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13 pages, 410 KiB

Open AccessArticle

C² Cubic Algebraic Hyperbolic Spline Interpolating Scheme by Means of Integral Values

by Salah Eddargani, Mohammed Oraiche, Abdellah Lamnii and Mohamed Louzar

Mathematics 2022, 10(9), 1490; https://doi.org/10.3390/math10091490 - 29 Apr 2022

Cited by 1 | Viewed by 1859

In this paper, a cubic Hermite spline interpolating scheme reproducing both linear polynomials and hyperbolic functions is considered. The interpolating scheme is mainly defined by means of integral values over the subintervals of a partition of the function to be approximated, rather than [...] Read more.

In this paper, a cubic Hermite spline interpolating scheme reproducing both linear polynomials and hyperbolic functions is considered. The interpolating scheme is mainly defined by means of integral values over the subintervals of a partition of the function to be approximated, rather than the function and its first derivative values. The scheme provided is

C^{2}

everywhere and yields optimal order. We provide some numerical tests to illustrate the good performance of the novel approximation scheme. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

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15 pages, 330 KiB

Open AccessArticle

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

by Abdelmonaim Saou, Driss Sbibih, Mohamed Tahrichi and Domingo Barrera

Mathematics 2022, 10(6), 893; https://doi.org/10.3390/math10060893 - 11 Mar 2022

Cited by 1 | Viewed by 1660

Abstract

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By [...] Read more.

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree

⩽ r - 1

, we obtain convergence order

2 r

for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are

3 r + 1

and

4 r

, respectively. Moreover, we show that the optimal convergence order

4 r

is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

17 pages, 2522 KiB

Open AccessArticle

Quadratic B-Spline Surfaces with Free Parameters for the Interpolation of Curve Networks

by Paola Lamberti and Sara Remogna

Mathematics 2022, 10(4), 543; https://doi.org/10.3390/math10040543 - 10 Feb 2022

Cited by 1 | Viewed by 2535

Abstract

In this paper, we propose a method for constructing spline surfaces interpolating a B-spline curve network, allowing the presence of free parameters, in order to model the interpolating surface. We provide a constructive algorithm for its generation in the case of biquadratic tensor [...] Read more.

In this paper, we propose a method for constructing spline surfaces interpolating a B-spline curve network, allowing the presence of free parameters, in order to model the interpolating surface. We provide a constructive algorithm for its generation in the case of biquadratic tensor product B-spline surfaces and bivariate B-spline surfaces on criss-cross triangulations. Finally, we present graphical results. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

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30 pages, 2441 KiB

Open AccessArticle

IGA-Energetic BEM: An Effective Tool for the Numerical Solution of Wave Propagation Problems in Space-Time Domain

by Alessandra Aimi and Ariel Surya Boiardi

Mathematics 2022, 10(3), 334; https://doi.org/10.3390/math10030334 - 22 Jan 2022

Cited by 1 | Viewed by 2298

Abstract

The Energetic Boundary Element Method (BEM) is a recent discretization technique for the numerical solution of wave propagation problems, inside bounded domains or outside bounded obstacles. The differential model problem is converted into a Boundary Integral Equation (BIE) in the time domain, which [...] Read more.

The Energetic Boundary Element Method (BEM) is a recent discretization technique for the numerical solution of wave propagation problems, inside bounded domains or outside bounded obstacles. The differential model problem is converted into a Boundary Integral Equation (BIE) in the time domain, which is then written into an energy-dependent weak form successively discretized by a Galerkin-type approach. Taking into account the space-time model problem of 2D soft-scattering of acoustic waves by obstacles described by open arcs by B-spline (or NURBS) parametrizations, the aim of this paper is to introduce the powerful Isogeometric Analysis (IGA) approach into Energetic BEM for what concerns discretization in space variables. The same computational benefits already observed for IGA-BEM in the case of elliptic (i.e., static) problems, is emphasized here because it is gained at every step of the time-marching procedure. Numerical issues for an efficient integration of weakly singular kernels, related to the fundamental solution of the wave operator and dependent on the propagation wavefront, will be described. Effective numerical results will be given and discussed, showing, from a numerical point of view, convergence and accuracy of the proposed method, as well as the superiority of IGA-Energetic BEM compared to the standard version of the method, which employs classical Lagrangian basis functions. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

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15 pages, 324 KiB

Open AccessArticle

Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

by Pedro González-Rodelas, Miguel Pasadas, Abdelouahed Kouibia and Basim Mustafa

Mathematics 2022, 10(2), 223; https://doi.org/10.3390/math10020223 - 12 Jan 2022

Cited by 5 | Viewed by 2988

Abstract

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland [...] Read more.

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

14 pages, 462 KiB

Open AccessArticle

Optimal Centers’ Allocation in Smoothing or Interpolating with Radial Basis Functions

by Pedro González-Rodelas, Hasan M. H. Idais, Mohammed Yasin and Miguel Pasadas

Mathematics 2022, 10(1), 59; https://doi.org/10.3390/math10010059 - 24 Dec 2021

Viewed by 2301

Abstract

Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using [...] Read more.

Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

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16 pages, 6756 KiB

Open AccessArticle

Weighted Quasi-Interpolant Spline Approximations of Planar Curvilinear Profiles in Digital Images

by Andrea Raffo and Silvia Biasotti

Mathematics 2021, 9(23), 3084; https://doi.org/10.3390/math9233084 - 30 Nov 2021

Cited by 4 | Viewed by 1920

Abstract

The approximation of curvilinear profiles is very popular for processing digital images and leads to numerous applications such as image segmentation, compression and recognition. In this paper, we develop a novel semi-automatic method based on quasi-interpolation. The method consists of three steps: a [...] Read more.

The approximation of curvilinear profiles is very popular for processing digital images and leads to numerous applications such as image segmentation, compression and recognition. In this paper, we develop a novel semi-automatic method based on quasi-interpolation. The method consists of three steps: a preprocessing step exploiting an edge detection algorithm; a splitting procedure to break the just-obtained set of edge points into smaller subsets; and a final step involving the use of a local curve approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), chosen for its robustness to data perturbation. The proposed method builds a sequence of polynomial spline curves, connected

C^{0}

in correspondence of cusps,

G^{1}

otherwise. To curb underfitting and overfitting, the computation of local approximations exploits the supervised learning paradigm. The effectiveness of the method is shown with simulation on real images from various application domains. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

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16 pages, 845 KiB

Open AccessArticle

A Reverse Non-Stationary Generalized B-Splines Subdivision Scheme

by Abdellah Lamnii, Mohamed Yassir Nour and Ahmed Zidna

Mathematics 2021, 9(20), 2628; https://doi.org/10.3390/math9202628 - 18 Oct 2021

Cited by 8 | Viewed by 2282

Abstract

In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized [...] Read more.

In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized multi-resolution mask based on their third-order subdivision filters. For the reverse of the fourth-order scheme, two methods are used; the first one is based on least-squares formulation and the second one is based on solving a linear optimization problem. Numerical examples are given to show the performance of the new schemes in reproducing different shapes of initial control polygons. Full article

(This article belongs to the Special Issue Spline Functions and Applications)

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