Related papers: Cubature Formulas for Symmetric Measures in Higher…

A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) +…

The nodes of certain minimal cubature rule are real common zeros of a set of orthogonal polynomials of degree $n$. They often consist of a well distributed set of points and interpolation polynomials based on them have desired convergence…

In this paper, we deal with several aspects of the universal Frolov cubature method, that is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable…

For the purpose of uncertainty quantification with collocation, a method is proposed for generating families of one-dimensional nested quadrature rules with positive weights and symmetric nodes. This is achieved through a reduction…

A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete…

We show that the spectral norm of a $d$-mode real or complex symmetric tensor in $n$ variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed…

We give a simple formula for finding the spectral norm of d-mode symmetric tensor in two variables over the complex or real numbers in terms of the complex or real roots of a corresponding polynomial in one complex variable. This result…

We consider quadrature formulas based on interpolation using the basis functions $1/(1+t_kx)$ $(k=1,2,3,\ldots)$ on $[-1,1]$, where $t_k$ are parameters on the interval $(-1,1)$. We investigate two types of quadratures: quadrature formulas…

In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions…

We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked out on the sphere of…

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many…

We introduce a new type of cubature formula for the evaluation of an integral over the disk with respect to a weight function. The method is based on an analysis of the Fourier series of the weight function and a reduction of the bivariate…

Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying…

We describe an algorithm for controlling the relative error in the numerical evaluation of a bivariate integral, without prior knowledge of the magnitude of the integral. In the event that the magnitude of the integral is less than unity,…

We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh,…

Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle…

We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain $[a,b]^2=[a,b]\times [a,b]$. Our modified cubature formulae use mixed type data: except evaluations of the…

We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyse the algebraic properties associated to flat…

Calabro et al. (2017) changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster.…

We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if S in R^d is…