×

On some methods of the construction of smoothing splines. (English) Zbl 0613.65009

Numerical approximation of partial differential equations, Sel. Pap. Int. Symp. Numer. Anal., Madrid/Spain 1985, North-Holland Math. Stud. 133, 141-150 (1987).
[For the entire collection see Zbl 0599.00010.]
The revised Weinert algorithm for the construction of the smoothing Lg- spline s(t) is discussed. The smoothing spline \(s\in H\) with respect to \(r\in E^ N\), L, \(\lambda_ j\), \(\rho_ j\), \(j=1,...,N\) is the solution of the extremal problem \[ \min_{f\in H}\{\int^{a}_{b}(Lf)^ 2dt+\sum^{N}_{j=1}(r_ j-\lambda_ jf)^ 2/\rho_ j\} \] where \(H=W^ n_ 2(a,b)\) is a Sobolev space, \(L=\sum^{n}_{j=0}a_ jD^ j\) is a differential operator \(H\to L_ 2(a,b)\), \(a_ j\in W^ j_ 2(a,b)\cap C(a,b)\), \(a_ n=1\), \(\{\lambda_ j\}^ N_{j=1}\) is linearly independent collection of continuous linear functionals on H, \(\rho_ j\) are strictly positive weights of measurement values \(r_ j\) at points \(t_ j\) of some mesh.
The construction of the smoothing Lg-splines is based on the reproducing kernel technique. The reproducing kernel K(t,\(\tau)\) in H has the dynamic model \(\partial /\partial t x(t,\tau)=A(t)x(t,\tau)+bG(\tau,t)\tau\),t\(\in [a,b]\), \(b=(0,...,0,1)^ T\), \(A(t)=\left( \begin{matrix} 0\\ -a_ 0(t)\end{matrix} \begin{matrix} I\\ -a_ 1(t)...-a_{n-1}(t)\end{matrix} \right)\), where I, G(t,\(\tau)\) denote the unit matrix and the Green function of the operator L, respectively. Weinert’s general recursive algorithm for the construction of smoothing splines [cf. H. Weinert, R. Byrd and G. Sidhu, J. Optimization Theory Appl. 30, 255-268 (1980; Zbl 0397.65008)] is not correct. A revised algorithm for the special case of the cubic smoothing spline (e.g. \(n=2\), \(L=D^ 2\) and \(\lambda_ jf=f(t_ j)=r_ j)\), is described. This corrected algorithm is compared with the algorithms of Reinsch, Wahba and de Boor for cubic smoothing splines. Numerical results show that this algorithm is much more effective.

MSC:

65D07 Numerical computation using splines
41A15 Spline approximation