Spline Quasi-interpolants Research Articles

Two methods based on spline quasi-interpolants to estimate volumes enclosed by parametric surfaces

In this study, we explore two methods based on spline quasi−interpolants (abbr. QI) that rely exclusively on point evaluations to estimate the volume of a closed parametric surface. The first method involves a division of this surface evenly into parallel slices in order to approximate the contours of each slice using quadratic QI. Subsequently, we calculate numerically the areas of the interior surfaces, which will be used in a quadrature formula to obtain the approximate volume. The second method uses a bivariate QI to calculate the entire area, and an integration formula is applied to estimate the desired volume. In both methods, we study the associated error estimates. Numerical examples are provided to illustrate our methods, as well as a real-world application, as well as a study of the effect of noise on the robustness of both volume estimation methods.

Local [formula omitted]-smooth spline quasi-interpolation methods

In this paper we construct new univariate local C2 quasi-interpolating splines having specific polynomial reproduction properties. The splines are directly determined by setting their Bernstein-Bézier coefficients to appropriate combinations of the given data values. In certain cases we obtain a family of quasi-interpolating operators satisfying the required conditions, so we fix some extra properties (interpolation of the vertices, extra locality, extra polynomial reproduction) in order to compute unique approximants. We also provide numerical results confirming the theoretical ones.

ENO and WENO cubic quasi-interpolating splines in Bernstein–Bézier form

In this paper we propose the use of C1-continuous cubic quasi-interpolation schemes expressed in Bernstein–Bézier form to approximate functions with jumps. The construction of these schemes is explicit and consists of directly attaching the Bernstein–Bézier coefficients to appropriate combinations of the given data values. This construction can lead to quasi-interpolation schemes with free parameters. This allows to write these schemes of optimal convergence order as a non-negative convex combination of certain quasi-interpolation schemes of lower convergence order. The idea behind that, is to divide the data set used to define a quasi-interpolant of optimal order into subsets, and then define the associated quasi-interpolants. The free parameters facilitate the choice of the convex combination weights. We then apply the WENO approach to the weights to eliminate the Gibbs phenomenon that occurs when we approximate in a non-smooth region. The proposed schemes are of optimal order in the smooth regions and near optimal order is achieved in the neighboring region of discontinuity.

Approximation of piecewise smooth functions by nonlinear bivariate C2 quartic spline quasi-interpolants on criss-cross triangulations

In this paper we focus on the space of C2 quartic splines on uniform criss-cross triangulations and we propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high-order accuracy in smooth regions. We analyse the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations

We investigate the construction of C2 cubic spline quasi-interpolants on a given arbitrary triangulation T to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of T and like a Bernstein basis for imposing smoothness across the edges of T. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of T. Different C2 cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.

A numerical calculation of arc length and area using some spline quasi-interpolants

In this paper, we propose two methods to approach numerically the length of curves and the area of surface of revolution created by rotating a curve around an axis. The first is based on an approximation of functions by quadratic spline discrete quasi-interpolant and calculating its exact length. The second method consists toapproximate the values of the first derivatives by those of cubic spline discrete quasiinterpolant. Those values are used to provide a quadrature formula to calculate the integral giving the length. In both methods, we prove that the order of convergence is O(h^4). The theoretical results given in this work are justified by some numerical examples.

Non-uniform WENO-based quasi-interpolating splines from the Bernstein–Bézier representation and applications

In this paper, we propose a family of C1 non-uniform cubic quasi-interpolation schemes. The construction used here is mainly based on directly establishing the BB-coefficients by a suitable combination of the data values. These combinations generate masks for each of the BB-coefficients. These masks can contain free parameters, which allow us to write a quasi-interpolation schemes defined from a large stencil as a non-negative convex combination of others defined from sub-stencils of small sizes, which coincide with the concept of WENO, which we will use the deal with non-smooth data, or data with jumps. We consider an application of the proposed technique for real measured data related to memristors fabricated with hafnium oxide as a dielectric.

Fully accurate approximation of piecewise smooth functions using corrected B-spline quasi-interpolants

This article is devoted to introducing new spline quasi-interpolants for the sharp approximation of data in one and several dimensions. The new construction aims to obtain accurate approximations close to singularities of the function from which the data are obtained. The technique relies in an accurate knowledge of the position of the singularity, which can be known or approximated, and that allows for the obtention of accurate approximations of the jumps of the function and its derivatives at this point. With this information, it is possible to compute correction terms for the B-spline bases of the spline that are affected by the presence of singularities. The result is a piecewise smooth reconstruction with full accuracy (meaning by full accuracy, the accuracy of the spline at smooth zones). The numerical experiments presented support the theoretical results obtained.

Enhancing mathematical simulation: a novel integro spline quasi-interpolation for nonlocal dynamical systems

ABSTRACT In this research, we present a novel and computationally streamlined method to tackle nonlocal initial value problems through a combination of modeling and simulation. Our approach utilizes the Integro spline quasi-interpolation technique, renowned for its effectiveness in approximating nonlocal integrals. To validate the efficacy of our proposed method, we conduct comprehensive comparisons with alternative approaches found in the existing literature for integrating nonlocal functions. Additionally, we successfully apply our suggested approach to address nonlocal Riccati, resistor-inductor circuit, and Airy oscillator dynamical systems. The numerical results obtained from these applications convincingly demonstrate the advantages and effectiveness of our developed approach in enhancing mathematical simulation.

Cutting-Edge Computational Approaches for Approximating Nonlocal Variable-Order Operators

This study presents an algorithmically efficient approach to address the complexities associated with nonlocal variable-order operators characterized by diverse definitions. The proposed method employs integro spline quasi interpolation to approximate these operators, aiming for enhanced accuracy and computational efficiency. We conduct a thorough comparison of the outcomes obtained through this approach with other established techniques, including finite difference, IQS, and B-spline methods, documented in the applied mathematics literature for handling nonlocal variable-order derivatives and integrals. The numerical results, showcased in this paper, serve as a compelling validation of the notable advantages offered by our innovative approach. Furthermore, this study delves into the impact of selecting different variable-order values, contributing to a deeper understanding of the algorithm’s behavior across a spectrum of scenarios. In summary, this research seeks to provide a practical and effective solution to the challenges associated with nonlocal variable-order operators, contributing to the applied mathematics literature.

Nonlinear 2D C1 Quadratic Spline Quasi-Interpolants on Triangulations for the Approximation of Piecewise Smooth Functions

The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Hermite–Birkhoff spline Quasi-Interpolation with application as dense output for Gauss–Legendre and Gauss–Lobatto Runge–Kutta schemes

Spline Quasi-Interpolation (QI) of even degree 2R on general partitions is introduced, where derivative information up to order R≥1 at the spline breakpoints is required and maximal convergence order can be proved. Relying on the B-spline basis with possible multiple inner knots, a family of quasi-interpolating splines with smoothness of order R is associated with each R≥1, since there is the possibility of using different local sequences of breakpoints to define each QI spline coefficient. By using suitable finite differences approximations of the necessary discrete derivative information, each QI spline in this family can be associated also with a twin approximant belonging to the same spline space, but requiring just function information at the breakpoints.Among possible different applications of the introduced QI scheme, a smooth continuous extension of the numerical solution of Gauss-Lobatto and Gauss-Legendre Runge-Kutta methods is here considered. When R>1, such extension is based on the use of the variant of the QI scheme with derivative approximation which preserves the approximation power of the original Runge-Kutta scheme.

Low-degree spline quasi-interpolants in the Bernstein basis

In this paper we propose the construction of univariate low-degree quasi-interpolating splines in the Bernstein basis, considering C1 and C2 smoothness, specific polynomial reproduction properties and different sets of evaluation points. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values. Moreover, we get quasi-interpolating splines with special properties, imposing particular requirements in case of free parameters. Finally, we provide numerical tests showing the performances of the proposed methods.

C1-Cubic Quasi-Interpolation Splines over a CT Refinement of a Type-1 Triangulation

C1 continuous quasi-interpolating splines are constructed over Clough–Tocher refinement of a type-1 triangulation. Their Bernstein–Bézier coefficients are directly defined from the known values of the function to be approximated, so that a set of appropriate basis functions is not required. The resulting quasi-interpolation operators reproduce cubic polynomials. Some numerical tests are given in order to show the performance of the approximation scheme.

Spline quasi-interpolation in the Bernstein basis on the Powell–Sabin 6-split of a type-1 triangulation

In this paper, we provide quasi-interpolation schemes defined on a uniform triangulation of type-1 endowed with a Powell–Sabin refinement. In contrast to the usual construction of quasi interpolation splines on the 6-split, the approach described in this work does not require a set of appropriate basis functions. The approximating splines are directly defined by setting their Bézier ordinates to suitable combinations of the given data values. The resulting quasi-interpolants are C1 continuous and reproduce quadratic polynomials. Some numerical tests are given to confirm the theoretical results.

On [formula omitted] cubic quasi-interpolating splines and their computation by subdivision via blossoming

We discuss the construction of C2 cubic spline quasi-interpolation schemes defined on a refined partition. These schemes are reduced in terms of degrees of freedom compared to those existing in the literature. Namely, we provide a rule for reducing them 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 coefficients associated with a finer partition from those associated with the former one.

On spline quasi-interpolation through dimensions

The approximation of functions and data in one and high dimensions is an important problem in many mathematical and scientific applications. Quasi-interpolation is a general and powerful approximation approach having many advantages. This paper deals with spline quasi-interpolants and its aim is to collect the main results obtained by the authors, also in collaboration with other researchers, in such a topic through spline dimension, i.e. in the 1D, 2D and 3D setting, highlighting the approximation properties and the reconstruction of functions and data, the applications in numerical integration and differentiation and the numerical solution of integral and differential problems.

Local Quartic C^2 Spline Quasi-interpolation on 3D Bounded Domains

Given a 3D bounded domain, in this paper we present new quasi-interpolating spline schemes, based on trivariate C^2 quartic box splines on type-6 tetrahedral partitions with approximation order four. They are of near-best type, i.e. with coefficient functionals obtained by minimizing an upper bound for their infinity norm. Such quasiinterpolants can be used for the reconstruction of gridded volume data and their higher smoothness is useful, for example, when functions have to be reconstructed with C^2 smoothness. Moreover, we give norm and error bounds. Finally, some numerical tests, illustrating the approximation properties of the proposed quasiinterpolants, and comparisons with other known spline methods are presented.

A class of C2 quasi-interpolating splines free of Gibbs phenomenon

In many applications, it is useful to use piecewise polynomials that satisfy certain regularity conditions at the joint points. Cubic spline functions emerge as good candidates having C2 regularity. On the other hand, if the data points present discontinuities, the classical spline approximations produce Gibbs oscillations. In a recent paper, we have introduced a new nonlinear spline approximation avoiding the presence of these oscillations. Unfortunately, this new reconstruction loses the C2 regularity. This paper introduces a new nonlinear spline that preserves the regularity at all the joint points except at the end points of an interval containing a discontinuity, and that avoids the Gibbs oscillations.

Generalized spline quasi-interpolants and applications to numerical analysis

In this work, we first establish a general Marsden’s identity for Unified and Extended B-splines (UE B-splines or Omega B-splines for short). Then, by using this result, we construct univariate omega spline quasi-interpolants on a bounded interval and we study their approximation errors. For particular values of omega, we refind some already developed quasi-interpolants. As a practical side of these operators, we give some applications to numerical analysis especially quadrature formulas, differentiation and numerical solutions of linear Fredholm integral equations, which are illustrated by some numerical examples.