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Marco Vianello: work on Padua Points and Lissajous sampling
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 studied and implemented during some collaboration
periods
at the University
of Padua and Verona within a group formed by Len Bos (Calgary--Verona),
Marco Caliari (Verona),
Stefano De Marchi (Verona--Padua), Alvise Sommariva (Padua),
Marco Vianello (Padua), Shayne Waldron (Auckland)
and Yuan Xu (Eugene).
One of the key features of the Padua Points is that they lie
on a particular Lissajous curve, therefore inserting in the framework
of multivariate approximation by Lissajous sampling.
Lagrange interpolation at the Padua points has been recently used
in several scientific and technological applications, for example in
Computational Chemistry (the
Fun2D subroutine of the
CP2K
simulation package for Molecular Dynamics, see
paper), in Image
Processing (algorithms for
image
retrieval
by colour indexing), in
Materials Science (Modelling of Composite Layered Materials,
see paper), in
Mathematical Statistics (Copula Density Estimation, see abstract by L.
Qu, p. 67), in Quantum Physics (Quantum State Tomography, see paper);
moreover,
it has been
added in the
Chebfun2 package.
Papers
-
Polynomial approximation on Lissajous curves in the d-cube
preprint - L. Bos, S. De Marchi and M. Vianello
Appl. Numer. Math. 116 (2017), 47--56
- Trivariate
polynomial approximation on Lissajous curves
preprint - L. Bos, S. De Marchi and M. Vianello
IMA J. Numer. Anal. 37 (2017), 519--541
-
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
-
A numerical code for fast interpolation and cubature at
the Padua points
preprint - M. Caliari, S. De Marchi, A. Sommariva and M. Vianello
Proceedings of the 9th CMMSE (2009), Vol. I,
218--228
-
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)
-
Nontensorial
Clenshaw-Curtis cubature
preprint - A. Sommariva, M. Vianello and R. Zanovello
Numer. Algorithms 49 (2008), 409--427
-
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
-
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
-
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 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
Codes
- Padua2D
(Fortran 77 code for
interpolation
at
Padua-like points
on rectangles, triangles and ellipses); the code is also in the Netlib
by M. Caliari, S. De Marchi and
M. Vianello (see paper) -
ACM Trans. Math.
Software 35-3 (2008)
note: a variant has been used in
Fun2D of the
CP2K
simulation package for molecular dynamics
by M. Guidon, J. Hutter and J. VandeVondele (see
paper)
- Padua2DM
(a Matlab/Octave code for interpolation and cubature at the Padua
points); the code is also in the Netlib
by M. Caliari, S. De Marchi, A. Sommariva and M.
Vianello (see paper) -
Numer. Algorithms 56
(2011)
-
hyperlissa
(Matlab code for hyperinterpolation on a Lissajous curve of the cube)
with a
demo
by S. De Marchi and M.
Vianello (see paper)
Third-party codes