Peter Alfeld, --- Department of Mathematics, --- College of Science --- University of Utah

Recent Papers by Peter Alfeld

You may also view a complete bibliography.

General Information

Following are brief paragraphs on some of my recent papers with links to postscript versions. These are also available via anonymous ftp (or send me e-mail).

The abstracts below are quoted almost verbatim from the paper. The papers are accessible as dvi and as ps files. The dvi files may be more legible over the network, but they do not contain figures.

Here are links to the home pages of some of my coworkers:

On the Dimensions of Piecewise Polynomial Functions

dvi file or postscript file. In D.E. Griffiths and G.A. Watson (eds), Numerical Analysis, Pitman Research Notes in Mathematics Series, No. 140, pp. 1-23, Proceedings of the Biennial Dundee Conference on Numerical Analysis, June 25-28, 1985, Langman Scientific and Technical. Lower bounds are given on the dimension of spaces of piecewise polynomial C1 and C2 functions defined on a tessellation of a polyhedral domain into tetrahedra. The analysis technique consists of embedding the space of interest in a larger space with a simpler structure, and then making appropriate adjustments. In the bivariate case, this approach reproduces the well-known bounds derived by Schumaker.

The Generic Dimension of the space of C1 splines of degree d >=8 on tetrahedral decompositions.

dvi file or postscript file . With Larry Schumaker and Walter Whiteley, SIAM JNA, v.~30, pp.~889--920, 1993. We consider the linear space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain which has been partitioned into tetrahedra. Combining Bernstein-Bezier methods and combinatorial and geometric techniques from rigidity theory, we give an explicit expression for the generic dimension of this space for sufficiently large polynomial degrees d >= 8. This is the first general dimension statement of its kind.

Upper and Lower Bounds on the Dimension of Multivariate Spline Spaces.

dvi file or postscript file . To appear in the SIAM Journal on Numerical Analysis. We give upper and lower bounds on the dimensions of multivariate spline spaces defined on triangulations in Rk. The bounds are optimal in a certain sense.

Scattered Data Interpolation in Three or More Variables

dvi file or postscript file . In Tom Lyche and Larry L.~Schumaker (eds), ``Mathematical Methods in Computer Aided Geometric Design'', Academic Press, 1989, 1--34. This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.

Bernstein-Bezier Polynomials on Spheres and Sphere-Like Surfaces

dvi file or postscript file . With Marian Neamtu and Larry Schumaker, to appear in the CAGD journal. In this paper we discuss a natural way to define barycentric coordinates on general sphere-like surfaces. This leads to a theory of Bernstein-Bezier polynomials which parallels the familiar planar case. Our constructions are based on a study of homogeneous polynomials on trihedra in R3. The special case of Bernstein-Bezier polynomials on a sphere is considered in detail.

Note: To see on-line images of the graphs of the Bernstein-Bézier functions click here .

Dimension and Local Bases of Homogeneous Spline Spaces

dvi file or postscript file . With Marian Neamtu and Larry Schumaker, to appear. Recently, we have introduced spaces of splines defined on triangulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein-Bezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangulations. Rather than working with splines on sphere-like surfaces directly, we instead investigate more general spaces of homogeneous splines in R3. In particular, we present formulae for the dimensions of such spline spaces, and construct locally supported bases for them.

Circular Bernstein-Bezier Polynomials

dvi file or postscript file . With Marian Neamtu and Larry Schumaker, to appear, in Mathematical Methods for Curves and Surfaces, M. Daehlen, T. Lyche, and L. Schumaker (eds.) Vanderbilt University Press, Nashville, 1995. We discuss a natural way to define barycentric coordinates associated with circular arcs. This leads to a theory of Bernstein-Bezier polynomials which parallels the familiar interval case, and which has close connections to trigonometric polynomials.

Fitting Scattered Data on Sphere-Like Surfaces using Spherical Splines

dvi file or postscript file . With Marian Neamtu and Larry Schumaker, to appear in the Journal of Computational and Applied Mathematics. Spaces of polynomial splines defined on planar triangulations are very useful tools for fitting scattered data in the plane. Recently, using homogeneous polynomials, we have developed analogous spline spaces defined on triangulations on the sphere and on sphere-like surfaces. Using these spaces, it is possible to construct analogs of many of the classical interpolation and fitting methods. Here we examine some of the more interesting ones in detail. For interpolation, we discuss macro-element methods and minimal energy splines, and for fitting, we consider discrete least squares and penalized least squares.

Click here to view some (otherwise unpublished) examples of our interpolants.

Dimensions of Multivariate Spline Spaces

postscript file. For many years, I have been maintaining a collection of actually computed dimensions of multivariate spline spaces. Some of these are listed in this set of notes. You are welcome to download the postscript file. If the examples help in your research I'd appreciate an acknowledgment and a note from you.

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Keywords for this page: multivariate splines, spline spaces, dimensions, interpolation, approximation, interpolation on the sphere, homogeneous splines, triangulations, finite elements, spherical splines, circular splines, sphere-like surfaces, tetrahedra, tetrahedral decompositions, spline, dimension, tetrahedron, Bernstein Polynomials, Bernstein-Bezier form.