Least squares approximation by splines with free knots
type: Item
instance of: scholarly article
title: Least squares approximation by splines with free knots
zbMATH Open document ID: 0847.65004
DOI: 10.1007/BF01732610
author: Hubert Schwetlick
author: Torsten Schütze
published in: BIT
publication date: 1996-02-01T00:00:00Z
review text: Let \(f\) be a function in the class \(W^q_2 [ a,b ]\), whose values on the discrete data \(a\leq x_1< \dots< x_m\leq b\), have been perturbed: \(y_i= f(x_i)+ \varepsilon_i\) \((i=1, \dots, m)\), \(\varepsilon_i\) being the error of the \(i\)th observation. In order to find information of the unknown function \(f\) the \(n\)-dimensional spline space \(S_{k,t}\) of all piecewise polynomials of order \(k\geq 1\), with knot sequence \(t= \{t_j \}\) can be considered. The quadratic pseudo-distance \(S= S(\{ y_i\}, \{s(x_i )\})\), \(s\in S_{k,t}\), can be used as a measure of fitness. The purpose of the paper is to develop a robust optimization algorithm to minimize \(S\) to calculate a good location of the free knots \(\{ t_j \}\). The introduction of the paper contains general remarks about function approximation by means of free knots splines and data reduction strategies. In the references 27 titles are listed. The method of approximating the noisy data \(\{ y_i\}\) does not suppose a Schoenberg-Whitney-like regularity condition but instead uses some type of regularization by a simple smoothing term which guarantees unique solvability independent of the position of knots. It also treats the linearly constrained nonlinear least squares problem directly, and is able to find a good location of only a subset of the inner knots. Some numerical examples show that the corresponding algorithm provides a robust and efficient tool for least squares approximation by splines with free knots.
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A29
zbMATH DE Number: 827432
zbMATH Keywords: data fitting
zbMATH Keywords: knot removal strategies
zbMATH Keywords: \(B\)-splines
zbMATH Keywords: robust optimization algorithm
zbMATH Keywords: free knots splines
zbMATH Keywords: data reduction
zbMATH Keywords: noisy data
zbMATH Keywords: smoothing
zbMATH Keywords: nonlinear least squares
zbMATH Keywords: numerical examples
reviewed by: J. Illán
describes a project that uses: FITPACK
MaRDI profile type: Publication
cites work: An efficient line search for nonlinear least squares
cites work: A practical guide to splines
cites work: Q3702408
cites work: Q3135817
cites work: Q4226179
cites work: The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate
cites work: An Algorithm for Constrained Non-Linear Least-Squares
cites work: An algorithm for data reduction using splines with free knots
cites work: Approximation to Data by Splines with Free Knots
cites work: A variable projection method for solving separable nonlinear least squares problems
cites work: A globalization scheme for the generalized Gauss-Newton method
cites work: Efficient implementation of a variable projection algorithm for nonlinear least squares problems
cites work: Q4879581
cites work: A Data-Reduction Strategy for Splines with Applications to the Approximation of Functions and Data
cites work: Algorithms for Separable Nonlinear Least Squares Problems
cites work: On Polya Frequency Function. III. The Positivity of Translation Determinants With an Application to the Interpolation Problem by Spline Curves
cites work: Q3895980
cites work: Spline smoothing under constraints on derivatives
cites work: Method of optimal variable-knot spline interpolation in the 2 discrete norm
cites work: A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations
cites work: Q3978908
cites work: Q3996207
full work available at URL: https://doi.org/10.1007/bf01732610
OpenAlex ID: W2024690006
description: scientific article; zbMATH DE number 827432