Variational bivariate interpolating splines with positivity constraints
type: Item
instance of: scholarly article
title: Variational bivariate interpolating splines with positivity constraints
zbMATH Open document ID: 1022.65011
DOI: 10.1016/S0168-9274(02)00174-5
published in: Applied Numerical Mathematics
publication date: 2003-03-10T00:00:00Z
review text: Positivity preserving interpolation methods are discussed in the context of Sobolev spaces \(H^m(\Omega) \) of order \( m \) for a finite set of interpolation points \(A\) given in a bounded subset \(\Omega \) of the plane \({\mathbb{R}} ^2 . \) For a data function \( f \in H^m(\Omega) \) the authors show the existence of a so-called \textit{positive} interpolation variational spline \(\sigma \in H^m(\Omega) \) which is the unique solution of the minimization problem \(|\sigma |\leq |v|\) where \( v\in H^m(\Omega) \) satisfies the positivity condition \( v(x) \geq 0 \) for all \( x\in \Omega \) and the interpolation condition \( v(a) = f(a) \) for all \( a \in A \); here the semi-norm \(|\cdot|\) is given through the \(L^2 \)-integral of the \(m\)-th derivative. Using finite elements methods a discretized version of the above problem is formulated and solved by an explicit algorithm. Convergence results show that the discretizations \(\sigma_h \) converges in the Sobolev norm to the positive interpolation variational spline \( \sigma . \) The authors illustrate their results by numerical and graphical examples.
reviewed by: Hermann Render
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 1879009
zbMATH Keywords: shape perserving interpolation
zbMATH Keywords: variational splines
zbMATH Keywords: finite element methods
zbMATH Keywords: approximation of surfaces
zbMATH Keywords: algorithm
zbMATH Keywords: convergence
zbMATH Keywords: numerical and graphical examples
author: A. Kouibia
author: M. Pasadas
MaRDI profile type: Publication
cites work: Approximation spline de surfaces de type explicite comportant des failles
cites work: Q4748596
cites work: Q4144698
cites work: An Analysis of Two Algorithms for Shape-Preserving Cubic Spline Interpolation
cites work: Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines
cites work: Positive quartic, monotone quintic \(C^ 2\)-spline interpolation in one and two dimensions
cites work: Approximation by shape preserving interpolation splines
cites work: Smoothing variational splines
cites work: Q3988921
cites work: Q3094428
cites work: \(S\)-convex, monotone, and positive interpolation with rational bicubic spline of \(C^ 2\)-continuity
cites work: Approximation in the finite element method
cites work: Q3726690
full work available at URL: s0168-9274(02)00174-5
OpenAlex ID: W2162453671
description: scientific article; zbMATH DE number 1879009