Lattices on simplicial partitions which are not simply connected

In this paper, (d+1)-pencil lattices on simplicial partitions in R^d, which are not simply connected, are studied. It is shown, how the fact that a partition is not simply connected can be used to increase the flexibility of a lattice. A local modification algorithm is developed also to deal with slight partition topology changes that may appear afterwards a lattice has already been constructed.     

Interpolation lattices in several variables

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

Multivariate polynomial interpolation under projectivities II: Neville-Aitken formulas

This is the second part of a note on interpolation by real polynomials of several real variables. For certain regular knot systems (geometric or regular meshes, tensor product grids), Neville-Aitken algorithms are derived explicitly. By application of a projectivity they can be extended in a simple way to arbitrary (k+1)-pencil lattices as recently introduced by Lee and Phillips. A numerical example is given.Partially supported by CICYT Res. Grant PS87-0060.Travel Grant Programa Europa 1991, CAI Zaragoza.     

Transfinite interpolation on the medians of a triangle and best L 1-approximation

Let be a triangle in and let be the set of its three medians. We construct interpolants to smooth functions using transfinite (or blending) interpolation on The interpolants are of type f(₁)+g(₂)+h(₃), where (₁,₂,₃) are the barycentric coordinates with respect to the vertices of . Based on an error representation formula, we prove that the interpolant is the unique best L¹-approximant by functions of this type subject the function to be approximated is from a certain convexity cone in C³().Received: 17 December 2003     

Computational aspects of multivariate polynomial interpolation: Indexing the coefficients

An algorithm is derived for generating the information needed to pass efficiently between multi-indices of neighboring degrees, of use in the construction and evaluation of interpolating polynomials and in the construction of good bases for polynomial ideals. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the generation of symmetric Lebesgue-like points in the triangle

We compute point sets on the triangle that have low Lebesgue constant, with sixfold symmetries and Gauss–Legendre–Lobatto distribution on the sides, up to interpolation degree 18. Such points have the best Lebesgue constants among the families of symmetric points used so far in the framework of triangular spectral elements.     

On Lagrange and Hermite interpolation in R k

Summary A method for the construction of a set of data of interpolation in several variables is given. The resulting data, which are either function values or directional derivatives values, give rise to a space of polynomials, in such a way that unisolvence is guaranteed. The interpolating polynomial is calculated using a procedure which generalizes the Newton divided differences formula for a single variable.     

Bivariate Hermite-Birkhoff interpolation and Vandermonde determinants

The concepts of Vandermonde determinant and confluent Vandermonde determinant are extended to the multidimensional setting by relating them to multivariate interpolation problems. With an approach different from that of other recent papers on this subject, the values of these determinants are computed, recovering and extending the results of those papers.Partially supported by Research Grant PS900121 DGICYT.     

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on Ω-splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.     

On computational aspects of simplicial splines

Some new results on multivariate simplex B-splines and their practical application are presented. New recurrence relations are derived based on [2] and [15]. Remarks on boundary conditions are given and an example of an application of bivariate quadratic simplex splines is presented. The application concerns the approximation of a surface which is constrained by a differential equation.Communicated by Charles Micchelli.     

Error bounds for a convexity-preserving interpolation and its limit function

Error bounds between a nonlinear interpolation and the limit function of its associated subdivision scheme are estimated. The bounds can be evaluated without recursive subdivision. We show that this interpolation is convexity preserving, as its associated subdivision scheme. Finally, some numerical experiments are presented.     

Least-squares polynomial approximation on weakly admissible meshes: Disk and triangle

We construct symmetric polar WAMs (weakly admissible meshes) with low cardinality for least-squares polynomial approximation on the disk. These are then mapped to an arbitrary triangle. Numerical tests show that the growth of the least-squares projection uniform norm is much slower than the theoretical bound, and even slower than that of the Lebesgue constant of the best known interpolation points for the triangle. As opposed to good interpolation points, such meshes are straightforward to compute for any degree. The construction can be extended to polygons by triangulation.     

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

Transfinite mean value interpolation in general dimension

Mean value interpolation is a simple, fast, linearly precise method of smoothly interpolating a function given on the boundary of a domain. For planar domains, several properties of the interpolant were established in a recent paper by Dyken and the second author, including: sufficient conditions on the boundary to guarantee interpolation for continuous data; a formula for the normal derivative at the boundary; and the construction of a Hermite interpolant when normal derivative data is also available. In this paper we generalize these results to domains in arbitrary dimension.     

Piecewise complementary Lidstone interpolation and error inequalities

The purpose of this paper is to develop piecewise complementary Lidstone interpolation in one and two variables and establish explicit error bounds for the derivatives in L_∞ and L₂ norms.     

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C¹ cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.     

On the bivariate Shepard-Lidstone operators

We propose a new combination of the bivariate Shepard operators (Coman and Trîmbi?a?, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given (Kraaijpoel and van Leeuwen, 2010 [13]). Numerical comparison with other well known combinations is presented.     

Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ?(x;α,h)=exp(−α²(x/h)²): in the limit h→0 for fixed α, the error does not converge to zero, but rather to E_S(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E_S(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α²)‖f‖_∞ — huge compared to the O(2π/α²)exp(−π²/[4α²]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α?1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π²/[4α²])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then .     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.