Back to "Padova-Verona research group on Constructive Approximation and Applications" (CAA) Home Page

Weakly Admissible polynomial Meshes


Polynomial Meshes are geometry-dependent discretizations of multidimensional compact sets, that are suitable for polynomial Least Squares approximation and contain discrete extremal subsets of Fekete and Leja type for polynomial interpolation (computable by standard numerical linear algebra algorithms).

Moreover, polynomial meshes and their discrete extremal subsets are a tool for numerical approximations in pluripotential theory (multivariate transfinite diameter, pluripotential Green function), and have been also used in the discretization of PDEs and in the framework of multivariate polynomial optimization.

Recently, approximate Fekete points extracted from polynomial meshes have been adopted in state of the art codes for the implementation of FIR digital filters on multi-interval domains (see paper).

Surveys

Software

Ph.D. Dissertation

Posters

Papers

  1. Tchakaloff polynomial meshes
    draft - L. Bos and M. Vianello
  2. Chebyshev-Dubiner norming webs on starlike polygons
    draft - M. Vianello
  3. Dubiner distance and stability of Lebesgue constants
    draft - M. Vianello
  4. Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies
    preprint - F. Piazzon and M. Vianello
    Optim. Lett., to appear
  5. Near optimal Tchakaloff meshes for compact sets with Markov exponent 2
    preprint
    Dolomites Res. Notes Approx. DRNA 11 (2018), to appear (Special Issue on Norm Levenberg's 60th birthday)
  6. Discrete norming inequalities on sections of sphere, ball and torus
    arXiv preprint 1802.01711 - A. Sommariva and M. Vianello
    J. Inequal. Spec. Funct., to appear
  7. Pluripotential Numerics
    arXiv preprint 1704.03411 - F. Piazzon
    Constr. Approx., published online 21 June 2018
  8. Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization
    preprint - M. Vianello
    Optim. Lett. 12 (2018), 1659--1667
  9. Global polynomial optimization by norming sets on sphere and torus
    preprint - M. Vianello
    Dolomites Res. Notes Approx. DRNA 11 (2018), 10--14
  10. Stability inequalities for Lebesgue constants via Markov-like inequalities
    preprint - F. Piazzon and M. Vianello
    Dolomites Res. Notes Approx. DRNA 11 (2018), 1--9
  11. Bernstein-Walsh theory associated to convex bodies and applications to multivariate approximation theory
    arXiv preprint 1701.05613 - L. Bos and N. Levenberg
    Comput. Methods Funct. Theory, published online 24 October 2017
  12. An elementary approach to polynomial optimization on polynomial meshes
    draft - M. Vianello
    J. Math. Fund. Sci. 50 (2018), 84--91
  13. A note on total degree polynomial optimization by Chebyshev grids
    preprint - F. Piazzon and M. Vianello
    Optim. Lett. 12 (2018), 63--71
  14. Caratheodory-Tchakaloff Least Squares
    preprint - F. Piazzon, A. Sommariva and M. Vianello
    Sampling Theory and Applications 2017, IEEE Xplore Digital Library, DOI: 10.1109/SAMPTA.2017.8024337
  15. Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere
    preprint - P. Leopardi, A. Sommariva and M. Vianello
    Dolomites Res. Notes Approx. DRNA 10 (2017), 18--24
  16. Caratheodory-Tchakaloff Subsampling
    F. Piazzon, A. Sommariva and M. Vianello
    Dolomites Res. Notes Approx. DRNA 10 (2017), 5--14
  17. Polynomial approximation and quadrature on geographic rectangles
    preprint - M. Gentile, A. Sommariva and M. Vianello
    Appl. Math. Comput. 297 (2017), 159--179
  18. Trivariate polynomial approximation on Lissajous curves
    draft - L. Bos, S. De Marchi and M. Vianello
    IMA J. Numer. Anal., published online 14 May 2016
  19. Jacobi norming meshes
    preprint - F. Piazzon and M. Vianello
    Math. Inequal. Appl. 19 (2016), 395--400
  20. Optimal Polynomial Admissible Meshes on Some Classes of Compact Subsets of R^d
    preprint - F. Piazzon
    J. Approx. Theory 207 (2016), 241--264
  21. Compressed sampling inequalities by Tchakaloff's theorem
    preprint - M. Vianello
    Math. Inequal. Appl. 19 (2016), 395--400
  22. Compression of multivariate discrete measures and applications
    preprint - A. Sommariva and M. Vianello
    Numer. Funct. Anal. Optim. 36 (2015), 1198--1223
  23. Polynomial fitting and interpolation on circular sections
    preprint - A. Sommariva and M. Vianello
    Appl. Math. Comput. 258 (2015), 410--424
  24. Constructing optimal polynomial meshes on planar starlike domains
    preprint, F. Piazzon and M. Vianello
    Dolomites Res. Notes Approx. DRNA 7 (2014), 22--25
  25. Norming meshes by Bernstein-like inequalities
    preprint - M. Vianello
    Math. Inequal. Appl. 17 (2014), 929--936
  26. Sub-optimal polynomial meshes on planar Lipschitz domains
    preprint - F. Piazzon and M. Vianello
    Numer. Funct. Anal. Optim. 35 (2014), 1467--1475
  27. Polynomial approximation on pyramids, cones and solids of rotation
    S. De Marchi and M. Vianello
    Dolomites Res. Notes Approx. DRNA 6 (2013), 20--26
  28. Small perturbations of polynomial meshes
    preprint - F. Piazzon and M. Vianello
    Appl. Anal. 92 (2013), 1063--1073
  29. Polynomial Interpolation and Approximation in C^d
    preprint - T. Bloom, L. Bos, J.-P. Calvi and N. Levenberg
    Ann. Polon. Math. 106 (2012), 53--81
  30. On the generation of symmetric Lebesgue-like points in the triangle
    preprint - F. Rapetti, A. Sommariva and M. Vianello
    J. Comput. Appl. Math. 236 (2012), 4925--4932
  31. Polynomial approximation and cubature at approximate Fekete and Leja points of the cylinder
    preprint - S. De Marchi, M. Marchioro and A. Sommariva
    Appl. Math. Comput. 218 (2012), 10617--10629
  32. Computing Fekete and Lebesgue points: simplex, square, disk
    preprint - M. Briani, A. Sommariva and M. Vianello
    J. Comput. Appl. Math. 236 (2012), 2477--2486
  33. Low cardinality admissible meshes on quadrangles, triangles and disks
    preprint - L. Bos and M. Vianello
    Math. Inequal. Appl. 15 (2012), 229--235
  34. On Multivariate Newton Interpolation at Discrete Leja Points
    preprint - L. Bos, S. De Marchi, A. Sommariva and M. Vianello
    Dolomites Res. Notes Approx. DRNA 4 (2011), 15--20
  35. Polynomial interpolation and cubature over polygons
    preprint - M. Gentile, A. Sommariva and M. Vianello
    J. Comput. Appl. Math. 235 (2011), 5232--5239
  36. Geometric Weakly Admissible Meshes, Discrete Least Squares Approximations and Approximate Fekete Points
    preprint - L. Bos, J.P. Calvi, N. Levenberg, A. Sommariva and M. Vianello
    Math. Comp. 80 (2011), 1601--1621
  37. Analytic transformations of admissible meshes
    preprint - F. Piazzon and M. Vianello
    East J. Approx. 16 (2010), 313--322
  38. Approximate Fekete points for weighted polynomial interpolation
    preprint - A. Sommariva and M. Vianello
    Electron. Trans. Numer. Anal. 37 (2010), 1--22
  39. Least-squares polynomial approximation on weakly admissible meshes: disk and triangle
    preprint - L. Bos, A. Sommariva and M. Vianello
    J. Comput. Appl. Math. 235 (2010), 660--668
  40. Computing multivariate Fekete and Leja points by numerical linear algebra
    L. Bos, S. De Marchi, A. Sommariva and M. Vianello
    SIAM J. Numer. Anal. 48 (2010), 1984--1999
  41. Computing approximate Fekete points by QR factorizations of Vandermonde matrices
    preprint - A. Sommariva and M. Vianello
    Comput. Math. Appl. 57 (2009), 1324--1336