The paper presents an in-depth exploration of the multinode Shepard interpolant on a regular rectangular grid, demonstrating its efficacy in reconstructing surfaces from DEM data. Additionally, we study the approximation order associated to this interpolant and present a detailed algorithm for reconstructing surfaces. Numerical tests showcase the effectiveness of the proposed algorithm.

Splines over triangulation are a fundamental tool in finite element analysis (FEM), due to their ability to perform local refinements and accurately represent complex geometries. An important example is the C^1 quadratic splines on the Powell-Sabin (PS) 6-split triangulations. This spline space is characterized by specifying discrete values and first derivative values at the vertices of the triangulation on which it is defined. However, a common challenge arises in many applications where only functional evaluations are known at equally spaced nodes, making the direct application of the Powell-Sabin finite element method impractical. To address this limitation in this setting, we present a novel technique for approximating derivative values at the set of vertices, essential for effectively defining the Powell-Sabin finite element. The idea of this technique lies in leveraging the provided functional evaluations and approximating the partial derivatives at the data points using an interpolation-regression operator.

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-B & eacute;zier 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.

In this paper we use cubic spline quasi-interpolant operator to numerically address a class of linear integro-differential equations with weakly singular kernel. As stated in Pedas and Tamme (2006), the exact solution of this equation lacks the desired level of smoothness and belongs to a particular function space. Then, in the first part of this paper, we analyze the approximation properties of the cubic spline quasi-interpolant operator in this particular space. Subsequently, we use these results in the analysis of the quasi-collocation method used to solve an integro-differential equation with weakly singular kernel. Also, some numerical tests are provided to confirm the theoretical results.

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.

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.

In this paper, a new non-stationary bivariate subdivision scheme is proposed to obtain a more flexible limit surface. By rewriting the univariate subdivision scheme (Fakhar et al., 2022) under two different methods, the tensor product and the subdivision rules for extraordinary points/faces are constructed. The new scheme produces continuous limit surfaces of higher order, except at extraordinary points/faces where the continuity is G1. Algorithms for generating sharp and semi-sharp features on the hybrid subdivision surface are presented. Numerical examples are also given to show the performance of these new schemes.

In this work, we introduce a numerical approach that utilizes spline quasi-interpolation operators over a bounded interval. This method is designed to provide a numerical solution for a class of Fredholm integro-differential equations with weakly singular kernels. We outline the computational components involved in determining the approximate solution and provide theoretical findings regarding the convergence rate. This convergence rate is analyzed in relation to both the degree of the quasi-interpolant and the grading exponent of the graded grid partition. Finally, we present numerical experiments that validate the theoretical findings.

This paper focuses on the approximation of solutions of second kind Fredholm integral equations using non-uniform spline quasi-interpolation. Our aim is to determine the most effective non-uniform partition that provides an optimal numerical solution to the integral equation. To achieve this, we introduce a solution approach based on genetic algorithms, using right approximation of the integral equation’s kernel. We present some numerical examples to show the method’s performance.

In this paper, the construction of C^1 cubic quasi-interpolants on a three-direction mesh of ℝ^2 is addressed. The quasi-interpolating splines are defined by directly setting their Bernstein-Bézier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Moreover, additional global properties are required. Finally, we provide some numerical tests confirming the approximation properties.

An efficient numerical method based on the Shepard method has been developed to deal with Fredholm integral equations of the second kind. The procedure involves expressing the unknown function as a linear combination of Shepard basis functions. Subsequently, a collocation method is used to get the matrix equation yielding the coefficients of such an approximate solution. The effectiveness of the proposed approach is illustrated by some examples.

A construction of Marsden's identity for UE-splines is developed and a complete proof is given. With the help of this identity, a new non-uniform quasi-interpolant that reproduces the spaces of polynomial, trigonometric and hyperbolic functions are defined. Efficient quadrature rules based on integrating these quasi-interpolation schemes are derived and analyzed. Then, a quadrature formula associated with non-uniform quasi-interpolation along with Nyström's method is used to numerically solve Hammerstein and Fredholm integral equations. Numerical results that illustrate the effectiveness of these rules are presented.

We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves the use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts.

In this paper we propose the construction of univariate low-degree quasi-interpolating splines in the Bernstein basis, considering C 1 and C 2 smoothness, specific polynomial re-production properties and different sets of evaluation points. The splines are directly de-termined by setting their Bernstein-Bezier coefficients to appropriate combinations of the given data values. Moreover, we get quasi-interpolating splines with special properties, im-posing particular requirements in case of free parameters. Finally, we provide numerical tests showing the performances of the proposed methods. & COPY; 2023 Elsevier Inc. All rights reserved.

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.

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.

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.

In this paper, we propose collocation type method, its iterated version and Nyström method based on discrete spline quasi-interpolating operators to solve Hammerstein integral equation. We present an error analysis of the approximate solutions and we show that the iterated solution of collocation type exhibits a superconvergence as in the case of the Galerkin method. Finally, we provide numerical tests, that confirm the theoretical results.

We have analyzed variability in resistive memories (Resistive Random Access Memories, RRAMs) making use of advanced numerical techniques to process experimental measurements and simulations based on the kinetic Monte Carlo technique. The devices employed in the study were fabricated using the TiN/Ti/HfO2/W stack. The switching parameters were obtained making use of new developed extraction methods. The appropriateness of the advanced parameter extraction methodologies has been checked by comparison to kinetic Monte Carlo simulations; in particular, the reset and set events have been studied and detected. The data obtained were employed to shed light on the resistive switching operation and the cycle-to-cycle variability. It has been shown that variability depends on the numerical technique employed to obtain the set and reset voltages, therefore, this issue must be taken into consideration in RS characterization and modeling studies. The proposed techniques are complementary and depending on the technology and the curves shape the features of a particular method could make it to be the most appropriate.