## The effective choice of the smoothing norm in regularization

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- by Jane Cullum PDF
- Math. Comp.
**33**(1979), 149-170 Request permission

## Abstract:

We consider ill-posed problems of the form \[ g(t) = \int _0^1 {K(t,s)f(s)ds,\quad 0 \leqslant t \leqslant 1,} \] where*g*and

*K*are given, and we must compute

*f*. The Tikhonov regularization procedure replaces (1) by a one-parameter family of minimization problems-Minimize $({\left \| {Kf - g} \right \|^2} + \alpha \Omega (f))$-where $\Omega$ is a smoothing norm chosen by the user. We demonstrate by example that the choice of $\Omega$ 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 $\Omega$ depends upon the character of the compactness of

*K*and upon the smoothness of the desired solution.

## References

- F. Smithies,
*Integral equations*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 49, Cambridge University Press, New York, 1958. MR**0104991**
A. N. TIKHONOV, "Solution of incorrectly formulated problems and the regularization method," - Joel N. Franklin,
*On Tikhonov’s method for ill-posed problems*, Math. Comp.**28**(1974), 889–907. MR**375817**, DOI 10.1090/S0025-5718-1974-0375817-5 - M. V. Aref′eva,
*Asymptotic estimates of the accuracy of optimal solutions of an equation of convolution type*, Ž. Vyčisl. Mat i Mat. Fiz.**14**(1974), 838–851, 1074 (Russian). MR**351129** - V. Ja. Arsenin and V. V. Ivanov,
*The solution of certain convolution type integral equations of the first kind by the method of regularization*, Ž. Vyčisl. Mat i Mat. Fiz.**8**(1968), 310–321 (Russian). MR**231155**
V. YA. ARSENIN & V. V. IVANOV, "The effect of regularization of order - V. Ja. Arsenin and T. I. Savelova,
*The application of the method of regularization to convolution type integral equations of first kind*, Ž. Vyčisl. Mat i Mat. Fiz.**9**(1969), 1392–1396 (Russian). MR**267801**
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," - J. H. Wilkinson,
*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422**
V. B. GLASKO, N. I. KULIK & A. N. TIKHONOV, "Determination of the geoelectric cross-section by the regularization method," - R. S. Anderssen and P. Bloomfield,
*Numerical differentiation procedures for non-exact data*, Numer. Math.**22**(1973/74), 157–182. MR**348976**, DOI 10.1007/BF01436965 - Jane Cullum,
*Numerical differentiation and regularization*, SIAM J. Numer. Anal.**8**(1971), 254–265. MR**290567**, DOI 10.1137/0708026 - John W. Hilgers,
*On the equivalence of regularization and certain reproducing kernel Hilbert space approaches for solving first kind problems*, SIAM J. Numer. Anal.**13**(1976), no. 2, 172–184. MR**471293**, DOI 10.1137/0713018
JANE CULLUM, - 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**471299**, DOI 10.1137/0714044

*Soviet Math. Dokl.*, v. 4, 1963, pp. 1035-1038. 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. 234-246.

*p*,"

*U.S.S.R. Computational Math. and Math. Phys.*, v. 8, 1968, pp. 221-225.

*U.S.S.R. Computational Math. and Math. Phys.*, v. 12 (3), 1972, pp. 243-254. ROBERT M. GRAY,

*Toeplitz and Circulant Matrices, A Review*, TR 6502-1, Stanford Electronics Lab., Stanford Univ., Stanford, Calif., 1971. H. WIDOM, "Toeplitz matrices,"

*Studies in Real and Complex Analysis*, Math. Assoc. Amer. Studies in Math. (I. I. Hirschmann, Editor), Prentice-Hall, Englewood Cliffs, N. J., 1965, pp. 179-209. JOHN MAKHOUL, "Linear prediction: A tutorial review,"

*Proc. IEEE*, v. 63 (4), 1975, pp. 561-580.

*U.S.S.R. Computational Math. and Math. Phys.*, v. 12 (1), 1972, pp. 174-186.

*Ill-Posed Problems, Regularization, and Singular Value Decompositions*, IBM Research Report RC 6465, Yorktown Heights, New York, 1977.

## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp.
**33**(1979), 149-170 - MSC: Primary 65R20; Secondary 41A25, 65D25
- DOI: https://doi.org/10.1090/S0025-5718-1979-0514816-1
- MathSciNet review: 514816