arXiv subjects

M. Vianello

Publications and source records attributed to M. Vianello.

6 recordsLinked to original sources

arXiv2026

Comparing algebraic cubature rules on spline curved elements

We compare four methods for the construction of algebraic cubature rules on planar elements, whose boundary is tracked by splines. The methods, that we have developed over the last two decades, are based on Green theorem together with some cornerstones of polynomial approximation theory: Gaussian quadrature,Tchakaloff theorem, discretized Chebyshev expansion (hyperinterpolation), Fekete-like interpolation. We discuss their advantages and drawbacks in view of the application to curved polytopal element methods. We have also made freely available at a single site the corresponding open-source Matlab codes.

math.NA↗

arXiv2024

Random sampling and polynomial-free interpolation by Generalized MultiQuadrics

We prove that interpolation matrices for Generalized MultiQuadrics (GMQ) of order greater than one are almost surely nonsingular without polynomial addition, in any dimension and with any continuous random distribution of sampling points. We also include a new class of generalized MultiQuadrics recently proposed by Buhmann and Ortmann.

math.NA↗

arXiv2024

Nonsingularity of unsymmetric Kansa matrices: random collocation by MultiQuadrics and Inverse MultiQuadrics

Unisolvence of unsymmetric Kansa collocation is still a substantially open problem. We prove that Kansa matrices with MultiQuadrics and Inverse MultiQuadrics for the Dirichlet problem of the Poisson equation are almost surely nonsingular, when the collocation points are chosen by any continuous random distribution in the domain interior and arbitrarily on its boundary.

math.NA↗

arXiv2023

Evaluating Lebesgue constants by Chebyshev polynomial meshes on cube, simplex and ball

We show that product Chebyshev polynomial meshes can be used, in a fully discrete way, to evaluate with rigorous error bounds the Lebesgue constant, i.e. the maximum of the Lebesgue function, for a class of polynomial projectors on cube, simplex and ball, including interpolation, hyperinterpolation and weighted least-squares. Several examples are presented and possible generalizations outlined. A numerical software package implementing the method is freely available online.

math.NA↗

arXiv2023

Numerical cubature on scattered data by adaptive interpolation

We construct cubature methods on scattered data via resampling on the support of known algebraic cubature formulas, by different kinds of adaptive interpolation (polynomial, RBF, PUM). This approach gives a promising alternative to other recent methods, such as direct meshless cubature by RBF or least-squares cubature formulas.

math.NA↗

arXiv2021

Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

math.NA↗