Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The effective choice of the smoothing norm in regularization

Author: Jane Cullum
Journal: Math. Comp. 33 (1979), 149-170
MSC: Primary 65R20; Secondary 41A25, 65D25
MathSciNet review: 514816
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider ill-posed problems of the form

$\displaystyle 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\Vert {Kf - g} \right\Vert^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 [Enhancements On Off] (What's this?)

  • [1] F. Smithies, Integral equations, Cambridge Tracts in Mathematics and Mathematical Physics, no. 49, Cambridge University Press, New York, 1958. MR 0104991
  • [2] A. N. TIKHONOV, "Solution of incorrectly formulated problems and the regularization method," Soviet Math. Dokl., v. 4, 1963, pp. 1035-1038.
  • [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. 234-246.
  • [4] Joel N. Franklin, On Tikhonov’s method for ill-posed problems, Math. Comp. 28 (1974), 889–907. MR 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 0351129
  • [6] 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 0231155
  • [7] 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. 221-225.
  • [8] 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 0267801
  • [9] 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. 243-254.
  • [10] ROBERT M. GRAY, Toeplitz and Circulant Matrices, A Review, TR 6502-1, Stanford Electronics Lab., Stanford Univ., Stanford, Calif., 1971.
  • [11] 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.
  • [12] JOHN MAKHOUL, "Linear prediction: A tutorial review," Proc. IEEE, v. 63 (4), 1975, pp. 561-580.
  • [13] J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
  • [14] V. B. GLASKO, N. I. KULIK & A. N. TIKHONOV, "Determination of the geoelectric cross-section by the regularization method," U.S.S.R. Computational Math. and Math. Phys., v. 12 (1), 1972, pp. 174-186.
  • [15] R. S. Anderssen and P. Bloomfield, Numerical differentiation procedures for non-exact data, Numer. Math. 22 (1973/74), 157–182. MR 0348976,
  • [16] Jane Cullum, Numerical differentiation and regularization, SIAM J. Numer. Anal. 8 (1971), 254–265. MR 0290567,
  • [17] 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 0471293,
  • [18] JANE CULLUM, Ill-Posed Problems, Regularization, and Singular Value Decompositions, IBM Research Report RC 6465, Yorktown Heights, New York, 1977.
  • [19] 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,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 41A25, 65D25

Retrieve articles in all journals with MSC: 65R20, 41A25, 65D25

Additional Information

Article copyright: © Copyright 1979 American Mathematical Society