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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. Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization
    draft - M. Vianello
  2. Global polynomial optimization by norming sets on sphere and torus
    draft - M. Vianello
  3. Pluripotential Numerics
    arXiv preprint 1704.03411 - F. Piazzon
  4. Discrete norming inequalities on sections of sphere, ball and torus
    draft - A. Sommariva and M. Vianello
  5. Mesh-based polynomial optimization on convex bodies
    draft - F. Piazzon and M. Vianello
  6. Bernstein-Walsh theory associated to convex bodies and applications to multivariate approximation theory
    arXiv preprint 1701.05613 - L. Bos and N. Levenberg
  7. Stability inequalities for Lebesgue constants via Markov-like inequalities
    draft - F. Piazzon and M. Vianello
  8. An elementary approach to polynomial optimization on polynomial meshes
    draft - M. Vianello
    J. Math. Fund. Sci. 50 (2018)
  9. Caratheodory-Tchakaloff Least Squares
    preprint - F. Piazzon, A. Sommariva and M. Vianello
    2017 International Conference on Sampling Theory and Applications (SampTA), IEEE Xplore Digital Library, DOI: 10.1109/SAMPTA.2017.8024337
  10. A note on total degree polynomial optimization by Chebyshev grids
    draft - F. Piazzon and M. Vianello
    Optim. Lett., published online 01 July 2017
  11. Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere
    P. Leopardi, A. Sommariva and M. Vianello
    Dolomites Res. Notes Approx. DRNA 10 (2017), 18--24
  12. Caratheodory-Tchakaloff Subsampling
    F. Piazzon, A. Sommariva and M. Vianello
    Dolomites Res. Notes Approx. DRNA 10 (2017), 5--14
  13. Polynomial approximation and quadrature on geographic rectangles
    preprint - M. Gentile, A. Sommariva and M. Vianello
    Appl. Math. Comput. 297 (2017), 159--179
  14. Trivariate polynomial approximation on Lissajous curves
    draft - L. Bos, S. De Marchi and M. Vianello
    IMA J. Numer. Anal., published online 14 May 2016
  15. Jacobi norming meshes
    preprint - F. Piazzon and M. Vianello
    Math. Inequal. Appl. 19 (2016), 395--400
  16. Optimal Polynomial Admissible Meshes on Some Classes of Compact Subsets of R^d
    preprint - F. Piazzon
    J. Approx. Theory 207 (2016), 241--264
  17. Compressed sampling inequalities by Tchakaloff's theorem
    preprint - M. Vianello
    Math. Inequal. Appl. 19 (2016), 395--400
  18. Compression of multivariate discrete measures and applications
    preprint - A. Sommariva and M. Vianello
    Numer. Funct. Anal. Optim. 36 (2015), 1198--1223
  19. Polynomial fitting and interpolation on circular sections
    preprint - A. Sommariva and M. Vianello
    Appl. Math. Comput. 258 (2015), 410--424
  20. Constructing optimal polynomial meshes on planar starlike domains
    preprint, F. Piazzon and M. Vianello
    Dolomites Res. Notes Approx. DRNA 7 (2014), 22--25
  21. Norming meshes by Bernstein-like inequalities
    preprint - M. Vianello
    Math. Inequal. Appl. 17 (2014), 929--936
  22. Sub-optimal polynomial meshes on planar Lipschitz domains
    preprint - F. Piazzon and M. Vianello
    Numer. Funct. Anal. Optim. 35 (2014), 1467--1475
  23. Polynomial approximation on pyramids, cones and solids of rotation
    S. De Marchi and M. Vianello
    Dolomites Res. Notes Approx. DRNA 6 (2013), 20--26
  24. Small perturbations of polynomial meshes
    preprint - F. Piazzon and M. Vianello
    Appl. Anal. 92 (2013), 1063--1073
  25. 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
  26. 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
  27. 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
  28. Computing Fekete and Lebesgue points: simplex, square, disk
    preprint - M. Briani, A. Sommariva and M. Vianello
    J. Comput. Appl. Math. 236 (2012), 2477--2486
  29. Low cardinality admissible meshes on quadrangles, triangles and disks
    preprint - L. Bos and M. Vianello
    Math. Inequal. Appl. 15 (2012), 229--235
  30. 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
  31. Polynomial interpolation and cubature over polygons
    preprint - M. Gentile, A. Sommariva and M. Vianello
    J. Comput. Appl. Math. 235 (2011), 5232--5239
  32. 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
  33. Analytic transformations of admissible meshes
    preprint - F. Piazzon and M. Vianello
    East J. Approx. 16 (2010), 313--322
  34. Approximate Fekete points for weighted polynomial interpolation
    preprint - A. Sommariva and M. Vianello
    Electron. Trans. Numer. Anal. 37 (2010), 1--22
  35. 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
  36. 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
  37. Computing approximate Fekete points by QR factorizations of Vandermonde matrices
    preprint - A. Sommariva and M. Vianello
    Comput. Math. Appl. 57 (2009), 1324--1336