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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.
People: S. De Marchi (coordinator), A. Martínez Calomardo, F. Marchetti , E. Perracchione, D. Poggiali (Padova)
Collaborators: M. Buhmann (Giessen), R. Cavoretto (Torino), A. De Rossi (Torino), B. Haasdonk (Stuttgart), A. Iske (Hamburg), M. Rossini (Milano), G. Santin (Stuttgart), R. Schaback (Goettingen), H. Wendland (Bayreuth).

Papers

  1. On the Convergence of the Rescaled Localized Radial Basis Function Method
    draft (July 2019), with H. Wendland. Appl. Math. Letters (2020) online
  2. Analysis of a new class of rational RBF expansions
    by M. Buhmann, S. De Marchi and E. Perracchione, Accepted by IMA J. Numer. Anal. (2018)
  3. Recursive POD expansion for advection-diffusion-reaction equation
    by M. Azaïez, T. Chácon Rebollo, E. Perracchione, J. M. Vega,
    to appear on Comm. Comput. Physics, 2018, doi: 10.4208/cicp.OA-2017-0257.
  4. Interpolation with uncoupled separable matrix-valued kernels by D. Wittwar, G. Santin, B. Haasdonk
    Dolomites Res. Notes Approx. 11, Special Issue SPAN2018, pp. 23--39 (2018)
  5. Greedy kernel methods for accelerating implicit integrators for parametric ODEs
    by T. Brünnette, G. Santin, and B. Haasdonk
    Proceedings of ENUMATH 2017. (2018)
  6. RBF-based partition of unity methods for elliptic PDEs: Adaptivity and stability issues via VSKs
    by S. De Marchi, A. Martinez, E. Perracchione and M. Rossini,
    J. Sci. Comput. 79(1) (2019), pp. 321-344. online version
  7. Fast and stable rational RBF-based Partition of Unity interpolation
    by S. De Marchi, A. Martinez, E. Perracchione. J. Comput. Appl. Math. 349 (2019), pp. 331-343 online
  8. Image Reconstruction from Scattered Radon Data by Weighted Positive Definite Kernel Functions
    by S. De Marchi, A. Iske and G. Santin, Calcolo 55(2) (2018), https://doi.org/10.1007/s10092-018-0247-6.
  9. 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.
  10. Optimal selection of local approximants in RBF-PU interpolation using bivariate LOOCV
    arXiv preprint 1703.04282 - R. Cavoretto, A. De Rossi and E. Perracchione
    J. Sci. Comput., to appear
  11. A rescaled method for RBF approximation
    by S. De Marchi, A. Idda and G. Santin
    Springer Proceedings on Mathematics and Statistics, Vol. 201 (2017), pp.39--59.
  12. 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
  13. 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
  14. Approximation of Eigenfunctions in Kernel-based Spaces
    by G. Santin and R. Schaback
    Adv. Comput. Math. 42(4) (2016), pp. 973--993.
  15. 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.
  16. 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.
  17. 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.
  18. 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".
  19. Nonstandard Kernels and their Applications
    by S. De Marchi and R. Schaback
    Dolomites Res. Notes on Approx. (DRNA) Vol. 2, (2009), pp. 16--43.
  20. 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.
  21. 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

  22. A stable meshfree PDE solver for source-type flows in porous media
    draft (Dec. 2018) with R. Campagna, S. Cuomo, E. Perracchione and G. Severino. Accepted by Appl. Num. Math.
  23. Learning with subsampled kernel-based methods: Environmental and financial applications
    by M. Aminian Shahrokhabadi, A. Neisy, E. Perracchione, M. Polato, Dolomites Res. Notes Approx. 12(2019), pp. 17-27.
  24. Greedy Kernel Approximation for Sparse Surrogate Modelling
    by B. Haasdonk and G. Santin
    . in Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing,
    W. Keiper, A. Milde, and S. Volkwein, Eds. Cham: Springer Int. Pub., 2018, pp. 21--45.
  25. Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario
    by M. Köppel, F. Franzelin, I. Kröker, G. Santin, D. Wittwar, S. Oladyshkin, A. Barth, B. Haasdonk, W. Nowak, D. Pflüger, C. Rohde,
    Accepted for publication in Comput. Geosci.
  26. Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods,
    by T. Köppl, G. Santin, B. Haasdonk, R. Helmig
    Int. J. Numer. Meth. Biomed. Engng., vol 34, 8 (2018), pg. e3095
  27. Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects
    by E. Perracchione, arXiv preprint arXiv:1802.01842, 2018. To appear in Comput. Appl. Math.
  28. 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

  29. On the Convergence of the Rescaled Localized Radial Basis Function Method
    by S. De Marchi and H. Wendland (May 2019)
  30. Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in MPI
    by S. De Marchi, W. Erb, F. Marchetti, E. Perracchione and M. Rossini (March 2109)
  31. A stable meshfree PDE solver for source-type flows in porous media
    R. Campagna, S. Cuomo, S. De Marchi, E. Perracchione and G. Severino, submitted (Dec. 2018)
  32. Greedy regularized kernel interpolation
    G. Santin, D. Wittwar, B. Haasdonk, submitted (2018).
  33. Jumping with Variably Scaled Discontinuos Kernels (VSDK)
    by S. De Marchi, F. Marchetti and E. Perracchione. submitted (Aug. 2018)

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 and E. Perracchione (2015), pp. 112

    Presentations

    1. S. De Marchi: New developments on rational RBF
    2. 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.
    3. 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.
    4. G. Santin: A fast algorithm for computing a truncated orthonormal basis for RBF native spaces
      Multivariate Approximation, Verona 29-30 November, 2013.
    5. 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