scholarly journals Analysis of a New Nonlinear Interpolatory Subdivision Scheme on σ Quasi-Uniform Grids

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1320
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Limit Function ◽  
Subdivision Scheme ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Piecewise Polynomials ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Nonlinear Subdivision ◽  
Subdivision Operator ◽  
Theoretical Results

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. The novelty of this work lies in the generalization of the already existing PPH subdivision scheme to the nonuniform case. We define the corresponding subdivision scheme and study some important issues related to subdivision schemes such as convergence, smoothness of the limit function, and preservation of convexity. In order to obtain general results, we consider σ quasi-uniform grids. We also perform some numerical experiments to reinforce the theoretical results.

scholarly journals A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 335
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Gibbs Phenomenon ◽  
Order Of Approximation ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Numerical Examples ◽  
Piecewise Polynomials ◽  
Jump Discontinuities ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Numerical Order

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to σ quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes.

scholarly journals On the Convexity Preservation of a Quasi C3 Nonlinear Interpolatory Reconstruction Operator on σ Quasi-Uniform Grids

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 310 ◽  
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Initial Data ◽  
Numerical Experiments ◽  
Approximation Order ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Convexity Properties ◽  
Second Degree

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for σ quasi-uniform grids with σ≤4, we get a quasi C3 reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.

scholarly journals STABILITY OF PIECEWISE POLYNOMIAL COLLOCATION FOR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

2001 ◽  
Vol 6 (2) ◽  
pp. 310-320
Author(s):  
P. Oja ◽  
M. Tarang
Keyword(s):  
Differential Equations ◽  
Numerical Stability ◽  
Stability Conditions ◽  
Piecewise Polynomial ◽  
Spline Collocation ◽  
Piecewise Polynomials ◽  
Numerical Tests ◽  
Test Equation ◽  
Theoretical Results ◽  
Polynomial Collocation

Numerical stability of the spline collocation method by piecewise polynomials for Volterra integro‐differential equations is investigated. Stability conditions depending on collocation parameters and also on parameters of certain test equation are obtained. Results of several numerical tests are presented supporting theoretical results.

Multivariate piecewise polynomials

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 65-109 ◽  
Author(s):  
C. de Boor
Keyword(s):  
Approximation Theory ◽  
Variational Approach ◽  
Polynomial Function ◽  
Piecewise Polynomial ◽  
Multivariate Splines ◽  
Piecewise Polynomial Function ◽  
Multivariate Spline ◽  
Piecewise Polynomials ◽  
The Subject ◽  
Function Of Several Arguments

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

scholarly journals Piecewise Legendre spectral-collocation method for Volterra integro-differential equations

2015 ◽  
Vol 18 (1) ◽  
pp. 231-249 ◽  
Author(s):  
Zhendong Gu ◽  
Yanping Chen
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Spectral Collocation Method ◽  
Piecewise Polynomial ◽  
Spectral Collocation ◽  
Numerical Errors ◽  
Piecewise Polynomial Collocation Method ◽  
Theoretical Results ◽  
Polynomial Collocation ◽  
Better Than

Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in$h^{m}N^{-m}$-version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

2005 ◽  
Vol 6 (2) ◽  
pp. 193-225 ◽  
Author(s):  
Sergio Amat ◽  
Rosa Donat ◽  
Jacques Liandrat ◽  
J. Carlos Trillo
Keyword(s):  
Image Processing ◽  
Subdivision Scheme ◽  
Nonlinear Subdivision ◽  
Nonlinear Subdivision Scheme

Solvability of cardinal spline interpolation problems

1983 ◽  
Vol 95 (1-2) ◽  
pp. 39-57
Author(s):  
T. N. T. Goodman
Keyword(s):  
Incidence Matrix ◽  
Bounded Solution ◽  
Spline Interpolation ◽  
Piecewise Polynomial ◽  
Piecewise Polynomials ◽  
Cardinal Spline ◽  
Interpolation Problems ◽  
Bounded Data ◽  
Derivatives Of ◽  
Cardinal Spline Interpolation

SynopsisWe consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points of a periodic vector x, specified by a periodic incidence matrix G. Similarly, we allow discontinuity of certain derivatives of the piecewise polynomial at certain points of x, specified by a periodic incidence matrix H. This generalises the well-known cardinal spline interpolation of Schoenberg. We investigate conditions on G, H and x under which there is a unique bounded solution for any given bounded data.

scholarly journals COLLOCATION APPROXIMATIONS FOR WEAKLY SINGULAR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 8 (4) ◽  
pp. 315-328 ◽  
Author(s):  
I. Parts ◽  
A. Pedas
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Order Of Convergence ◽  
Piecewise Polynomial ◽  
Volterra Type ◽  
Weakly Singular ◽  
Uniform Grids ◽  
Attainable Order ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

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