Least squares approximation by splines with free knots
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.
scientific article; zbMATH DE number 827432
scientific article