Minimal cubature rules of degree $4n-1$ for the weight functions $$ W_{\a,\b,\pm \frac12}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are constructed explicitly and are shown to be closed related to the Gaussian cubature rules in a domain bounded by two lines and a parabola. Lagrange interpolation polynomials on the nodes of these cubature rules are constructed and their Lebesgue constants are determined.
In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D Laplace boundary value problems, using a collocation method within the Isogeometric Analysis paradigm. In such setting the regular part of the integrand can be defined as the product of a tensor product B-spline and a general function. The rules are derived by using first the spline quasi-interpolation approach to approximate such function and then the extens…
Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. This paper introduces a new criterion for selecting which nodes to eliminate that is based on a linearization of the moment equation. In addition, a penalized iterative solver is introduced, that ensures that weights are positive and nodes are inside the integration domain. A strategy for constructing an initial quadra…
73 new cubature rules are found for three standard multidimensional integrals with spherically symmetric regions and weights, using direct search with a numerical zero-finder. All but four of the new rules have fewer integration points than known rules of the same degree, and twenty are within three points of M{ö}ller's lower bound. Most have all positive coefficients and most have some symmetry, including some supported by one or two concentric spheres. They include degree 7 formulas for integration over the sphere and Gaussian-weighted integrals over all space, each in 6 and 7 dimensions, wi…
We present the Continuous Empirical Cubature Method (CECM), a novel algorithm for empirically devising efficient integration rules. The CECM aims to improve existing cubature methods by producing rules that are close to the optimal, featuring far less points than the number of functions to integrate. The CECM consists on a two-stage strategy. First, a point selection strategy is applied for obtaining an initial approximation to the cubature rule, featuring as many points as functions to integrate. The second stage consists in a sparsification strategy in which, alongside the indexes and corres…
A mixed interpolation-regression method for function approximation on certain planar domains
La Laguna University, Spain（拉帕尔马大学）；IMAG and Departamento de Matemática Aplicada, Universidad de Granada, Spain（格拉纳达大学）；Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain（马德里卡洛斯三世大学）
Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions
KU Leuven（荷语鲁汶大学）；VITO, Energy Technology（维托能源技术研究所）；Philipps-University Marburg, Faculty of Mathematics and Computer Science（马尔堡菲利普斯大学数学与计算机学院）；University of Siegen, Department of Mathematics（锡根大学数学系）
On exponential convergence of Chebyshev polynomial approximation for multivariate analytic functions
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
On the role of weak Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases
School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）；Department of Mathematics "Tullio Levi-Civita", University of Padova（帕多瓦大学蒂利奥·莱维-奇维塔数学系）
Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for …
Optimization approaches to quadrature: new characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions
University of Tromsø（特罗姆瑟大学）；Universität Konstanz（康斯坦茨大学）
A rule for constructing interpolation nodes for $n$th degree polynomials on the simplex is presented. These nodes are simple to define recursively from families of 1D node sets, such as the Lobatto-Gauss-Legendre (LGL) nodes. The resulting nodes have attractive properties: they are fully symmetric, they match the 1D family used in construction on the edges of the simplex, and the nodes constructed for the $(d-1)$-simplex are the boundary traces of the nodes constructed for the $d$-simplex. When compared using the Lebesgue constant to other explicit rules for defining interpolation nodes, the n…
The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related polytopes), numerical interpolation procedures (usually orthonormal polynomial bases), and numerical integration procedures (usually quadrature rules). It is well known that each of these areas ha…
