Convergence rates for regularized solutions

Author:
Mark A. Lukas

Journal:
Math. Comp. **51** (1988), 107-131

MSC:
Primary 65R20; Secondary 41A25, 45L05

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942146-8

MathSciNet review:
942146

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a first-kind integral equation

*W*of the regularization functional . It is shown that in any one of a wide class of norms, which includes , if in a certain way as , then converges to the true solution . Convergence rates are obtained and are used to estimate the optimal regularization parameter .

**[1]**N. Aronszajn, "Theory of reproducing kernels,"*Trans. Amer. Math. Soc.*, v. 68, 1950, pp. 337-404. MR**0051437 (14:479c)****[2]**J. Aubin,*Applied Functional Analysis*, Wiley, New York, 1979. MR**549483 (81a:46083)****[3]**C. Bennett & J. E. Gilbert, "Homogeneous algebras on the circle: II. Multipliers, Ditkin conditions,"*Ann. Inst. Fourier*(*Grenoble*), v. 22, 1972, pp. 21-50. MR**0338783 (49:3547)****[4]**D. D. Cox, "Asymptotics for*M*-type smoothing splines,"*Ann. Statist.*, v. 11, 1983, pp. 530-551. MR**696065 (84h:62097)****[5]**D. D. Cox, "Multivariate smoothing spline functions,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 789-813. MR**749371 (86b:41018)****[6]**D. D. Cox,*Approximation of Method of Regularization Estimators*, Technical Report No. 723, Dept. of Statistics, Univ. of Wisconsin-Madison, 1983.**[7]**P. Craven & G. Wahba, "Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation,"*Numer. Math.*, v. 31, 1979, pp. 377-403. MR**516581 (81g:65018)****[8]**J. 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. 172-184. MR**0471293 (57:11030)****[9]**E. Hille, "Introduction to general theory of reproducing kernels,"*Rocky Mountain J. Math.*, v. 2, 1972, pp. 321-368. MR**0315109 (47:3658)****[10]**M. Z. Nashed & G. Wahba, "Generalized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations,"*SIAM J. Math. Anal.*, v. 5, 1974, pp. 974-987. MR**0358405 (50:10871)****[11]**F. Natterer, "Error bounds for Tikhonov regularization in Hilbert scales,"*Applicable Anal.*, v. 18, 1984, pp. 29-37. MR**762862 (86e:65081)****[12]**J. Rice & M. Rosenblatt, "Integrated mean squared error of a smoothing spline,"*J. Approx. Theory*, v. 33, 1981, pp. 353-369. MR**646156 (83k:41010)****[13]**J. Rice & M. Rosenblatt, "Smoothing splines: regression, derivatives and deconvolution,"*Ann. Statist.*, v. 11, 1983, pp. 141-156. MR**684872 (84j:62042)****[14]**F. Riesz & B. Sz.-Nagy,*Functional Analysis*, Ungar, New York, 1955. MR**0071727 (17:175i)****[15]**P. Speckman, "The asymptotic integrated mean square error for smoothing noisy data by splines,"*Numer. Math.*(To appear.)**[16]**U. Tippenhauer, "Methoden zur Bestimmung von Reprokernen,"*J. Approx. Theory*, v. 21, 1977, pp. 394-410. MR**0467272 (57:7135)****[17]**H. Triebel,*Interpolation Theory, Function Spaces, Differential Operators*, North-Holland, New York, 1978. MR**503903 (80i:46032b)****[18]**F. Utreras Diaz, "Sur le choix du paramètre d'adjustement dans le lissage par fonctions spline,"*Numer. Math.*, v. 34, 1980, pp. 15-28. MR**560791 (83c:65024)****[19]**F. Utreras, "Natural spline functions, their associated eigenvalue problem,"*Numer. Math.*, v. 42, 1983, pp. 107-117. MR**716477 (86c:65013)****[20]**G. Wahba, "Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind,"*J. Approx. Theory*, v. 7, 1973, pp. 167-185. MR**0346453 (49:11178)****[21]**G. Wahba, "Practical approximate solutions to linear operator equations when the data are noisy,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 651-667. MR**0471299 (57:11036)****[22]**G. Wahba, "Constrained regularization for ill posed linear operator equations, with applications in meteorology and medicine," in*Statistical Decision Theory and Related Topics*: III, Vol. 2 (S. S. Gupta and J. O. Berger, eds.), Academic Press, New York, 1982. MR**705326 (85e:45011)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20,
41A25,
45L05

Retrieve articles in all journals with MSC: 65R20, 41A25, 45L05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942146-8

Article copyright:
© Copyright 1988
American Mathematical Society