ATA 2017-18
Approximation theory and applications:
"Introduction to RBF and Kernel methods
and multivariate polynomial approximation "

  • Class: Wednesday 14:30-16:00, Room 2AB45
  • Class: Thursday 12:30-14:30, Room 1BC45
  • Lab: 5 Tuesdays starting on 21 November, in the NumLab . See below for details.

    The timetable of lectures can be found here.

    There will be some lectures done by prof. Martin Buhmann , University of Giessen, in the week 30/10-3/11/2017. The schedule and the rooms are as follows
    31 Oct. 2017: 10:30-12:00, room 2BC30: Introductory material
    2 Nov. 2017: 10:30-12:00, room 2BC30: Interpolation and Wavelets with Radial Basis Functions
    2 Nov. 2017: 12:45-14:15, room 1BC45: Quasi-interpolation
    3 Nov. 2017: 11:30-13:00, room 2BC30: New Results on Interpolation with Multiply Monotone Functions.


    First part

    October 4. Recall of some basic facts concerning polynomial interpolation in 1d: Lagrange polynomials, error bounds. Existence of best polynomial approximation, Lebesgue constant. Weierstrass theorem and Bernstein operator. (6)
    October 5. Stone_Weierstrass theorem and some consequences. Modulus of continuity. Existence and uniqueness of the best approximating polynomial. Properties of BAP (alternating set) and Remez algorithm. (4)
    October 11. Lebesgue constant and its fundamental properties. Growth of Lebesgue constant for equispaced points, Chebyshev points, Extended Chebyshev, Jacobi and Fekete points. (5)
    October 12. Leja sequences. Optimal interpolation points. Haar spaces and Haar-Mairhuber-Curtis theorem. Unisolvency. Chung-Yao points. Tensor product points and Padua points. (5)
    October 18. Padua points: theory and construction (5)
    October 19. Polynomial inequalities and (weakly) admissible meshes: constructions in 1 and 2 dimensions. (5)
    October 25. From splines to RBF. Radial basis function interpolation system and solution in a simple case. (5)
    October 26. Distance matrix and its invertibility. Positive definite matrices and functions. The case of gaussian and the cosine. Schoenberg's characterization. (4)
    October 31. (Buhmann). Introdution to RBF approximation: examples of bases functions. (4)
    November 2 (4h). (Buhmann). Interpolation and quasi-interpolation with RBF, wavelets, RBF compression and under tension. (4)
    November 3. (Buhmann). Multiply monotone and logaritmically monotone functions: recent results. (3)
    November 8. Radial basis functions interpolation: recall. Positive definite functions: characterization and properties. Examples: gaussian, gaussian-laguerre, poisson, matern. (5)
    November 9. Truncted power kernels, whittaker, integral kernels. Completely and k-times monotone functions: definitions, properties and examples. Polynomial reproduction and conditionally positive definite functions. (5)
    November 15. Conditionally positive definite matrices and functions. Examples of CPD functions: IM, Powers, TPS. (5)
    November 16. CPD functions of order 1. De monte' e de descente operators. Compactly supported functions: Wendland, Wu, Gneiting and Buhmann. (5)
    November 22. Reproducing Kernel Hilbert Spaces. Native spaces for RBF. Cardinal functions, power function and error estimation. (5)
    November 23. Error bounds involving the power function and the fill-distance. Trade-off principle. Bounds on the condition number. Trail and error, power function and LOOCV methods on changing the shape parameter. (4)
    November 29. Optimality principles of RBF interpolants. Least-squares and moving least-squares. The Shepard's method. (5)
    December 6. Backus-Gilbert approach and its equivalence to the moving least square. Construction of high order Shepard's methods. (5)
    December 7. Hermite interpolation. Generalized Hermite interpolation. Kansa unsymmetric collocation method for elliptic PDEs. Fasshauer symmetric method. Meshless Galerkin approach for elliptic pdes. (5)
    December 13. Partition of unity and its application in interpolation and collocation. Solution of time dependent PDES by the Method of Lines. (5)
    December 20. Convergence results for the meshless Galerkin. Spectral and pseudo-spectral approximation with RBF. (5)
    January 10. Meshfree approximation of multi-assets option pricing (Black-Scholes equation) by collocation (5)

    Lab seminars (in NumLab)

    Dates and topics
    Lissajous curves : 21/11/17 (3),
    RKHS basic theory : 28/11/17 (5),
    A new stable basis for RBF interpolation: 12/12/17 (5),
    RBF-based partition of unity method for elliptic PDEs: Adaptivity and stability issues via VSKs: 19/12/17 (4),
    Radial basis function network for ODEs: application to diabetes and insulin therapy models : 9/1/2018 (5).
    Final test
    The final test this year consists of two parts: the presentation of a seminar and solutions of homeworks, discussed aftwards with the teacher.
    Here the homeworks .

    Dates and rooms
    25 Jan. 2018: 14-17, room 2AB40
    19 Feb. 2018: 10-13, room 2AB40
    5 Jun. 2018: 10-13, room 2AB40
    18 Sep. 2018: 10-13, room 2AB40

    Stefano De Marchi: Lectures on multivariate polynomial approximation
    Stefano De Marchi: 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
    Robert Schaback: A practical guide to Radial Basis Functions , lectures notes, Goettingen 2007.
    Wen Shen, C.S.Shen and Zhuo-Jia Fu Recent Advances in Radial Basis Function Collocation Methods

    (last update: 10 January 2018).