Approximation by spline functions on Powell-Sabin triangulations

Abstract

Smooth splines on triangulations are the subject of many applications in various elds, among them approximation theory, computer-aided geometric design, entertainment industry, etc. Smooth spline spaces with a lower degree are the classical choice, which is extremely di cult to achieve in arbitrary triangulations. An alternative is to use macro elements of lower degree that split each triangle into a number of macro-triangles. In particular, Powell-Sabin (PS-) split which divides each triangle into six macro-triangles. In this thesis, we deal with the approximation by quartic PS-splines. Namely, we start by solving a Hermite interpolation problem in the space of C1 quartic PS-splines and providing several local quasi-interpolation schemes reproducing quartic polynomials and not requiring the resolution of any linear system. The provided schemes are constructed with the help of Marsden's identity. Then, we address the geometric characterization of Powell-Sabin triangulations allowing the construction of bivariate quartic splines of class C2. Quasi-interpolation in a space of sextic PS-splines has also been considered. These spline functions are C2 continuous on the whole domain but fourth-order regularity is required at vertices and C3 smoothness conditions are imposed across the edges of the re ned triangulation and also at the interior point chosen to de ne the re nement. An algorithm is proposed to de ne the Powell-Sabin triangles with small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden's identity from a more explicit version of the control polynomials introduced some years ago in the literature. Examining the applicability of PS-splines the numerical quadratures, we have proved that any Gaussian quadrature formula exact on the space of quadratic polynomials de ned on a triangle T endowed with a speci c PS-re nement integrates also the functions in the space of C1 quadratic PS-splines de ned on T. This extends the existing results in the literature, where the inner split point Z chosen to de ne the split had to lie on a very speci c subset of the T. Now Z can be freely chosen inside T. Sometimes, when dealing with Digital Elevation Models in engineering, the construction of normalized basis functions could be extremely expensive regarding time and memory needed, which is caused by the treatment of big data. To avoid this problem, we provide quasiinterpolation schemes de ned on a uniform triangulation of type-1 endowed with a PS-split. The spline schemes are generated by setting their B ezier ordinates to suitable combinations of the given data values. Inspiring from bivariate PS-splines theory, we de ne a family of univariate many knot spline spaces of arbitrary degree de ned on an initial partition that is re ned by adding a point in each sub-interval. For an arbitrary smoothness r, splines of degrees 2r and 2r + 1 are considered by imposing additional regularity when necessary. For an arbitrary degree, a B-spline-like basis is constructed by using the Bernstein-B ezier representation. Blossoming is then used to establish a Marsden's identity from which several quasi-interpolation operators having optimal approximation orders are de ned. Finally, we address the approximation by C2 cubic splines via two approaches. In the rst one, we discuss the construction of C2 cubic spline quasi-interpolation schemes de ned on a re ned partition. These schemes are reduced in terms of the degree of freedom compared to those existing in the literature. Namely, we provide a recipe for reducing the degree of freedom by imposing super-smoothing conditions while preserving full smoothness and cubic precision. In addition, we provide subdivision rules by means of blossoming. The derived rules are designed to express the B-spline coe cients associated with a ner partition from those associated with the former one. While in the second approach, we construct a novel normalized B-spline-like representation for C2 continuous cubic spline space de ned on an initial partition re ned by inserting two new points inside each sub-interval. Thus, we derive several families of super-convergent quasi-interpolation operators.

El objetivo general de esta tesis es la construcción de espacios de funciones spline sobre particiones de Powell-Sabin, tanto en un sentido clásico como en una situación univariada.