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CAA: Stable kernel-based approximation and applications


Kernel-based approximation have become particularly popular in the last decades especially with the use of RBF (Radial Basis Functions). Our research is addressed primarily to the analysis and construction of stable bases and their use in image reconstruction from Radon data and meshless approximations.

Papers

  1. Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation
    by G. Santin and B. Haasdonk, Dolomites Res. Notes on Approx. 10 (2017), pp. 68-78.
  2. A rescaled method for RBF approximation
    by S. De Marchi, A. Idda and G. Santin, to appear on the proceedings of "Approximation Theory 15".
  3. Partition of unity interpolation using stable kernel-based techniques
    by R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione and G. Santin
    Appl. Numer. Math. 116 (2017), pp. 95-107 online, http://dx.doi.org/10.1016/j.apnum.2016.07.005
  4. Kernel-based Image Reconstruction from Scattered Radon Data
    by S. De Marchi, A. Iske and A. Sironi
    Dolomites Res. Notes on Approx. Vol 9 (2016), special issue for the workshop "Kernel-based methods and function approximation", pp. 19-31.
    available as Hamburger Beitraege zur Angewandten Mathematik 2016-11
  5. Approximation of Eigenfunctions in Kernel-based Spaces
    by G. Santin and R. Schaback
    Adv. Comput. Math. 42(4) (2016), pp. 973--993.
  6. RBF approximation of large datasets by partition of unity and local stabilization
    by R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione and G. Santin
    Proceedings CMMSE (2015), Vol. I-II-III-IV, pp. 317--326.
  7. Fast computation of orthonormal basis for RBF spaces through Krylov space methods
    by S. De Marchi and G. Santin
    BIT Numerical Mathematics 55(4) (2015), pp. 949--966.
  8. A new stable basis for radial basis function interpolation
    by S. De Marchi and G. Santin
    J. Comp. Appl. Math., Vol. 253 (2013), pp. 1--13.
  9. Stability of Kernel-Based Interpolation
    by S. De Marchi and R. Schaback
    Adv. Comput. Math. Vol. 32(2), 2010, p. 155-161
    Examples and more:
    these are examples and figures illustrating the results of the paper "Stability of Kernel-Based Interpolation".
  10. Nonstandard Kernels and their Applications
    by S. De Marchi and R. Schaback
    Dolomites Res. Notes on Approx. (DRNA) Vol. 2, (2009), pp. 16--43.
  11. Univariate Radial Basis Functions with Compact Support Cardinal Functions
    by L. Bos and S. De Marchi
    East J. Approx., Vol. 14(1) 2008, pp. 69--80.
  12. Near-Optimal Data-independent Point Locations for Radial Basis Function Interpolation
    by with R. Schaback and H. Wendland
    Adv. Comput. Math. 23(3) (2005), pp. 317--330.

    Papers on applications

  13. Approximating basins of attraction for dynamical systems via stable radial bases
    preprint - R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione and G. Santin
    AIP Conference Proceedings, 1738, 390003 (2016); doi:10.1063/1.4952177 online

    Submitted

  14. RBF-based partition of unity method for elliptic PDEs: Adaptivity and stability issues via VSKs
    by S. De Marchi, A. Martinez Calomardo, E. Perracchione and M. Rossini (June 2017).
  15. Greedy Kernel Approximation for Sparse Surrogate Modelling
    by B. Haasdonk and G. Santin (June 2017).
  16. Image Reconstruction from Scattered Radon Data by Weighted Positive Definite Kernel Functions
    by S. De Marchi, A. Iske and G. Santin (Jan. 2017)

Posters

Software

Books

Meshfree Approximation for Multi-Asset European and American Option Problems
by Stefano De Marchi, Maddalena Mandarà and Anna Viero
ISBN: 9788854851511 (2012), pp. 92.

Lecture Notes

  • Four Lectures on radial basis functions by S. De Marchi (2015), pp. 58.

    Presentations

    1. S. De Marchi: Kernel-based Image Reconstruction from scattered Radon data by (anisotropic) positive definite functions
      Kernel-based methods and function approximation - Department of Mathematics, University of Torino (Italy), February 5th, 2016.
    2. S. De Marchi: On a new orthonormal basis for RBF native spaces and its fast computation
      Colloqium at the Department of Mathematics, University of Torino (Italy), on June 11th, 2014.
    3. G. Santin: A fast algorithm for computing a truncated orthonormal basis for RBF native spaces
      Multivariate Approximation, Verona 29-30 November, 2013.
    4. S. De Marchi: On a new orthonormal basis for RBF native spaces
      San Diego (USA), SIAM Annual meeting: July 8th, 2013.

    PhD theses

    Master theses

    Degree theses