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CAA: the Padua Points
The Padua points are the first known example of optimal points for
total degree polynomial interpolation
in two variables,
with a Lebesgue constant increasing like log square of the degree.
They have been discovered and studied by our group during some collaboration
periods
at the University
of Padua
with Len Bos (Calgary), Shayne Waldron (Auckland)
and Yuan Xu (Eugene).
Lagrange interpolation at the Padua points has been recently used
in some scientific and technological applications, for example in
Computational Chemistry (the Fun2D
tool of the
CP2K
simulation package for molecular dynamics, see
paper), and in Image
Processing (algorithms for
image
retrieval
by colour indexing).
- Brief description of
the Padua points, with figure.
- Poster on interpolation and cubature at
the Padua points.
- Software for
interpolation and cubature at the Padua
points.
Publications, preprints, abstracts and codes concerning or related to
the Padua points
-
Bivariate polynomial
interpolation on the square at new nodal sets
preprint - M. Caliari, S. De Marchi and M. Vianello
Appl. Math. Comput. 165/2 (2005), 261--274
-
Bivariate Lagrange interpolation at the
Padua points: the
generating curve approach
preprint - L. Bos, M. Caliari, S. De Marchi, M. Vianello and Y. Xu
J. Approx. Theory 143 (2006), 15--25
-
Bivariate Lagrange interpolation at the
Padua points:
computational aspects
M. Caliari (with S. De Marchi, R. Montagna and M. Vianello)
poster presented at the "1st Dolomites Workshop on Constructive
Approximation and Applications"
(Alba di Canazei, Trento (Italy),
September 2006)
-
Bivariate Lagrange interpolation at the
Padua points: the
ideal theory approach
preprint - L. Bos, S. De Marchi, M. Vianello and Y. Xu
Numer. Math. 108 (2007), 43--57
-
Hyperinterpolation at Xu points and
interpolation at Padua points in the square:
computational
aspects
graduate
thesis - R. Montagna (supervisors S. De Marchi
and M. Caliari)
University of Verona, 2007
-
Some Recent
Advances in Multivariate
Polynomial Interpolation
J.M. Carnicer and M. Gasca
International Conference of
Numerical Analysis and Applied Mathematics. AIP Conference Proceedings,
Volume 936, pp. 7-9 (2007)
-
Exact order of the Lebesgue constants for bivariate Lagrange
interpolation at certain node systems
B. Della Vecchia, G. Mastroianni and P. Vertesi
Studia Sci. Math. Hungar., published online 2 September 2008
-
Bivariate Lagrange interpolation at the Padua
points:
computational aspects
preprint - M. Caliari, S. De Marchi and M. Vianello
J. Comput. Appl. Math. 221 (2008), 284--292
-
Nontensorial Clenshaw-Curtis cubature
preprint - A. Sommariva, M. Vianello and R. Zanovello
Numer. Algorithms 49 (2008), 409--427
-
Algorithm 886: Padua2D: Lagrange Interpolation at Padua
Points on Bivariate
Domains
preprint - M. Caliari, S. De Marchi and M. Vianello
ACM Trans. Math. Software 35-3 (2008)
-
Histogram Compression and Image Retrieval Through Padua Points
Interpolation
R. Montagna and G. Finlayson (University of East Anglia, UK)
abstract of a talk presented at the "4th European Conference
on
Colour in
Graphics, Imaging, and Vision"
(Barcelona, June 2008)
-
Near-optimal
interpolation and
quadrature in two variables: the Padua points
M. Vianello (with M. Caliari, S. De Marchi, A. Sommariva)
poster presented at the "5th European Congress of
Mathematics" (Amsterdam, July 2008)
-
Efficient implementation of bivariate interpolation
and cubature at Padua points
M. Caliari (with S. De Marchi, A. Sommariva, M. Vianello)
abstract of a talk presented at the "SIMAI 2008 (in cooperation with SIAM)"
(Rome, September 2008)
-
Optimal interpolation and Lp norm minimisation in colour
indexing
G. Finlayson and R. Montagna (University of East Anglia, UK)
abstract of a poster presented at the
"Sixteenth Color Imaging Conference"
(Portland, November 2008)
-
Issues in multivariate polynomial interpolation
C. de Boor (Madison)
abstract of a talk presented at "Perspectives in Numerical
Analysis" (Helsinki, May 2008)
and
at the "7th Australia-New Zealand Mathematics Convention" (Christchurch, December 2008)
-
Padua points: theory, computation and applications
S. De Marchi
slides of a Guest Lecture at the CMA (Center of
Mathematics for Applications), University of Oslo (April 2009)
-
A numerical code for fast interpolation and cubature at
the Padua points
M. Caliari (with S. De Marchi, A. Sommariva and M. Vianello)
Proceedings of the 9th CMMSE (Gijon, Spain, July 2009), Editors: P.
Alonso, S. Oharu, E. Venturino and B. A. Wade, Vol. I,
218--228
-
On the Vandermonde Determinant of Padua-like Points
L. Bos, S. De Marchi and S. Waldron
Dolomites Research Notes on Approximation 2 (2009), 1--15
-
Robust Periodic Hartree-Fock Exchange for Large-Scale
Simulations Using Gaussian Basis Sets
M. Guidon, J. Hutter, J. VandeVondele
J. Chem. Theory Comput. 5 (2009), 3010--3021
-
A method to compute recurrence relation coefficients for bivariate
orthogonal polynomials by unitary matrix transformations
M. Van Barel and A. Chesnokov
Numer. Algorithms 55 (2010), 383--402
-
Bivariate Lagrange interpolation at the Chebyshev nodes
L.A. Harris
Proc. Amer. Math. Soc. 138 (2010), 4447--4453
-
Padua2DM: fast interpolation and cubature at the Padua points in
Matlab/Octave
preprint - M. Caliari, S. De Marchi, A. Sommariva and M. Vianello
Numer. Algorithms 56 (2011), 45--60
- Chebyshev
lattices, a unifying framework for cubature with Chebyshev weight
function
R. Cools and K. Poppe
BIT Numerical Mathematics 51 (2011), 275--288