TAA AA 2015-2016
Corso di Teoria dell'Approssimazione e Applicazioni
"Approssimazione polinomiale multivariata ed
introduzione alle funzioni radiali di base e metodi kernel"

Le lezioni del corso di Teoria dell'Approssimazione e Applicazioni AA 2015-16, tenute dal prof. Stefano De Marchi quest'anno saranno svolte in INGLESE per la presenza di studenti stranieri

  • Class: Monday 14:30-16:15, Aula 2AB45
  • Class: Tuesday 14:30-16:15, Aula 2AB45
  • Lab: Monday 11:30-13:15 at the NumLab (Numerical Lab) 4th floor.

    nei giorni di seguito indicati

  • October 2015
    5, 6, 12, 13, 19, 20, 26, 27: class
    12, 19, 27 : lab

  • Novembre 2015
    [2, 3] (*), 9,10,16,17,23,34,30: class
    9, 16, 30: lab
    (*) The lectures of 2 and 3 November will be done later during the semester.

  • December 2015
    1 (**) , 7 (**) , 14, 15 : class
    14: lab
    (**) The lecture of December 1st is cancelled by the academic governing bodies for a student's assembly. The lectures of December 7th are cancelled too because the whole University is closed for vacation.

  • January 2016
    7 (***) , 11, 12, 18, 19: class
    11, 18: lab
    (***) The lecture will be done in 2AB40 from 11.30 to 13.30.


  • 5 October 2015. Introduction to the topics of the course. 1-dimensional interpolation: Vandermonde system, Lagrange form and barycentric form. Interpolation error.
  • 6 October 2015. Bernstein's approximant and Weirstrass's theorem. Properties of Bernstein polynomials and Bernstein approximant. Modulo of Continuity and its properties.
  • 12 October 2015. Jackson's theorem and some corollaries. Best polynomial approximant: characterization and computation. Remez algorithm.
  • 13 October 2015. Lebesgue constant: definition and some properties. Analysis of Lebegsue constants for equispaced, Chebyshev (classical, extended, extrema), Jacobi and Fekete points.
  • 19 October 2015. Leja sequences and optimal interpolation points. Haar space and Haar-Meihuber-Curtis theorem for multivariate polynomial interpolation. Unisolvency. Though the construction of good points on the square. Dubiner metric.
  • 20 October 2015. Dubiner metric on the square. Points equally distributed w.r.t. the Dubiner metric: Morrow-Patterson and Padua points. Padua points definitions and some properties.
  • 26 October 2015. Reproducing kernels and properties. Padua points: how to construct the elementary Lagrange polynomials and the interpolant. Implementation issues.
  • 27 October 2015. Padua points: computational aspects both for the construction of the interpolant and the cubature formulas. Admissible and Weakly Admissible Meshes. Definition and the 10 properties.
  • 9 November 2015. WAMs: constructions on 2dimensional and 3dimensional domains.
  • 10 November 2015. Introduction to RBF via splines. Polynomial splines: definition and construction. B-splines.
  • 16 November 2015. From splines to RBF. Interpolation problem with RBF. Distance matrices and their invertibility.
  • 17 November 2015. Positive Definite Functions (PD): definition, integral representation (Bochner theorem) and properties. Radiality.
  • 23 November 2015. Examples of (S)PD functions: Gaussian, Laguerre, Poisson, Matérn, truncated powers, potentials and integration w.r.t. kernel. Completely Mnotone (CM) functions: definitions, examples and properties. Characterization's theorems.
  • 24 November 2015. Multiply Monotone (MM) functions: definition and their properties. Summary of integral representations of PD, CM and MM functions. Reproduction of constants by means of Conditionally Positive Definite functions. Existence and uniqueness of the solution of the corresponding linear system.
  • 30 November 2015. Conditionally Positive Definite (CPD) Functions : definition, characterization and properties. Examples: generalized multiquadrics, power functions and thin plate splines. Connection between CPD, completely monotone and multiply monotone functions.
  • 14 December 2015. Strictly positive definite functions of order 1. Operators "de monteé" and "descente". Compactly supported functions of Wendland, Wu, Gneiting, "Euclid hat" and Buhmann: definitions, construction and examples.
  • 15 December 2015. Reproducing kernel Hilbert spaces and RBF. Native spaces. Power function: definition(s), computation and error pointwise error estimates. Error estimate based in the fill-distance for SPD kernels.
  • 7 January 2016. Bounds on the condition number of RBF matrices. The trade-off principle. Stability as function of the shape parameter. Methods for finding the "optimal" shape parameter: trial and error, power function, cross-validation and contour Padé.
  • 11 January 2016.Optimality principles of RBF. Least-squares and Moving Least-Square (MLS) with RBF approximants.
  • 12 January 2016.Some recalls about least-squares and wighted LS. More about MLS and Backus-Gilbert (BG) approach for quasi-interpolation.
  • 18 January 2016.Equivalence between MLS e BG. Examples: Shepard's methods and some extension. MLS error analysis. Conditioning and change of basis.
  • 19 January 2016. Again on the change of basis. Generalized Hermite interpolation. Solution of elliptic PDEs with Kansa (non symmetric) and Fasshauer (symmetric) methods. Some comments on the Galerkin method with RBF.

    Lab exercises

  • 12 October 2015. exercises .
  • 19 October 2015. exercises .
  • 26 October 2015. exercises .
  • 9 November 2015. In this lab we continue to work on the last lab exercises.
  • 16 November 2015. exercises .
  • 30 November 2015. exercises .
  • 14 December 2015. exercises .
  • 11 January 2016. exercises .
  • 18 January 2016. exercises .
    Final test
    The final test consists of two parts: written test and oral discussion.
    At the oral, students must present the solution of lab excercises.

    Dates and rooms
    08/02/2016 11:00-12:30 2AB45. Test.
    24/02/2016 10:00-13:00 2AB45 Test.
    27/06/2016 10:00-13:00 2AB45
    25/07/2016 10:00-13:00 2AB45
    The last exam: TBA

    Stefano De Marchi: Lectures on multivariate polynomial interpolation
    Stefano De Marchi: Four lectures on radial basis functions
    Gregory E. Fasshauer: Meshfree Approximation Methods with MATLAB
    Gregory E. Fasshauer and Michael Mc Court: Kernel-based Approximation Methods using Matlab

    (last update: 19 January 2016).