#
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.

Return to Peter Alfeld's Home Page.

###
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.

[16-Aug-1996]