Rossana Morandi

Monotone and convex cubic spline interpolation

Given a set of monotone and convex data, we present a necessary and sufficient condition for the existence of cubic differentiable interpolating splines which are monotone and convex. Further, we discuss their approximation properties when applied to the interpolation of functions having preassigned degree of smoothness.

Piecewise monotone quadratic histosplines

A method is presented for constructing a quadratic spline function satisfying area-matching conditions and local monotonicity constraints, according to the frequencies on the class intervals and to the shape of a given histogram. Such a function is “as close as possible” to the quadratic spline that satisfies the area-matching conditions and the minimum curvature property and generally exhibits a visually pleasing graph.

Local hybrid approximation for scattered data fitting with bivariate splines

We suggest a local hybrid approximation scheme based on polynomials and radial basis functions, and use it to improve the scattered data fitting algorithm of (Davydov, O., Zeilfelder, F., 2004. Scattered data fitting by direct extension of local polynomials to bivariate splines. Adv. Comp. Math. 21, 223-271). Similar to that algorithm, the new method has linear computational complexity and is therefore suitable for large real world data. Numerical examples suggest that it can produce high quality artifact-free approximations that are more accurate than those given by the original method where pure polynomial local approximations are used.

An algorithm for computing shape-preserving cubic spline interpolation to data

We present an algorithm for the construction of shape-preserving cubic splines interpolating a set of data point. The method is based upon some existence properties recently developed. Graphical examples are given.

Piecewise cubic monotone interpolation with assigned slopes

An algorithm to construct a monotonicity preserving cubicC1 interpolant without modification of the assigned slopes is proposed. AnO(h4) convergence result is obtained when exact function and derivative values are available andO(hp) convergence can be obtained withp=min(4,q) forO(hq) accurate function and derivative values. Numerical experiments carried out on data coming from functions with very different behaviours are presented. The results show that the method can interpolate monotone data in a visually pleasing way, even for data which present rapid variations.ZusammenfassungEs wird ein Algorithmus angegeben, der ohne Modifikation der vorgeschriebenen Ableitungen eine monotonieerhaltende kubischeC1-Interpolierende erzeugt. Wenn die Funktions- und Ableitungswerte exakt sind, erhält manO(h4)-Konvergenz, undO(hp)-Konvergenz mitp=min(4,q) beiO(hq)-genauen Funktions- und Ableitungswerten. Es werden numerische Experimente mit Daten von Funktionen sehr verschiedenen Verhaltens präsentiert. Man erkennt, daß das Verfahren monotone Daten auf visuell befriedigende Weise interpolieren kann, sogar im Fall starker Variation der Daten.

Piecewise C 1 -shape-preserving Hermite interpolation

A local scheme for piecewiseC1-Hermite interpolation is presented. The interpolant is obtained patching together cubic with quadratic polynomial segments; it is co-monotone and/or co-convex with the data. Under appropriate assumptions the method is fourth-order accurate.ZusammenfassungEin lokaler Ansatz zur stückweisenC1-Hermite-Interpolation wird vorgestellt. Die Interpolierende besteht aus zusammengesetzten kubischen und quadratischen Segmenten; sie erhält die Monotonie und/oder die Konvexität der Daten. Unter geeigneten Voraussetzungen approximier sie von vierter Ordnung.

An algebraic-elliptic algorithm for boundary orthogonal grid generation

To produce grids conforming to the boundary of a physical domain with boundary orthogonality features, algebraic methods like transfinite interpolating schemes can be profitably used. Moreover, the coupling of Hermite-type (also interpolating prescribed boundary direction) schemes with elliptic methods turns out to be effective to overcome the drawback of both algebraic and elliptic strategies. Thus, in this paper, we present an algorithm for the generation of boundary orthogonal grids which couples a mixed Hermite algebraic method with a boundary orthogonal elliptic scheme. Numerical tests on domains with classical geometries show satisfactory performances of the algorithm and coupling effectiveness in achieving grid boundary orthogonality and smoothness.

Cubic Spline Data Reduction Choosing the Knots from a Third Derivative Criterion

This paper presents a data reduction method for functional data. Starting with noisy or not noisy data, we first define a function f, called the “reference function”, as a cubic smoothing spline which is supposed to have the global form of the data. This reference function is then used to locate the knots of the final approximating spline by using a criterion based on the third derivative of f. Then, the least-squares spline approximating all the data is derived with these knots. Numerical results show the effectiveness of the method.

Locally tensor product functions in algebraic grid optimization

Abstract This paper deals with an a posteriori local improvement of numerical grids generated trough a new algebraic mixed scheme. The improvement is achieved by an optimization strategy where only a small number of new variables is used to provide an “optimal” algebraic grid. The proposed new mixed scheme is based on a special family of functions that is tensor product functions in sub-regions, and it controls the position of the grid points by moving a subset of control points of the scheme.

P 1 -reproducing shape-preserving quasi-interpolant using positive quadratic splines with local support

In this paper a strategy is presented to construct a shape-preserving quasi-interpolant function expressed as a linear combination of quadratic splines with local support where the coefficients are given by the data. The quasi-interpolant is shown to be linear-reproducing, monotone and/or convex conforming to the data.