A. Kouibia

Approximation by discrete variational splines

In this paper we present a numerical approximation of curves and surfaces from a given scattered data set. An approximating curve or surface problem is obtained by minimizing a quadratic functional in a parametric finite element space, its solution is called a discrete smoothing variational spline. The existence and uniqueness of this problem are shown, as long as the convergence of the method is established. Finally some particular examples are given.

Approximation of surfaces by fairness bicubic splines

In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method.

Variational bivariate interpolating splines with positivity constraints

This paper deals with a shape preserving method of interpolating positive data at points of the plane in R2. We formulate a problem in order to define a positive interpolation variational spline in the Sobolev space Hm (Ω), where Ω is a non-empty bounded set of R2, by minimizing the semi-norm of order m, and we discrete such problem in a finite element space. An algorithm allows us to compute the resulting function. Some convergence theorems are established. The error is of the order O(1/N)m when N tends to +∞, being N the number of the interpolating points. Some numerical and graphical examples are given in order to test the validity of this method.

Approximation by shape preserving interpolation splines

In this paper we present a shape preserving method of interpolation for scattered data defined in the form of some constraints such as convexity, monotonicity and positivity. We define a k-convex interpolation spline function in a Sobolev space, by minimizing a semi-norm of order k + 1, and we discretize it in the space of piecewise polynomial spline functions. The shape preserving condition that we consider here is the positivity of the derivative function of order k. We present an algorithm to compute the resulting function and we show its convergence. Some convergence theorems are established. The error is of order o(1/N) k+1 ,w hereN is the number of the Lagrangian data. Finally, we analyze some numerical and graphical examples.  2001 IMACS. Published by Elsevier Science B.V. All rights reserved.

Approximation of curves by fairness splines with tangent conditions

We present in this paper an approximation method of curves from sets of Lagrangian data and vectorial tangent subspaces. We define a discrete smoothing fairness spline with tangent conditions by minimizing certain quadratic functional on finite element spaces. Convergence theorem is established and some numerical and graphical examples are analyzed in order to show the validity and the effectiveness of this paper.