Spline quasi-interpolation in the Bernstein basis on the Powell–Sabin 6-split of a type-1 triangulation
Introduction
Spline interpolation and quasi-interpolation have become indispensable mathematical tools for the approximation of functions and data. They are topics widely developed in the literature, which includes many different ways of construction of approximating splines. In the uniform case, standard quasi-interpolation produces approximants in the space spanned by a family of non-negative compactly supported functions forming a partition of unity (for instance, B-splines or box splines) and no linear system resolution is required.
Following the idea used in [1], a new procedure was introduced in [2] and [3] based on the definition of the Bernstein–Bézier (BB-) coefficients of the spline on each triangle in the uniform partition. They are set directly from specific point values in a neighbourhood of the triangle so that continuity is achieved, in addition to the reproduction of the polynomials of a specific degree.
The aim of the method addressed in [3] is to construct quartic splines on a type-1 triangulation in such a way that the cubic polynomials are reproduced. Simple rules to produce the BB-coefficients of the quasi-interpolant on each triangle of the partition are provided. The values of the quasi-interpolated function at the domain points of order four relative each triangle of the triangulation are assumed to be known. For a triangle with barycentric coordinates , they are of the form ,with non negative integers such that . The splines obtained from these rules interpolate the point values at vertices.
In [4] a general study of this problem is carried out in order to determine all possible rules for defining BB-coefficients giving continuity and exactness on the space of polynomials of total degree equal to three. It is shown that there exists a multi-parametric family of rules, having nineteen degrees of freedom, and then the reduction of the number of evaluations needed to compute the BB-coefficients is addressed. Moreover, it is proved that there exists a family of rules based on evaluation at vertices and midpoints of edges of triangles depending on only three parameters. The resulting quasi-interpolating splines also interpolate the point values at vertices. Both in [3] and [4] the used rules have symmetries, so the computational cost is reduced.
A similar methodology is used in [5] to construct cubic quasi-interpolants that reproduce quadratic polynomials when the values at the vertices and midpoints are known. In this case, quasi-interpolants do not interpolate the data values at vertices. Moreover, different rules correspond to different domain points. There are no symmetries applicable. However, there exists only one solution, i.e. a set of rules that allow the objectives to be met: continuity and reproduction of the quadratic polynomials.
It would be natural, therefore, to construct quadratic quasi-interpolants in the same way on type-1 triangulations, reproducing polynomials of degree 1 at most, but this is not possible. Only constants can be reproduced. Consequently, we propose to construct quadratic quasi-interpolants on a type-1 triangulation endowed with a Powell–Sabin refinement [6] to achieve the optimal approximation order. quadratic quasi-interpolation on general Powell–Sabin triangulations is studied in [7] and [8] by defining appropriate bases of B-spline-like functions. Super-convergent quadratic quasi-interpolants are defined in [9] and quadratic quasi-interpolation is also studied on the sphere [10].
The rest of this work is organized as follows: in Section 2, we give some preliminaries on the Bernstein–Bézier form of quadratic polynomials on type-1 triangulations and we introduce the Powell–Sabin split and some useful notations used throughout the paper. In Section 3, we provide the unique quasi-interpolating spline that reproduces quadratic polynomials. It is defined from the prescribed values at the vertices and midpoints of the edges of the triangulation. In Section 4, we propose some numerical tests that confirm the theoretical results established in Section 3. Another scheme exact on the space of polynomials of degree less than or equal to two is provided in Section 5, but now defined only from the point values at vertices. Numerical tests supporting the theoretical results of Section 5 are also included. We conclude the paper with a conclusion section.
Section snippets
Bernstein-Bézier form of quadratic splines
For , the vectors and define the lattice , where . These vertices define the faces of the lattice, that can be decomposed into the triangles and , so that a type-1 triangulation is obtained, namely . In general, these triangles will be referred to as macro-triangles and any one of them will be represented by the capital letter , without specifying what type it is.
Let be the set of
quadratic quasi-interpolating splines on a 6-split
In this work, we aim to construct a quasi-interpolation operator exact on , that is to say, for all . The quasi-interpolant of will be defined from the values of at the vertices and the midpoints of the edges, and this will be done by directly setting its BB-coefficients for all micro-triangles to appropriate combinations of the values at these points. In the rest of this section, the values of at the vertices and the midpoints of the edges are then
Numerical tests
In order to illustrate the performance of the quasi-interpolating spline we have defined, we consider the two test functions defined on the unit square: They are the Franke and Nielson functions, respectively [14], [15]. Their typical plots are shown in Fig. 8.
The quasi-interpolation error is estimated as and being equally spaced points
Quasi-interpolation from point values at vertices
In [12] new approximating splines were constructed by application of a preprocessing to the quasi-interpolating splines defined in [4] from the values at vertices and midpoints: firstly, the values of the given function at -points are replaced by the ones obtained after one step of a subdivision algorithm suitable for type-1 triangulated data, and then the resulting values are used jointly with the values to get a quasi-interpolant whose BB-coefficients only involve values at the
Conclusion
In this work, we have introduced two kinds of quasi-interpolation schemes. Both kinds are generated by setting their B-ordinates to suitable combinations of the given data values, instead of being defined as linear combinations of a set of bivariate functions and they do not require derivative values. The first kind involves the values at the vertices and midpoints of the type-1 triangulation. While, the second one is restricted to the values prescribed at the set of its vertices. The presented
Acknowledgements
The authors wish to thank the anonymous referees for their very pertinent and useful comments which helped them to improve the original manuscript. The first author acknowledges partial financial support by the IMAG–María de Maeztu grant CEX2020-001105-M/AEI / 10.13039/501100011033. The second author would like to thank the University of Granada for the financial support for the research stay during which this work was partially carried out. The third author is a member of the research group
References (15)
- et al.
Quasi-interpolation by quadratic piecewise polynomials in three variables
J. Comput. Aided Geom. Des.
(2005) - et al.
An explicit quasi-interpolation scheme based on quartic splines on type-1 triangulations
Comput. Aided Geom. Design
(2008) - et al.
Quasi-interpolation by quartic splines on type-1 triangulations
J. Comput. Appl. Math.
(2019) - et al.
Point and differential quasi-interpolation on three direction meshes
J. Comput. Appl. Math.
(2019) - et al.
Polar forms and quadratic spline quasi-interpolants on Powell–Sabin partitions
Appl. Numer. Math.
(2009) - et al.
Superconvergent quadratic spline quasi-interpolants on Powell–Sabin partitions
Appl. Numer. Math.
(2015) - et al.
Quadratic spherical spline quasi-interpolants on Powell–Sabin partitions
Appl. Numer. Math.
(2009)