Spline Quasi-interpolants Research Articles

Multilevel quadratic spline integration

In this paper we present new quadratures based on both quasi-interpolation and multilevel methods by using bivariate quadratic B-spline functions, defined on simple and multiple knot type-2 triangulations, improving classical quadratures, based on quasi-interpolating splines. We also prove some symmetry properties that simplify their expression, study their approximation performances, propose some numerical results and a comparison with other known multilevel spline quadratures.

Nyström methods for approximating the solutions of an integral equation arising from a problem in mathematical biology

The paper deals with an integral equation arising from a problem in mathematical biology. We propose approximating its solution by Nyström methods based on Gaussian rules and on product integration rules according to the smoothness of the kernel function. In particular, when the latter is weakly singular we propose two Nyström methods constructed by means of different product formulas. The first one is based on the Lagrange interpolation while the second one is based on discrete spline quasi-interpolants. The stability and the convergence of the proposed methods are proved in uniform spaces of continuous functions. Finally, some numerical tests showing the effectiveness of the methods and the sharpness of the obtained error estimates are given.

Non-Uniform Spline Quasi-Interpolation to Extract the Series Resistance in Resistive Switching Memristors for Compact Modeling Purposes

An advanced new methodology is presented to improve parameter extraction in resistive memories. The series resistance and some other parameters in resistive memories are obtained, making use of a two-stage algorithm, where the second one is based on quasi-interpolation on non-uniform partitions. The use of this latter advanced mathematical technique provides a numerically robust procedure, and in this manuscript, we focus on it. The series resistance, an essential parameter to characterize the circuit operation of resistive memories, is extracted from experimental curves measured in devices based on hafnium oxide as their dielectric layer. The experimental curves are highly non-linear, due to the underlying physics controlling the device operation, so that a stable numerical procedure is needed. The results also allow promising expectations in the massive extraction of new parameters that can help in the characterization of the electrical device behavior.

Quadrature formulas based on spline quasi-interpolation for hypersingular integrals arising in IgA-SGBEM

In this paper, we study the construction of quadrature rules for the approximation of hypersingular integrals that occur when 2D Neumann or mixed Laplace problems are numerically solved using Boundary Element Methods. In particular the Galerkin discretization is considered within the Isogeometric Analysis setting and spline quasi-interpolation is applied to approximate integrand factors, then integrals are evaluated via recurrence relations. Convergence results of the proposed quadrature rules are given, with respect to both smooth and non smooth integrands. Numerical tests confirm the behavior predicted by the analysis. Finally, several numerical experiments related to the application of the quadrature rules to both exterior and interior differential problems are presented.

Integro spline quasi-interpolants and their super convergence

This paper gives a type of new spline quasi-interpolants where the entry values are integral values of successive intervals, rather than the usual function values at the knots. They are called integro spline quasi-interpolants. Also, their super convergence property in approximating function values/derivative values at the knots/mid-knots are proved. Numerical experiments show that the integro spline quasi-interpolants possess super convergence.

Quasi-interpolant operators in Bernstein basis

The aim of this paper is to present a family of polynomial quasi-interpolants in Bernstein basis. More precisely, we will combine the strong features of the polar forms and the symmetric polynomials to derive the coefficients of the quasi-interpolant in the Bernstein basis representation. We also derive a collection of spline quasi-interpolants that reproduce polynomial functions up to degree 2. Numerical examples support the theoretical results and show that the proposed scheme is simple and effective.

Multilevel quadratic spline quasi-interpolation

In this paper we present new approximation schemes based on both quasi-interpolation and multilevel methods by using bivariate quadratic B-spline functions, defined on simple and multiple knot type-2 triangulations, improving the classical quasi-interpolating spline results. We also prove polynomial reproduction, optimal approximation order and propose some numerical results and applications.

A trivariate near-best blending quadratic quasi-interpolant

In this paper, we construct a new trivariate spline quasi-interpolation operator. It is expressed as blending sum of univariate and bivariate C1 quadratic spline quasi-interpolants and it is of near-best type, i.e. it has a small infinity norm and the coefficients functionals defining it are determined by minimizing an upper bound of the operator infinity norm, derived from the Bernstein-Bézier coefficients of its Lebesgue function.

Estimation of the reset voltage in resistive RAMs using the charge–flux domain and a numerical method based on quasi-interpolation and discrete orthogonal polynomials

Resistive RAMs (RRAMs) are the most promising devices for the near future in terms of non-volatile memory applications. This new technology requires advances in all the fronts that need to be addressed prior to industrialization. One of them is connected with compact modeling, i.e, the development of analytical expressions to account for the most important physical effects that are needed to calculate the current, capacitance, transient response, etc. The device models should be accurate and this issue is achieved by implementing the correct physics in a flexible and robust mathematical architecture. We will focus on this latter problem here since we will deal with a good numerical approximation of experimental data based on spline quasi-interpolation to perform the integrals of the current and voltage as function of time. We do so to transform the usual modeling domain, consisting of a current–voltage representation, to a charge–flux domain; i.e., the time integral of the current and voltage measured variables. In this domain we introduced a new method to obtain the reset voltage of RRAMs and avoid the effects of the usual measurement noise. The main features of the mathematical technique we propose along with practical examples built upon real experimental data will be explained. The numerical stability of this new technique is of great interest for the implementation in automatic measurement environments for industrial applications.

Cubic quasi-interpolation spline collocation method for solving convection–diffusion equations

In this paper, we use a cubic spline collocation method to solve a two dimensional convection–diffusion equation. More precisely, we approximate first and second order partial derivatives by those of cubic spline quasi-interpolants to produce a system of first order ordinary differential equations. The resulting system can be solved using MATLAB’s ode solver. Error estimates of quasi-interpolants which are used are given with full discussion. Furthermore, numerical examples are presented to show the validity of our methods.

Point and differential C1 quasi-interpolation on three direction meshes

In this paper we construct and analyse C 1 cubic and quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data, without using minimal determining sets and without defining the approximating splines as linear combinations of compactly supported bivariate spanning functions. In particular, the C 1 cubic splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values without using prescribed derivatives at any point of the domain, in such a way that the C 1 -smoothness conditions are satisfied and approximation order three is guaranteed, for smooth functions. We also propose some numerical tests that confirm the theoretical results. Then, from the above C 1 cubic splines we obtain C 1 quartic splines exact on P 3 , achieving approximation order four. The associated differential quasi-interpolation operator involves the values of the first partial derivatives in its definition.

Algebraic hyperbolic spline quasi-interpolants and applications

In this paper, a construction of Marsden’s identity for UAH B-splines (i.e. Uniform Algebraic Hyperbolic B-splines) is developed and a clear proof is given. With the help of this identity, quasi-interpolant schemes that produce the space of algebraic hyperbolic functions are derived. Efficient quadrature rules, based on integrating some of these quasi-interpolant schemes, are constructed and studied. Numerical results that illustrate the effectiveness of these rules are presented.

Quasi-interpolation by C1 quartic splines on type-1 triangulations

In this paper we construct two new families of C 1 quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain. The quasi-interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation, reproduce cubic polynomials and yield approximation order four for smooth functions. We also propose some numerical tests that confirm the theoretical results.

Efficient quadrature rules based on spline quasi-interpolation for application to IGA-BEMs

We propose a new class of quadrature rules for the approximation of weakly and strongly singular integrals, based on the spline quasi-interpolation scheme introduced in Mazzia and Sestini (2009). These integrals in particular occur in the entries of the stiffness matrix coming from Isogeometric Boundary Element Methods (IgA-BEMs). The presented formulas are efficient, since they combine the locality of any spline quasi-interpolation scheme with the capability to compute the modified moments for B-splines, i.e. the weakly or strongly singular integrals of such functions. No global linear system has to be solved to determine the quadrature weights, but just local systems, whose size linearly depends on the adopted spline degree. The rules are preliminarily tested in their basic formulation, i.e. when the integrand is defined as the product of a singular kernel and a continuous function g. Then, such basic formulation is compared with a new one, specific for the approximation of the singular integrals appearing in the IgA-BEM context, where a B-spline factor is explicitly included in g. Such a variant requires the usage of the recursive spline product formula given in Mørken (1991), and it is useful when the ratio between g and its B-spline factor is smooth enough.

Two methods based on bivariate spline quasi-interpolants for solving Fredholm integral equations

For solving a Fredholm integral equation of the second kind, we approximate its kernel by two types of bivariate spline quasi-interpolants, namely the tensor product and the continuous blending sum of univariate spline quasi-interpolants. We give the construction of the approximate solutions, and we prove some theoretical results related to the approximation errors of these methods. We illustrate the obtained results by some numerical tests giving a comparison with several methods in the literature.

A spline quasi-interpolation based method to obtain the reset voltage in Resistive RAMs in the charge-flux domain

Resistive RAMs (RRAMs) are the most promising devices for near future in terms of non-volatile memory applications. This new technology needs advances in all the fronts that have to be addressed prior to industrialization. One of them is connected with compact modeling, i.e., the development of analytical expressions to account for the most important physical effects that are needed to calculate the current, capacitance, transient response, etc. The device models should be accurate and this issue is achieved by implementing the correct physics in a flexible and robust mathematical architecture. We will focus on this latter problem here since we will deal with a good numerical approximation of experimental data based on spline quasi-interpolation to perform the integrals of the current and voltage as function of time. We do so to transform the usual modeling domain, consisting of a current–voltage representation, to a charge-flux domain; i.e., the time integral of the current and voltage measured variables. In this domain we introduced a new method to obtain the reset voltage of RRAMs and avoid the effects of the usual measurement noise. The main features of the mathematical technique we propose along with practical examples built upon real experimental data will be explained. The numerical stability of this new technique is of great interest for the implementation in automatic measurement environments for industrial applications.

Discrete septic spline quasi-interpolants for solving generalized Fredholm integral equation of the second kind via three degenerate kernel methods

Three main contributions are presented in this paper. First, the septic quasi-interpolants are calculated with all their coefficients. Second, we explore the results to solve a generalized and broad class of Fredholm integral equations of the second kind. Finally, we present three degenerate kernel methods; the latter is a combination of the two previously established methods in the literature. Moreover, we provide a convergence analysis and we give new error bounds. Finally, we exhibit some numerical examples and compare them with previous results in the literature.

Multiquadric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective

Based on multiquadric trigonometric spline quasi-interpolation, the paper proposes a scheme for numerical differentiation of noisy data, which is a well-known ill-posed problem in practical applications. In addition, in the perspective of kernel regression, the paper studies its large sample properties including optimal bandwidth selection, convergence rate, almost sure convergence, and uniformly asymptotic normality. Simulations are provided at the end of the paper to demonstrate features of the scheme. Both theoretical results and simulations show that the scheme is simple, easy to compute, and efficient for numerical differentiation of noisy data.

Local cubic splines on non-uniform grids and real-time computation of wavelet transform

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of signals and multidimensional data arrays. The designed splines serve as a source for generating real-time wavelet transforms to apply to signals in scenarios where the signal’s samples subsequently arrive one after the other at random times. The wavelet transforms are executed by six-tap weighted moving averages of the signal’s samples without delay. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects arise.

Superconvergent spline quasi-interpolants and an application to numerical integration

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.