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LISSA: 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 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 subroutine of the CP2K simulation package for molecular dynamics, see paper), in Image Processing (algorithms for image retrieval by colour indexing), in Electromagnetic Modelling (see paper); moreover, it has been added in the Chebfun2 package. One of the key features of the Padua Points is that they lie on a particular Lissajous curve. The more general topic of multivariate Polynomial Approximation on Lissajous Curves turned out to be of interest in the emerging field of Magnetic Particle Imaging (MPI) (see, e.g., some recent publications below and the activities of the scientific network MathMPI). Lissajous sampling seems to be relevant also in the field of Atomic Force Microscopy (AFM).

Papers

  1. Lissajous sampling and spectral filtering in MPI applications: the reconstruction algorithm for reducing the Gibbs phenomenon
    Extended Abstract submitted to SampTA 2017 - S. De Marchi, W. Erb and F. Marchetti
  2. Spectral filtering for the reduction of the Gibbs phenomenon in MPI applications by Lissajous sampling
    draft - S. De Marchi, W. Erb and F. Marchetti
  3. 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
  4. A Simple Recipe for Modelling a d-cube by Lissajous curves
    L. Bos
    Dolomites Res. Notes Approx. DRNA 10 (2017), 1--4
  5. Trivariate polynomial approximation on Lissajous curves
    preprint - L. Bos, S. De Marchi and M. Vianello
    IMA J. Numer. Anal. 37 (2017), 519--541
  6. On certain multivariate Vandermonde determinants whose variables separate
    preprint - S. De Marchi and K. Usevich
    Linear Algebra Appl. 449 (2014), 17--27
  7. 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
  8. 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
  9. 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
  10. 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)
  11. Nontensorial Clenshaw-Curtis cubature
    preprint - A. Sommariva, M. Vianello and R. Zanovello
    Numer. Algorithms 49 (2008), 409--427
  12. 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
  13. 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
  14. 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
  15. 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


Third-party codes