Weakly Admissible Meshes & Discrete Extremal Sets
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
-
WAM (Matlab
package for multivariate polynomial fitting and
interpolation
on Weakly Admissible Meshes)
version 2.0
by S. De Marchi, F. Piazzon, A. Sommariva and M. Vianello
note: contains the algorithms to compute Discrete Extremal Sets of
Fekete and Leja type and to estimate
Lebesgue constants of
interpolation sets
- WAM generators for quadrangles and disks are being used in the Matlab
package
GaussQR by G. Fasshauer and M. McCourt
Ph.D. Dissertation
Posters
Papers
-
Chebyshev admissible meshes and Lebesgue constants of complex
polynomial projections
draft - L. Bialas-Ciez, D. Kenne, A. Sommariva and M. Vianello
-
Numerical cubature on scattered data by
adaptive interpolation
arXiv:2307.07203 - R. Cavoretto, A. De Rossi, F. Dell'Accio,
F. Di Tommaso, N. Siar, A. Sommariva and M. Vianello
-
DISC: an adaptive numerical Differentiator by local polynomial Interpolation
on SCattered data
F. Dell'Accio, F. Di Tommaso, N. Siar and M. Vianello
Dolomites Res. Notes Approx. DRNA 15 (2022) (Special Issue
"Software for Approximation 2022", 92--100)
-
Numerical differentiation
on scattered data through multivariate polynomial interpolation
arXiv preprint 2105.09262 - F. Dell'Accio, F. Di Tommaso, N. Siar
and M. Vianello
BIT Numer. Math. 62 (2022), 773--801
-
Near-optimal polynomial interpolation
on spherical triangles
preprint - A. Sommariva and M. Vianello
Mediterr. J. Math. 19 (2022)
-
Near G-optimal Tchakaloff designs
preprint - L. Bos, F. Piazzon and M. Vianello
Comput. Statistics 35 (2020), 803--819
-
Near optimal polynomial regression on polynomial meshes
preprint - L. Bos, F. Piazzon and M. Vianello
Sampling Theory and Applications 2019, IEEE Xplore Digital Library,
DOI: 10.1109/SampTA45681.2019.9030910
-
Chebyshev-Dubiner norming webs
on starlike polygons
preprint - M. Vianello
J. Inequal. Spec. Funct. 10-3 (2019), 26--32
-
Tchakaloff polynomial meshes
preprint - L. Bos and M. Vianello
Ann. Polon. Math. 122 (2019), 221--231
-
Dubiner distance and stability of Lebesgue constants
preprint - M. Vianello
J. Inequal. Spec. Funct. 10-2 (2019), 49--60
-
Pluripotential Numerics
arXiv preprint 1704.03411 -
F. Piazzon
Constr. Approx. 49 (2019), 227--263
-
Markov
inequalities, Dubiner
distance, norming meshes and polynomial optimization
on convex bodies
preprint - F. Piazzon and M. Vianello
Optim. Lett. 13 (2019), 1325--1343
-
Near optimal Tchakaloff meshes for compact sets
with Markov exponent 2
preprint - M. Vianello
Dolomites Res. Notes Approx. DRNA 11 (2018), 92--96 (Special Issue on Norm
Levenberg's 60th birthday)
-
Subperiodic Dubiner distance, norming meshes and trigonometric
polynomial optimization
preprint - M. Vianello
Optim. Lett. 12 (2018), 1659--1667
-
Discrete
norming inequalities on sections of sphere, ball and torus
preprint - A. Sommariva and M. Vianello
J. Inequal. Spec. Funct. 9-4 (2018), 113--121
-
Global polynomial optimization by norming sets on sphere and torus
preprint - M. Vianello
Dolomites Res. Notes Approx. DRNA 11 (2018), 10--14
-
Stability inequalities for Lebesgue constants via Markov-like
inequalities
preprint - F. Piazzon and M. Vianello
Dolomites Res. Notes Approx. DRNA 11 (2018), 1--9
-
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
-
An
elementary approach to polynomial optimization on polynomial
meshes
draft - M. Vianello
J. Math. Fund. Sci. 50 (2018), 84--91
-
A note on total degree polynomial optimization by Chebyshev grids
preprint - F. Piazzon and M. Vianello
Optim. Lett. 12 (2018), 63--71
-
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
-
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
-
Caratheodory-Tchakaloff
Subsampling
F. Piazzon, A. Sommariva and M. Vianello
Dolomites Res. Notes Approx. DRNA 10 (2017), 5--14
- Polynomial
approximation and quadrature on
geographic rectangles
preprint - M. Gentile, A. Sommariva and M. Vianello
Appl. Math. Comput. 297 (2017), 159--179
- Trivariate
polynomial
approximation on Lissajous curves
draft - L. Bos, S. De Marchi and M. Vianello
IMA J. Numer. Anal., published online 14 May 2016
-
Jacobi norming meshes
preprint - F. Piazzon and M. Vianello
Math. Inequal. Appl. 19 (2016), 395--400
- Optimal
Polynomial Admissible Meshes on Some Classes of Compact Subsets of
R^d
preprint - F. Piazzon
J. Approx. Theory 207 (2016), 241--264
- Compressed
sampling
inequalities by Tchakaloff's theorem
preprint - M. Vianello
Math. Inequal. Appl. 19 (2016), 395--400
-
Compression of multivariate discrete measures and applications
preprint - A. Sommariva and M. Vianello
Numer. Funct. Anal. Optim. 36 (2015), 1198--1223
- Polynomial
fitting and interpolation on
circular sections
preprint - A. Sommariva and M. Vianello
Appl. Math. Comput. 258 (2015), 410--424
- Constructing
optimal polynomial meshes
on planar starlike domains
preprint, F. Piazzon and M. Vianello
Dolomites Res. Notes Approx. DRNA 7 (2014), 22--25
-
Norming meshes by Bernstein-like inequalities
preprint - M. Vianello
Math. Inequal. Appl. 17 (2014), 929--936
- Sub-optimal polynomial
meshes on planar Lipschitz domains
preprint - F. Piazzon and M. Vianello
Numer. Funct. Anal. Optim. 35 (2014), 1467--1475
- Polynomial
approximation on pyramids, cones and solids of rotation
S. De Marchi and M. Vianello
Dolomites Res. Notes Approx. DRNA 6 (2013), 20--26
-
Small perturbations of polynomial meshes
preprint - F. Piazzon and M. Vianello
Appl. Anal. 92 (2013), 1063--1073
-
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
-
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
-
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
-
Computing Fekete and Lebesgue points: simplex, square, disk
preprint - M. Briani, A. Sommariva and M. Vianello
J. Comput. Appl. Math. 236 (2012), 2477--2486
-
Low cardinality admissible meshes on quadrangles, triangles and
disks
preprint - L. Bos and M.
Vianello
Math. Inequal. Appl. 15 (2012), 229--235
-
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
-
Polynomial interpolation and cubature over polygons
preprint - M. Gentile, A. Sommariva and M.
Vianello
J. Comput. Appl. Math. 235 (2011), 5232--5239
-
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
-
Analytic transformations of admissible meshes
preprint - F. Piazzon and M.
Vianello
East J. Approx. 16 (2010), 313--322
-
Approximate Fekete points for weighted polynomial interpolation
preprint - A. Sommariva and M. Vianello
Electron. Trans. Numer. Anal. 37 (2010), 1--22
-
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
-
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
-
Computing approximate Fekete points by QR factorizations of
Vandermonde matrices
preprint - A. Sommariva and M. Vianello
Comput. Math. Appl. 57 (2009), 1324--1336