Domingo Barrera

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on @W-splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.

A general spline differential quadrature method based on quasi-interpolation

The differential quadrature method is a numerical discretization technique for the approximation of derivatives. The classical method is polynomial-based, and there is a natural restriction in the number of grid points involved. A general spline-based method is proposed to avoid this problem. For any degree a Lagrangian spline interpolant is defined having a fundamental function with small support. A quasi-interpolant is used to achieve the optimal approximation order. That two-stage scheme is detailed for the cubic, quartic, quintic and sextic cases and compared with another methods that appear in the literature.

Increasing the approximation order of spline quasi-interpolants

Abstract In this paper, we show how by a very simple modification of bivariate spline discrete quasi-interpolants, we can construct a new class of quasi-interpolants which have remarkable properties such as high order of regularity and polynomial reproduction. More precisely, given a spline discrete quasi-interpolation operator Q d , which is exact on the space P m of polynomials of total degree at most m , we first propose a general method to determine a new differential quasi-interpolation operator Q r D which is exact on P m + r . Q r D uses the values of the function to be approximated at the points involved in the linear functional defining Q d as well as the partial derivatives up to the order r at the same points. From this result, we then construct and study a first order differential quasi-interpolant based on the C 1 cubic B-spline on the equilateral triangulation with a hexagonal support. When the derivatives are not available or extremely expensive to compute, we approximate them by appropriate finite differences to derive new discrete quasi-interpolants Q d . We estimate with small constants the quasi-interpolation errors f − Q r D [ f ] and f − Q d [ f ] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.

On spline-based differential quadrature

In the paper Barrera et al. (2014), a boolean sum differential quadrature method (DQM) was proposed by combining a spline interpolation operator having a fundamental function with minimal compact support and a spline quasi-interpolation operator reproducing the polynomials in the spline space. It is a general framework that provides results that differ from the ones obtained by defining specific schemes with structures which depend on the degree of the B-spline to be considered. The main drawback of these boolean sum DQMs is that the number of evaluation points increases quickly with the degree of the B-spline due to the use of a quasi-interpolation operator. We propose a different construction avoiding this problem and derive explicit results for low degree B-splines.

Near-best operators based on a C2 quartic spline on the uniform four-directional mesh

We present some results about the construction of quasi-interpolant operators based on a special C^2 quartic B-spline. We show that these operators, called near-best quasi-interpolants, have the best approximation order and small infinity norms. They are obtained by solving a minimization problem that admits always a solution. We give an error bound of these quasi-interpolants and we illustrate our results by a numerical example.

Construction techniques for multivariate modified quasi-interpolants with high approximation order

In this paper, we propose several approximations of a multivariate function by quasi-interpolants on non-uniform data and we study their properties. In particular, we characterize those that preserve constants via the partition of unity approach. As one of the main results, we show how by a very simple modification of a given quasi-interpolant it is possible to construct new quasi-interpolants with remarkable properties. We also provide some results regarding bivariate C^2 quintic spline quasi-interpolation. Finally, numerical tests are presented to show the approximation power of these quasi-interpolants.

On the construction of trivariate near-best quasi-interpolants based on C 2 quartic splines on type-6 tetrahedral partitions

The construction of new quasi-interpolants (QIs) having optimal approximation order and small infinity norm and based on a trivariate C 2 quartic box spline is addressed in this paper. These quasi-interpolants, called near-best QIs, are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the QIs. We show that this problem has always a unique solution, which is explicitly given. We also prove that the sequence of the resulting near-best quasi-interpolants converges in the infinity norm to the Schoenberg operator.

Hermite spline interpolation on a three direction mesh from Powell-Sabin and Hsieh-Clough-Tocher finite elements

In this paper we develop a general local method to define Hermite interpolants of prescribed order r ź 1 and global class C s on the three direction mesh of the real plane. They are defined from Powell-Sabin and Hsieh-Clough-Tocher finite elements in such a way that the interpolation operators have fundamental functions with compact support and reproduce a given space P m of polynomials included in the spline space.

Polynomial pattern finding in scattered data

A new numerical procedure to extract the threshold voltage in MOSFET transistors has been developed by means of polynomial pattern recognition. The technique proposed here is based on the use of particular properties of discrete orthogonal Chebyshev polynomials, it allows the extraction of polynomial curves of different degrees within a set of experimental or simulated data. For the MOSFET threshold voltage determination we have detected linear patterns in the logarithmic representation of MOSFET transfer characteristics (drain current versus gate voltage curves). The results have been compared with the threshold voltage obtained with a classical technique, the transconductance change method, where the maximum of the drain current second derivative is assumed as the threshold voltage. Reasonable and comparable results are obtained. The new technique has shown more immunity to numerical and measurement noise, which is an important feature in the current industrial context.

Trivariate near-best blending spline quasi-interpolation operators

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated.