Scattered data interpolation using minimum energy Powell-Sabin elements and data dependent triangulations
The author notes that in the interpolation problem in two variables, patching together a function interpolating given data points \((x_ i,y_ i,f_ i)\) it is best to let the triangulation of the point set \((x_ i,y_ i)\) depend on the problem. His main tool is a theorem of \textit{C. L. Lawson} [Discrete Math. 3, 365-372 (1972; Zbl 0253.05116)] that any triangulation of a finite planar set can be transformed into any other triangulation of the same set by looking at the convex quadrilaterals formed by two adjacent triangles and replacing one diagonal by the other. Given an interpolation function \(S(x,y)\) and starting with a given triangulation, in particular, starting with the Delaunay triangulation and using the Powell-Sabin piecewise quadratic interpolant [\textit{M. J. D. Powell} and \textit{M. A. Sabin} [ACM Trans. Math. Software 3, 316-325 (1977; Zbl 0375.41010)] the author uses as measure of smoothness the total (euclidean) norm of the Hessian \(H(S)\) of \(S\), \(C = \Sigma\int \| H(S)\|^ 2 dxdy\), the integral taken over each triangle, the sum over the triangulation. Since \(S\) is locally quadratic, the entire operation is algebraic. For each Lawson operation, the change of \(C\) is computed and the triangulation with a (local) minimum of \(C\) is the final result. Some numerical examples are given showing locally optimal triangulations with very slim triangles.
Powell-Sabin piecewise quadratic interpolant
scientific article; zbMATH DE number 510230
scientific article