We present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that the number of quadrature points $N$ be equal to the dimension of a lower dimensional polynomial space. Quadrature forumulas are presented for up to degree $d=25$, all which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results.
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 rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of poi…
The numerical integration of an analytical function $f(x)$ using a finite set of equidistant points can be performed by quadrature formulas like the Newton-Cotes. Unlike Gaussian quadrature formulas however, higher-order Newton-Cotes formulas are not stable, limiting the usable order of such formulas. Existing work showed that by the use of orthogonal polynomials, stable high-order quadrature formulas with equidistant points can be developed. We improve upon such work by making use of (orthogonal) Gram polynomials and deriving an iterative algorithm, together allowing us to reduce the space-co…
Numerical Integration in Multiple Dimensions with Designed Quadrature
Scientific Computing and Imaging Institute, University of Utah（科学计算与成像研究所，犹他大学）；School of Computing, University of Utah（计算机学院，犹他大学）；Department of Mathematics, University of Utah（数学系，犹他大学）
Efficient Numerical Integration for Finite Element Trunk Spaces in 2D and 3D using Machine Learning: A new Optimisation Paradigm to Construct Application-Specific Quadrature Rules
The problem of estimating single- and multi-dimensional integrals, with or without end-point singularities, is prevalent in all fields of scientific research, and in particular in physics. Although tanh-sinh quadrature is known to handle most of these cases excellently, its use is not widely spread among physicists. Moreover, while most calculations are limited by the use of finite-precision floating-point arithmetic, similar considerations for tanh-sinh quadrature are mostly lacking in literature, where infinite-precision floating-point numbers are often assumed. Also little information is av…
Numerical integration on spheres, including the computation of the areas of spherical triangles, is a core computation in geomathematics. The commonly used techniques sometimes suffer from instabilities and significant loss of accuracy. We describe a new algorithm, called ARPIST, for accurate and stable integration of functions on spherical triangles. ARPIST is based on an easy-to-implement transformation to the spherical triangle from its corresponding linear triangle via radial projection to achieve high accuracy and efficiency. More importantly, ARPIST overcomes potential instabilities in c…
In this technical note, we explain how to construct Gaussian quadrature rules for efficiently and accurately computing integrals of S(x)f(x) where S(x) is the solar irradiance function tabulated in the ASTM standard and f(x) is an arbitary application-specific smooth function. This allows the integral to be computed accurately with a relatively small number of f(x) evaluations despite the fact that S(x) is non-smooth and wildly oscillatory. Julia software is provided to compute solar-weighted quadrature rules for an arbitrary bandwidth or number of points. We expect that this technique will be…
RiemCirc is a C++ program which allocates points inside the unit circle for numerical quadrature on the circle, aiming at homogeneous equidistant distribution. The weights of the quadrature rule are computed by the area of the tiles that surround these nodes. The shapes of the areas are polygonal, defined by Voronoi tessellation.
In this work, semi-analytical formulae for the numerical evaluation of surface integrals occurring in Galerkin boundary element methods (BEM) in 3D are derived. The integrals appear as the entries of BEM matrices and are formed over pairs of surface triangles. Since the integrands become singular if the triangles have non-empty intersection, the transformation presented by Sauter and Schwab is used to remove the singularities. It is shown that the resulting integrals admit analytical formulae if the triangles are identical or share a common edge. Moreover, the four-dimensional integrals are re…
An error estimate for the Gauss-Lobatto quadrature formula for integration over the interval $[-1, 1]$, relative to the Jacobi weight function $w^{α,β}(t)=(1-t)^α(1+t)^β$, $α,β>-1$, is obtained. This estimate holds true for functions belonging to some Sobolev-type subspaces of the weighted space $L_{w^{α,β}}^1([-1,1])$.
Numerical integration is encountered in all fields of numerical analysis and the engineering sciences. By now, various efficient and accurate quadrature rules are known; for instance, Gauss-type quadrature rules. In many applications, however, it might be impractical---if not even impossible---to obtain data to fit known quadrature rules. Often, experimental measurements are performed at equidistant or even scattered points in space or time. In this work, we propose stable high order quadrature rules for experimental data, which can accurately handle general weight functions.
We revisit the problem of extending quadrature formulas for general weight functions, and provide a generalization of Patterson's method for the constant weight function. The method can be used to compute a nested sequence of quadrature formulas for integration with respect to any continuous probability measure on the real line with finite moments. The advantages of the method include that it works directly with the moments of the underlying distribution, and that for distributions with rational moments the existence of the formulas can be verified by exact rational arithmetic.
Numerical integration is a classical problem emerging in many fields of science. Multivariate integration cannot be approached with classical methods due to the exponential growth of the number of quadrature nodes. We propose a method to overcome this problem. Tensor-train decomposition of a tensor approximating the integrand is constructed and used to evaluate a multivariate quadrature formula. We show how to deal with singularities in the integration domain and conduct theoretical analysis of the integration accuracy. The reference open-source implementation is provided.
A method is presented for the evaluation of integrals on tetrahedra where the integrand has an integrable singularity at one vertex. The approach uses a transformation to spherical polar coordinates which explicitly eliminates the singularity and facilitates the evaluation of integration limits. The method is also implemented in an adaptive form which gives convergence to a required tolerance. Results from the method are compared to the output from an exact analytical method for one tetrahedron and show high accuracy. In particular, when the adaptive algorithm is used, highly accurate results …
In this paper, we propose and study a fast multilevel dimension iteration (MDI) algorithm for computing arbitrary $d$-dimensional integrals based on tensor product approximations. It reduces the computational complexity (in terms of the CPU time) of a tensor product method from the exponential order $O(N^d)$ to the polynomial order {\color{black} $O(d^3N^2)$ or better}, where $N$ stands for the number of quadrature points in each coordinate direction. As a result, the proposed MDI algorithm effectively circumvents the curse of the dimensionality of tensor product methods for high dimensional n…
Tübingen AI Center（图宾根人工智能中心）；University of Tübingen（图宾根大学）；School of Engineering Sciences（工程科学学院）；Lappeenranta–Lahti University of Technology（拉佩伦塔-拉赫蒂技术大学）
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…
Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Curtin University（科廷大学）；Commonwealth Scientific and Industrial Research Organization (CSIRO)（联邦科学与工业研究组织）
Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate ho…
