Convergence rates for regularized solutions
Author:
Mark A. Lukas
Journal:
Math. Comp. 51 (1988), 107131
MSC:
Primary 65R20; Secondary 41A25, 45L05
MathSciNet review:
942146
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Given a firstkind integral equation with discrete noisy data , , let be the minimizer in a Hilbert space 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. 68 (1950), 337–404. MR 0051437
(14,479c), http://dx.doi.org/10.1090/S00029947195000514377
 [2]
JeanPierre
Aubin, Applied functional analysis, John Wiley & Sons, New
YorkChichesterBrisbane, 1979. Translated from the French by Carole
Labrousse; With exercises by Bernard Cornet and JeanMichel Lasry. MR 549483
(81a:46083)
 [3]
Colin
Bennett and John
E. Gilbert, Homogeneous algebras on the circle. II. Multipliers,
Ditkin conditions, Ann. Inst. Fourier (Grenoble) 22
(1972), no. 3, 21–50 (English, with French summary). MR 0338783
(49 #3547)
 [4]
Dennis
D. Cox, Asymptotics for 𝑀type smoothing splines, Ann.
Statist. 11 (1983), no. 2, 530–551. MR 696065
(84h:62097)
 [5]
Dennis
D. Cox, Multivariate smoothing spline functions, SIAM J.
Numer. Anal. 21 (1984), no. 4, 789–813. MR 749371
(86b:41018), http://dx.doi.org/10.1137/0721053
 [6]
D. D. Cox, Approximation of Method of Regularization Estimators, Technical Report No. 723, Dept. of Statistics, Univ. of WisconsinMadison, 1983.
 [7]
Peter
Craven and Grace
Wahba, Smoothing noisy data with spline functions. Estimating the
correct degree of smoothing by the method of generalized
crossvalidation, Numer. Math. 31 (1978/79),
no. 4, 377–403. MR 516581
(81g:65018), http://dx.doi.org/10.1007/BF01404567
 [8]
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
(57 #11030)
 [9]
Einar
Hille, Introduction to general theory of reproducing kernels,
Rocky Mountain J. Math. 2 (1972), no. 3,
321–368. MR 0315109
(47 #3658)
 [10]
M.
Z. Nashed and Grace
Wahba, Generalized inverses in reproducing kernel spaces: an
approach to regularization of linear operator equations, SIAM J. Math.
Anal. 5 (1974), 974–987. MR 0358405
(50 #10871)
 [11]
Frank
Natterer, Error bounds for Tikhonov regularization in Hilbert
scales, Applicable Anal. 18 (1984), no. 12,
29–37. MR
762862 (86e:65081), http://dx.doi.org/10.1080/00036818408839508
 [12]
John
Rice and Murray
Rosenblatt, Integrated mean squared error of a smoothing
spline, J. Approx. Theory 33 (1981), no. 4,
353–369. MR
646156 (83k:41010), http://dx.doi.org/10.1016/00219045(81)900666
 [13]
John
Rice and Murray
Rosenblatt, Smoothing splines: regression, derivatives and
deconvolution, Ann. Statist. 11 (1983), no. 1,
141–156. MR
684872 (84j:62042)
 [14]
Frigyes
Riesz and Béla
Sz.Nagy, Functional analysis, Frederick Ungar Publishing Co.,
New York, 1955. Translated by Leo F. Boron. MR 0071727
(17,175i)
 [15]
P. Speckman, "The asymptotic integrated mean square error for smoothing noisy data by splines," Numer. Math. (To appear.)
 [16]
Ulrich
Tippenhauer, Methoden zur Bestimmung von Reprokernen, J.
Approximation Theory 21 (1977), no. 4, 394–410
(German, with English summary). MR 0467272
(57 #7135)
 [17]
Hans
Triebel, Interpolation theory, function spaces, differential
operators, NorthHolland Mathematical Library, vol. 18,
NorthHolland Publishing Co., AmsterdamNew York, 1978. MR 503903
(80i:46032b)
 [18]
F.
Utreras Diaz, Sur le choix du paramètre d’ajustement
dans le lissage par fonctions spline, Numer. Math. 34
(1980), no. 1, 15–28 (French, with English summary). MR 560791
(83c:65024), http://dx.doi.org/10.1007/BF01463995
 [19]
Florencio
Utreras, Natural spline functions, their associated eigenvalue
problem, Numer. Math. 42 (1983), no. 1,
107–117. MR
716477 (86c:65013), http://dx.doi.org/10.1007/BF01400921
 [20]
Grace
Wahba, Convergence rates of certain approximate solutions to
Fredholm integral equations of the first kind, J. Approximation Theory
7 (1973), 167–185. MR 0346453
(49 #11178)
 [21]
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)
 [22]
Grace
Wahba, Constrained regularization for illposed linear operator
equations, with applications in meteorology and medicine, Statistical
decision theory and related topics, III, Vol. 2 (West Lafayette, Ind.,
1981) Academic Press, New York, 1982, pp. 383–418. MR 705326
(85e:45011)
 [1]
 N. Aronszajn, "Theory of reproducing kernels," Trans. Amer. Math. Soc., v. 68, 1950, pp. 337404. 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. 2150. MR 0338783 (49:3547)
 [4]
 D. D. Cox, "Asymptotics for Mtype smoothing splines," Ann. Statist., v. 11, 1983, pp. 530551. MR 696065 (84h:62097)
 [5]
 D. D. Cox, "Multivariate smoothing spline functions," SIAM J. Numer. Anal., v. 21, 1984, pp. 789813. MR 749371 (86b:41018)
 [6]
 D. D. Cox, Approximation of Method of Regularization Estimators, Technical Report No. 723, Dept. of Statistics, Univ. of WisconsinMadison, 1983.
 [7]
 P. Craven & G. Wahba, "Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized crossvalidation," Numer. Math., v. 31, 1979, pp. 377403. 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. 172184. MR 0471293 (57:11030)
 [9]
 E. Hille, "Introduction to general theory of reproducing kernels," Rocky Mountain J. Math., v. 2, 1972, pp. 321368. 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. 974987. MR 0358405 (50:10871)
 [11]
 F. Natterer, "Error bounds for Tikhonov regularization in Hilbert scales," Applicable Anal., v. 18, 1984, pp. 2937. MR 762862 (86e:65081)
 [12]
 J. Rice & M. Rosenblatt, "Integrated mean squared error of a smoothing spline," J. Approx. Theory, v. 33, 1981, pp. 353369. MR 646156 (83k:41010)
 [13]
 J. Rice & M. Rosenblatt, "Smoothing splines: regression, derivatives and deconvolution," Ann. Statist., v. 11, 1983, pp. 141156. 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. 394410. MR 0467272 (57:7135)
 [17]
 H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, NorthHolland, 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. 1528. MR 560791 (83c:65024)
 [19]
 F. Utreras, "Natural spline functions, their associated eigenvalue problem," Numer. Math., v. 42, 1983, pp. 107117. 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. 167185. 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. 651667. 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)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65R20,
41A25,
45L05
Retrieve articles in all journals
with MSC:
65R20,
41A25,
45L05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809421468
PII:
S 00255718(1988)09421468
Article copyright:
© Copyright 1988
American Mathematical Society
