Rudolf M.J. van Damme

Convexity preservation of the four-point interpolatory subdivision scheme

In this note we examine the convexity preserving properties of the (linear) four-point interpolatory subdivision scheme of Dyn, Gregory and Levin when applied to functional univariate strictly convex data. Conditions on the tension parameter guaranteeing preservation of convexity are derived. These conditions depend on the initial data. The resulting scheme is the four-point scheme with tension parameter bounded from above by a bound smaller than 1/16. Thus the scheme generates C1 limit functions and has approximation order two.

Shape Preserving Interpolatory Subdivision Schemes for Nonuniform Data

This article is concerned with a class of shape preserving four-point subdivision schemes which are stationary and which interpolate nonuniform univariate data {(xi, fi)}. These data are functional data, i.e., xi xj if i j. Subdivision for the strictly monotone x-values is performed by a subdivision scheme that makes the grid locally uniform. This article is concerned with constructing suitable subdivision methods for the f-data which preserve convexity; i.e., the data at the kth level, {x^(^k^)i, fi(k)} is a convex data set for all k provided the initial data are convex. First, a sufficient condition for preservation of convexity is presented. Additional conditions on the subdivision methods for convergence to a C^1 limit function are given. This leads to explicit rational convexity preserving subdivision schemes which generate continuously differentiable limit functions from initial convex data. The class of schemes is further restricted to schemes that reproduce quadratic polynomials. It is proved that these schemes are third order accurate. In addition, nonuniform linear schemes are examined which extend the well-known linear four-point scheme to the case of nonuniform data. Smoothness of the limit function generated by these linear schemes is proved by using the well-known smoothness criteria of the uniform linear four-point scheme.

Monotonicity preserving interpolatory subdivision schemes

A class of local nonlinear stationary subdivision schemes that interpolate equidistant data and that preserve monotonicity in the data is examined. The limit function obtained after repeated application of these schemes exists and is monotone for arbitrary monotone initial data. Next a class of rational subdivision schemes is investigated. These schemes generate limit functions that are continuously differentiable for any strictly monotone data. The approximation order of the schemes is four. Some generalisations, such as preservation of piecewise monotonicity and application to homogeneous grid refinement, are briefly discussed.

Bivariate Hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in

Curve interpolation with constrained length

mathbb{R}^3

Transient periodic behaviour related to a saddle-node bifurcation

is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points

A Linear Approach to Shape Preserving Spline Approximation

{(x_i, y_i)}_{i=1}^N

Complexes of block copolymers in solution: a graph-theoretical approach

from which none of the pairs

Complexes of block copolymers in solution: tree approximation

(x_i,y_i)

Influence of Stochastic Contact Resistances on Coupling Losses

and