Spline Quasi-interpolants Research Articles

Development of immersion subunit vaccine against grouper iridovirus (GIV)

In this paper, we present a new technique to get superconvergence phenomenon of spline quasi-interpolants at the knots of the partition. This method gives rise to good approximation not only at these knots but also on the whole domain of definition. Moreover, we give an application to numerical integration. Numerical results are given to illustrate the theoretical ones.

Bivariate hierarchical Hermite spline quasi-interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.

Construction of univariate spline quasi-interpolants with symmetric functions

A new approach to construct univariate spline quasi-interpolants on arbitrary partitions of bounded intervals is developed. In the first part of this paper, we give some results about the symmetric functions of the difference of two finite sets. These results are used, in the second part of this work, to construct explicitly different types of quasi-interpolants. We revise the definition of a uniformly bounded quasi-interpolant and we propose some results on this subject. Some numerical examples are given to illustrate our theoretical results.

Formula omitted] Superconvergent quasi-interpolation based on polar forms

In this paper, we use C1 cubic B-splines to construct the Hermite interpolant of any polynomial in terms of their blossom. Consequently, a simple method is presented to get superconvergence phenomenon of cubic spline quasi-interpolants at the knots of a uniform partition. Thanks to this phenomenon, the cubic spline quasi-interpolant provides an interesting approximation very accurate at the superconvergence points. Numerical results are given to illustrate the theoretical ones.

Trivariate spline quasi-interpolants based on simplex splines and polar forms

In this work we describe an approximating scheme based on simplex splines on a tetrahedral partition using volumetric data. We use the trivariate simplex B-spline representation of polynomials or piecewise polynomials in terms of their polar forms to construct several differential or discrete quasi interpolants which have an optimal approximation order.

A superconvergent cubic spline quasi-interpolant and application

In this paper, we present a new technique to get superconvergence phenomenon of cubic spline quasi-interpolants at the knots of an uniform partition. This method gives rise to good approximation not only at these knots but also on the whole domain of definition. Moreover, We give an application to numerical integration. Numerical results are given to illustrate the theoretical ones.

Near-best $$C^2$$ C 2 quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains

In this paper, we present new quasi-interpolating spline schemes defined on 3D bounded domains, based on trivariate $C^2$ quartic box splines on type-6 tetrahedral partitions and with approximation order four. Such methods can be used for the reconstruction of gridded volume data. More precisely, we propose near-best quasi-interpolants, i.e. with coefficient functionals obtained by imposing the exactness of the quasi-interpolants on the space of polynomials of total degree three and minimizing an upper bound for their infinity norm. In case of bounded domains the main problem consists in the construction of the coefficient functionals associated with boundary generators (i.e. generators with supports not completely inside the domain), so that the functionals involve data points inside or on the boundary of the domain. We give norm and error estimates and we present some numerical tests, illustrating the approximation properties of the proposed quasi-interpolants, and comparisons with other known spline methods. Some applications with real world volume data are also provided.

Superconvergent $$C^1$$ C 1 cubic spline quasi-interpolants on Powell-Sabin partitions

In this paper we introduce a B-spline representation of the cubic Hermite Powell-Sabin interpolant of any polynomial or any piecewise polynomial, over Powell-Sabin partitions of class at least $$C^1$$ , in terms of their polar forms. We use this B-spline representation for constructing several superconvergent discrete cubic spline quasi-interpolants which approximate a function $$f$$ better than the superconvergent quadratic ones developed in one of our recent published papers. The new results presented in this work are an improvement and a generalization of those studied recently in the literature. We also illustrate by numerical examples that global errors and cubature rules based on these cubic Powell-Sabin spline quasi-interpolants are positively affected by the superconvergence phenomenon.

Computing the range of values of real functions using B-spline form

In this paper, we present an easy and efficient method for computing the range of a function by using spline quasi-interpolation. We exploit the close relationship between the spline function and its control polygon and use tight subdivision technique in order to obtain monotonic splines which make the range of the spline easy to compute. The proposed method is useful in case of given scattered data generated by some (unknown) function f or scientific measurements. Several numerical examples are given, for cubic and quintic quasi-interpolant approximant, to illustrate the efficiency and the performance of our method.

Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

In this paper we consider the space generated by the scaled translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals.We give weights and nodes of the above rules and we analyse their properties.Finally, some numerical tests and comparisons with other known integration formulas are presented.

Iteration methods for Fredholm integral equations of the second kind based on spline quasi-interpolants

In this paper, we propose an efficient iteration algorithm for Fredholm integral equations of the second kind based on spline quasi-interpolants (abbr. QIs). We show that for every iteration step we obtain superconvergence rates. A superconvergent method called functional approximation method based on QIs is also developed. We illustrate our results by numerical experiments.

Trivariate quartic splines on type-6 tetrahedral partitions and their applications

In this paper we develop a 3D quasi-interpolating spline scheme, on a bounded domain, based on trivariate quartic C 2 box splines on type-6 tetrahedral partitions. Then, we describe some applications related both to the reconstruction of gridded volume data and to numerical integration. We also provide some numerical tests.

Construction of quintic Powell–Sabin spline quasi-interpolants based on blossoming

By blossoming Marsden’s identity, we investigate local quasi-interpolation schemes forC2-continuous quintic Powell–Sabin splines represented with a normalized B-spline basis. As applications, various families of discrete and differential quasi-interpolants reproducing quintic polynomials are presented.

Increasing the approximation order of spline quasi-interpolants

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 Qd, which is exact on the space Pm of polynomials of total degree at most m, we first propose a general method to determine a new differential quasi-interpolation operator QrD which is exact on Pm+r. QrD uses the values of the function to be approximated at the points involved in the linear functional defining Qd 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 C1 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 Q̃d. We estimate with small constants the quasi-interpolation errors f−QrD[f] and f−Q̃d[f] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.

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 C2 quintic spline quasi-interpolation. Finally, numerical tests are presented to show the approximation power of these quasi-interpolants.

Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in ℝ2. By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.

Error bounds on the approximation of functions and partial derivatives by quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of a rectangular domain

Given a non-uniform criss-cross triangulation of a rectangular domain Ω, we consider the approximation of a function f and its partial derivatives, by general C1 quadratic spline quasi-interpolants and their derivatives. We give error bounds in terms of the smoothness of f and the characteristics of the triangulation. Then, the preceding theoretical results are compared with similar results in the literature. Finally, several examples are proposed for illustrating various applications of the quasi-interpolants studied in the paper.

High-order quadrature rules based on spline quasi-interpolants and application to integral equations

In this paper, we present a class of quadrature rules with endpoint corrections based on integrating spline quasi-interpolants. The correction weights are obtained as solutions of certain systems of linear algebraic equations. We give a comparison between the rules obtained here and the Gregory rules of the same order. Furthermore, an application of these quadrature rules to the numerical solution of Fredholm integral equations of the second kind is worked out in detail. Numerical examples illustrating the theory are given.

A multinomial spline approximation scheme using spline quasi-interpolants

This paper presents a multinomial spline approximation scheme based on spline quasi-interpolants. The scheme can be considered as an extension of the usual Bernstein approximation for complex exponentials. Error estimates and numerical examples are given to show that this new scheme could produce highly accurate results.

Quasi-interpolation operators based on the trivariate seven-direction C 2 quartic box spline

This paper investigates the space \(\mathcal{S}(X)\) generated by the integer translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions that forms a regular partition of the space into tetrahedra.In \(\mathcal{S}(X)\) local spline quasi-interpolants are defined by linear combinations of integer translates of B {B α ,α∈ℤ3} and local linear functionals \(\{\mathbb{D} f(\alpha), \alpha\in\mathbb{Z}^{3} \}\). First a quasi-interpolant of differential type is proposed and its coefficient functionals are linear combinations of values of f with its partial derivatives at the center of the support of B α . Then, by convenient discretisations of the above differential coefficients, three quasi-interpolants of discrete type with different properties are defined. In this case the coefficient functionals are linear combinations of values of f at specific points in the support of B α .Upper bounds both for the norm of the discrete quasi-interpolation operators and for the approximation error are given.Finally, some numerical examples, illustrating the approximation properties of the proposed quasi-interpolants, are presented.