zbMATH — the first resource for mathematics

Convergence rates for multivariate smoothing spline functions. (English) Zbl 0646.41006
Given the set \(\{t_ i\}\) \(n_ 1\) of scattered data points in an open bounded subset \(\Omega\) of the d-dimensional space \({\mathbb{R}}^ d \)and the values \(\{z_ i\}\) \(n_ 1\), the thin plate smoothing spline \(\sigma_{\lambda}\) is defined as the unique minimizer of the expression \(\lambda | u|^ 2_{m,\Omega}+(1/n)\sum^{n}_{i-1}(u(t_ i)-z_ i)^ 2,\) Here \(| \cdot |\) \(2_{m,\Omega}\) is the Sobolev semi-norm in H m(\(\Omega)\). The author studies the approximation of functions \(f\in H\) m(\(\Omega)\) by a thin plate smoothing spline \(\sigma_{\lambda}\) based on the values \(z_ i=f(t_ i)+\epsilon_ i\), \(i=1,...,n\) of f dt n scattered points in \(\Omega\) known with random errors \(\epsilon_ i\). He obtains the following estimate \[ E[| f- \sigma_{\lambda}|^ 2_{k,\Omega}]\leq C[\lambda^{(m-k)/m}| f|^ 2_{m,\Omega}+D/(n\lambda^{(2k+d)/2m})],\quad k=0,...,m-1 \] provided the boundary of \(\Omega\) is smooth and the points satisfy a “quasi-uniform” condition.
Reviewer: B.Bojanov

41A15 Spline approximation
41A25 Rate of convergence, degree of approximation
65D10 Numerical smoothing, curve fitting
Full Text: DOI
[1] Adams, R.A., Sobolev spaces, (1975), Academic Press London/New York · Zbl 0186.19101
[2] Agmon, S., Lectures on elliptic boundary value problems, (1965), Van Nostrand Princeton, NJ · Zbl 0151.20203
[3] Atteia, M., Existence et determination des fonctions spline a plusieurs variables, C. R. acad. sci. Paris, 262, 575-578, (1966) · Zbl 0168.35002
[4] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[5] Courant, R.; Hilbert, D., Methods of mathematical physics, (1953), Interscience New York · Zbl 0729.35001
[6] Cox, D.D., Multivariate smoothing spline functions, SIAM J. numer. anal., 21, No. 4, (August 1984)
[7] Craven, P.; Wahba, G., Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation, Numer. math., 31, 377-403, (1979) · Zbl 0377.65007
[8] Duchon, J., Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, RAIRO anal. numér., 10, No. 12, (December 1976)
[9] Duchon, J., Splines minimizing rotation invariant semi-norms in Sobolev spaces, () · Zbl 0342.41012
[10] Duchon, J., Sur l’erreur d’interpolation des fonctions de plusieurs variables par LES D^{m}-splines, RAIRO anal. numér., 12, No. 4, (1978) · Zbl 0403.41003
[11] Paihua, L., Quelques méthodes numériques pour le calcul de fonctions spline à une et plusiers variables, Docteur de spécialité, thesis, (1978), Grenoble
[12] Paihua, L.; Utreras, F., Un ensemble de programmes pour l’interpolation des fonctions, par des fonctions spline du type plaque mince, Université scientifique et Médicale de Grenoble, rapport de recherche no. 140, (October 1978)
[13] Ragozin, D., Error bounds for derivative estimates based on spline smoothing of exact or noisy data, J. approx. theory, 37, 335-355, (1983) · Zbl 0531.41011
[14] Schwartz, L., Theorie des distributions, (1966), Hermann Paris
[15] Utreras, F., Cross-validation techniques for smoothing spline functions in one or two dimensions, () · Zbl 0447.65005
[16] Utreras, F.I., On the eigenvalue problem associated with cubic splines: the arbitrary spaced knots case, Sigma, 6, No. 3, (1980)
[17] Utreras, F.I., Natural spline functions: their associated eigenvalue problem, Numer. math., 42, 107-117, (1983) · Zbl 0522.41011
[18] Wahba, G., Convergence rates of “thing plate” smoothing splines when the data are noisy, () · Zbl 0449.65003
[19] Wahba, G.; Wendelberger, J., Some new mathematical methods for variational objective analysis using splines and cross-validation, ()
[20] Weinberger, H.F., Variational methods for eigenvalue approximation, () · Zbl 0089.08402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.