV, "Solution of incorrectly formulated problems and the regularization method," Soviet Math. Dokl., v. 4, 1963, pp. 10351038.
 [3]
 A. V. CHECHK1N, "A. N. Tikhonov's special regularizer for integral equations of the first kind," U.S.S.R. Computational Math. and Math. Phys., v. 10, 1970, pp. 234246.
 [4]
 JOEL N. FRANKLIN, "On Tikhonov's method for illposed problems," Math. Comp., v. 28, 1974, pp. 889907. MR 0375817 (51:12007)
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 M. V. AREF'EVA, "Asymptotic estimates for the accuracy of optimal solutions of equations of the convolution type," U.S.S.R. Computational Math. and Math. Phys., v. 14 (4), 1974, pp. 1933. MR 0351129 (50:3618)
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 V. YA. ARSENIN & V. V. IVANOV, "The solution of certain convolution type integral equations of the first kind by the regularization method," U.S.S.R. Computational Math. and Math. Phys., v. 8, 1968, pp. 88106. MR 0231155 (37:6710)
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 V. YA. ARSENIN & V. V. IVANOV, "The effect of regularization of order p," U.S.S.R. Computational Math. and Math. Phys., v. 8, 1968, pp. 221225.
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 V. YA. ARSENIN & T. I. SAVELOVA, "The application of the method of regularization to integral equations of the first kind of the convolution type," U.S.S.R. Computational Math. and Math. Phys., v. 9, 1969, p. 13921396. MR 0267801 (42:2703)
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 A. V. GONCHARSKIĬ, A. S. LEONOV & A. G. YAGOLA, "Some estimates of the rate of convergence of regularized approximations for equations of the convolution type," U.S.S.R. Computational Math. and Math. Phys., v. 12 (3), 1972, pp. 243254.
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 H. WIDOM, "Toeplitz matrices," Studies in Real and Complex Analysis, Math. Assoc. Amer. Studies in Math. (I. I. Hirschmann, Editor), PrenticeHall, Englewood Cliffs, N. J., 1965, pp. 179209.
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 R. S. ANDERSSEN & PETER BLOOMFIELD, "Numerical differentiation procedures for nonexact data," Numer. Math., v. 22, 1974, pp. 157182. MR 0348976 (50:1470)
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 JANE CULLUM, "Numerical differentiation and regularization," SIAM J. Numer. Anal., v. 8 (2), 1971, pp. 254265. MR 0290567 (44:7747)
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 JOHN W. HILGERS, "On the equivalence of regularization and certain reproducing kernel Hilbert space approaches for solving first kind problems," SIAM J. Numer. Anal., v. 13, 1976, pp. 172184. MR 0471293 (57:11030)
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 JANE CULLUM, IllPosed Problems, Regularization, and Singular Value Decompositions, IBM Research Report RC 6465, Yorktown Heights, New York, 1977.
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 GRACE 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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DOI:
http://dx.doi.org/10.1090/S00255718197905148161
PII:
S 00255718(1979)05148161
Article copyright:
© Copyright 1979
American Mathematical Society
