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Marco Vianello

Publications and source records attributed to Marco Vianello.

At least 19 recordsLinked to original sources
arXiv2026

On the role of weak Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases

We compute numerically the $L^2$ Marcinkiewicz-Zygmund constants of cubature rules, with a special attention to their role in polynomial approximation by orthogonal bases. We test some relevant rules on domains such as the interval, the square, the disk, the triangle, the cube and the sphere. The approximation power of the corresponding least squares (LS) projection is compared with standard hyperinterpolation and its recently proposed ``exactness-relaxed'' version. The Matlab codes used for these tests are available in open-source form.

math.NA↗
arXiv2025

ORTHOCUB: integral and differential cubature rules by orthogonal moments

We discuss a numerical package, named ORTHOCUB, for the computation of linear functionals of both integral and differential type on multivariate polynomial spaces. The weighted sums corresponding to such integral and differential cubatures are implemented via orthogonal polynomial moments and auxiliary near-minimal algebraic cubature in a bounding box, with no conditioning issue since no matrix inversion or factorization is needed. The whole computational process indeed reduces to moment computation and dense matrix-vector products of relatively small size. The Matlab and Python codes are freely available, to be used as building blocks for integral and differential problems.

cs.MS↗
arXiv2025

Effective numerical integration on complex shaped elements by discrete signed measures

We discuss a cheap and stable approach to polynomial moment-based compression of multivariate measures by discrete signed measures. The method is based on the availability of an orthonormal basis and a low-cardinality algebraic quadrature formula for an auxiliary measure in a bounding set. Differently from other approaches, no conditioning issue arises since no matrix factorization or inversion is needed. We provide bounds for the sum of the absolute values of the signed measure weights, and we make two examples: efficient quadrature on curved planar elements with spline boundary (in view of the application to high-order FEM/VEM), and compression of QMC integration on 3D elements with complex shape.

math.NA↗
arXiv2025

Unisolvence of randomized MultiQuadric Kansa collocation for convection-diffusion with mixed boundary conditions

We make a further step in the open problem of unisolvence for unsymmetric Kansa collocation, proving that the MultiQuadric Kansa method with fixed collocation points and random fictitious centers is almost surely unisolvent, for stationary convection-diffusion equations with mixed boundary conditions on general domains. For the purpose of illustration, the method is applied in 2D with fictitious centers that are local random perturbations of predetermined collocation points.

math.NA↗
arXiv2025

Cheap and stable quadrature on polyhedral elements

We discuss a cheap tetrahedra-free approach to the numerical integration of polynomials on polyhedral elements, based on hyperinterpolation in a bounding box and Chebyshev moment computation via the divergence theorem. No conditioning issues arise, since no matrix factorization or inversion is needed. The resulting quadrature formula is theoretically stable even in the presence of some negative weights.

math.NA↗
arXiv2024

Unisolvence of Kansa collocation for elliptic equations by polyharmonic splines with random fictitious centers

We make a further step in the unisolvence open problem for unsymmetric Kansa collocation, proving nonsingularity of Kansa matrices with polyharmonic splines and random fictitious centers, for second-order elliptic equations with mixed boundary conditions. We also show some numerical tests, where the fictitious centers are local random perturbations of predetermined collocation points.

math.NA↗
arXiv2024

Unisolvence of unsymmetric random Kansa collocation by Gaussians and other analytic RBF vanishing at infinity

We give a short proof of almost sure invertibility of unsymmetric random Kansa collocation matrices by a class of analytic RBF vanishing at infinity, for the Poisson equation with Dirichlet boundary conditions. Such a class includes popular Positive Definite instances such as Gaussians, Generalized Inverse MultiQuadrics and Matern RBF. The proof works on general domains in any dimension, with any distribution of boundary collocation points and any continuous random distribution of internal collocation points.

math.NA↗
arXiv2024

Unisolvence of random Kansa collocation by Thin-Plate Splines for the Poisson equation

Existence of sufficient conditions for unisolvence of Kansa unsymmetric collocation for PDEs is still an open problem. In this paper we make a first step in this direction, proving that unsymmetric collocation matrices with Thin-Plate Splines for the 2D Poisson equation are almost surely nonsingular, when the discretization points are chosen randomly on domains with analytic boundary.

math.NA↗
arXiv2024

Polynomial-free unisolvence of polyharmonic splines with odd exponent by random sampling

In a recent paper almost sure unisolvence of RBF interpolation at random points with no polynomial addition was proved, for Thin-Plate Splines and Radial Powers with noninteger exponent. The proving technique left unsolved the case of odd exponents. In this short note we prove almost sure polynomial-free unisolvence in such instances, by a deeper analysis of the interpolation matrix determinant and fundamental properties of analytic functions.

math.NA↗
arXiv2023

Compressed QMC volume and surface integration on union of balls

We discuss an algorithm for Tchakaloff-like compression of Quasi-MonteCarlo (QMC) volume/surface integration on union of balls (multibubbles). The key tools are Davis-Wilhelmsen theorem on the so-called Tchakaloff sets for positive linear functionals on polynomial spaces, and Lawson-Hanson algorithm for NNLS. We provide the corresponding Matlab package together with several examples.

math.NA↗
arXiv2022

TetraFreeQ: tetrahedra-free quadrature on polyhedral elements

In this paper we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape. The key tools are the notion of Tchakaloff discretization set and the solution of moment-matching equations by Lawson-Hanson iterations for NonNegative Least-Squares. Several numerical tests are presented. The method is implemented in Matlab as open-source software.

math.NA↗