\(L_1\) spline methods for scattered data interpolation and approximation
For finding a smooth surface which interpolates or approximates the given scattered data, the authors use bivariate spline spaces of degree \(d\) and smoothness \(r\) (with \(r<d\)) over any triangulation. They generalize the \(L_1\) spline methods proposed by \textit{J. E. Lavery} in [Comput. Aided Geom. Des. 17, 715--727 (2000; Zbl 0997.65014) and ibid. 18, 321--343 (2001; Zbl 0971.68177)]. Lavery suggested the \(L_2\) norm in the usual minimal quadratic energy to be replaced by \(L_1\) norms. The authors extend this method to scattered data sets, they use more general spline spaces and present a detailed mathematical analysis for the \(L_1\) spline methods. On a number of numerical examples they illustrate the advantage of the \(L_1\) spline method over the minimal energy method. Finally they discuss some extensions of the \(L_1\) spline methods for other surface design problems.
scientific article; zbMATH DE number 2085731