Approximation by spline functions on Powell-Sabin triangulations
Metadatos
Mostrar el registro completo del ítemAutor
Eddargani, SalahEditorial
Universidad de Granada
Departamento
Universidad de Granada. Programa de Doctorado en Física y MatemáticasMateria
Powell-Sabin split Bernstein–Bézier form Quasi-interpolation schemes Hermite interpolation Marsden's identity Many knot spline spaces Normalized representation
Fecha
2021Fecha lectura
2021-12-08Referencia bibliográfica
Eddargani, Salah. Approximation by spline functions on Powell-Sabin triangulations. Granada: Universidad de Granada, 2021. [http://hdl.handle.net/10481/72080]
Patrocinador
Tesis Univ. Granada.; University of Granada; CNRST Excellence Scholarship ProgrammeResumen
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.