Cubature Formulas Research Articles

On the uniqueness and computation of commuting extensions

A tuple (Z1,…,Zp) of matrices of size r×r is said to be a commuting extension of a tuple (A1,…,Ap) of matrices of size n×n if the Zi pairwise commute and each Ai sits in the upper left corner of a block decomposition of Zi (here, r and n are two arbitrary integers with n<r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:(i)Theorems on the uniqueness of commuting extensions for three matrices or more.(ii)Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1.(iii)A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

Function recovery on manifolds using scattered data

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the Lγ(M)-average of the geodesic distance to the point set and determine the value of γ∈(0,∞]. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the nth minimal worst case error for Lq(M)-approximation for all 1≤q≤∞.Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ<∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

The Approximate Integrals by the Reproducing Kernel Method on the Elliptical Region Eₙ In the Space Rⁿ

In this paper, we applied the reproducing kernel method on elliptical region, we establish the formulas of the reproducing kernel from several degrees and generalized those formulas whatever the dimension of space, we obtained all surfaces, points, and their constants for compensation in the cubature formula, tables are included to show our results and examples to show the approximation solution in this method.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

Perfect Quadratic Forms Connected With a Lattice and Cubature Formulas

In the present work, a new improved Voronoi algorithm is proposed for calculating the Voronoi neighborhood of a perfect form in many variables, and using this algorithm, all non-equivalent adjacent perfect forms in five variables are calculated.

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog Dm(hβ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D_{m}(h\\beta )$\\end{document} of the differential operator d2mdx2m+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\frac{d^{2m}}{dx^{2m}}+1$\\end{document} designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space is demonstrated. The errors of the optimal quadrature formula in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space and in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space.

On optimization of cubature formulae for Sobolev classes of functions defined on star domains

We find asymptotically optimal methods of recovery of the integration operator given values of the function at a finite number of points for a class of multivariate functions defined on a bounded star domain that have bounded in $L_p$ norm of their distributional gradient. Thus we generalize the known solution of this optimization problem in the case, when the domain of the functions is convex. Let $Q\subset \mathbb{R}^d$, $d\in\mathbb{N}$, be a nonempty bounded open set. By $W^{1,p}(Q)$, $p\in [1,\infty]$, we denote the Sobolev space of functions $f\colon Q\to \mathbb{R}$ such that $f$ and all their (distributional) partial derivatives of the first order belong to $L_p(Q)$. For $x=(x^1,\dots, x^d)\in \mathbb{R}^d$ and $q\in [1,\infty)$ we set$|x|_q:= \Big(\sum_{k=1}^d|x^k|^q\Big)^\frac {1}{q},$ $|x|_\infty:= \max\{|x^k|\colon k\in\{1,\ldots, d\}\}$, and $W^{\infty}_{p}(Q):=\{f\in W^{1,p}(Q)\colon \|\,|\nabla f|_1\,\|_{L_p(Q)}\leq 1\},$ where $\nabla f=(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_d})$, $p\in[1,\infty]$. In particular we prove the following statement: Let $d\geq 2$, $p\in(d,\infty]$ and $Q$ be a bounded star domain. Then$\displaystyle E_n\Big(W_{p}^{\infty}(Q)\Big)=c(d,p)\Big(\frac {\mathop{mes} Q}{2^d}\Big)^{\frac 1 d +\frac 1 {p'}}\cdot \frac{1+o(1)} {n^{\frac 1 d}}$ $(n\to\infty),$ where $E_n(X):=\inf\Big\{\inf\big\{ e(X,\Phi,x_1,\dots,x_n)\colon\, \Phi\colon\mathbb{R}^n\to\mathbb{R}\big\}\colon x_1,\dots,x_n\in Q\big\},$$e(X, \Phi, x_1,\dots,x_n):= \sup\Big\{\Big|\,\int\limits_{Q}f(x)dx - \Phi(f(x_1),\ldots,f(x_n))\Big|\colon f\in X\Big\}$for $X=W_{p}^{\infty}(Q)$, and $c(d,p)\in \mathbb{R}$ depends only on $d$ and $p$.

Efficient Methods for Reliability Sensitivity Analysis of Distribution Parameters and Their Application

To efficiently evaluate the influence of the distribution parameters of the input variables on the failure probability of engineering structures and improve the reliability and safety of engineering structures in a targeted manner, new methods for the global reliability sensitivity analysis (RSA) of distribution parameters are proposed in this study based on the cubature formula (CF), surrogate sampling probability density function (SSPDF), and quasi-Monte Carlo (QMC) method. By introducing CF, the proposed methods can effectively improve the computational efficiency of the nested expectation and variance operators in the reliability sensitivity indices of the distribution parameters. Based on the concept of SSPDF, a surrogate importance sampling probability density function was developed. This not only overcomes the problem of the computational effort of propagating parameter uncertainty to the failure probability function (FPF), which depends on the dimensionality of the parameters; it also further improves the efficiency of the RSA of the parameters in the case of a small failure probability. Finally, by incorporating the idea of the QMC method, the process of calculating the reliability sensitivity indices of the parameters is reduced from a double-loop to a single-loop one. Three engineering examples are used in this study to demonstrate the efficiency and accuracy of the new algorithms.

Weighted Optimal Formulas for Approximate Integration

Solutions to problems arising from much scientific and applied research conducted at the world level lead to integral and differential equations. They are approximately solved, mainly using quadrature, cubature, and difference formulas. Therefore, in the current work, we consider a discrete analogue of the differential operator 1−12π2d2dx2m in the Hilbert space H2μR, called Dmβ. We modify the Sobolev algorithm to construct optimal quadrature formulas using a discrete operator. We provide a weighted optimal quadrature formula, using this algorithm for the case where m=1. Finally, we construct an optimal quadrature formula in the Hilbert space H2μR for the weight functions px=1 and px=e2πiωx when m=1.

Extended homogeneous bivariate orthogonal polynomials: symbolic and numerical Gaussian cubature formula

Extended homogeneous bivariate orthogonal polynomials: symbolic and numerical Gaussian cubature formula

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.

Особенности построения инвариантных кубатурных формул высокой степени точности для приближенного интегрирования по поверхности тора

Invariant cubature formulas of 11th and 13th degrees of accuracy for a torus in R3 are constructed. An algorithm for constructing such formulas is described.

Высокоточная схема для уравнения переноса в задаче нейтронной защиты

The paper considers the problems of calculating the stationary transport of high-energy neutrons from an external source in a hollow metal structure. The spatial grid for calculation consists of tetrahedra, which allows to model complicated geometric shapes of objects. The calculated area is highly heterogeneous, since it consists of metal parts with strong absorption and scattering and practically non-absorbing internal parts. A scheme based on a minimal stencil, within a single tetrahedron for the calculation. The scheme has a high approximation order and is based on characteristic approaches. For a smooth solution and a constant absorption coefficient in each cell, the scheme has a third order of convergence. The cells are traversed using the graph topological sorting algorithm. A new version of the parallelization of the calculation for the OpenMP standard has been proposed. For discretization by an angular variable, a cubature formula of the direct product type over two angles with an algebraic order of accuracy of nine is used. The angular grid is constructed in such a way as to correctly calculate the moments for narrowly directed beams of external radiation. The standard 26-group approximation of the BNAB library is used for energy sampling. A decrease in the neutron flux inside the structure is shown for groups with an energy above 1 MeV. Graphs of the angular distribution inside the structure are given.

Methods for constructing invariant cubature formulas for integrals over the surface of a torus in R^3

Methods for constructing invariant cubature formulas for integrals over the surface of a torus in R^3

Optimal cubature formulas for approximate integrals of functions defined on a sphere in three-dimensional space

The development of effective methods of approximate calculation of integrals using optimal cubature formulas and optimal quadrature formulas with trigonometric weights for defined functions on a sphere, the creation of new algorithms for approximate calculation of trigonometric weighted integrals in different classes of functions, as well as the assessment of their errors, is the goal of this work.

Compound cubature formulas on a lattice

In the present paper lattice optimal cubature formulas are constructed by the variational method in the Sobolev space. In addition, the square of the norm of the error functional of the constructed lattice optimal cubature formulas in the conjugate Sobolev space is explicitly calculated.

A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains

This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good conditioning of the method are proved in suitable subspaces of continuous functions of Sobolev type. The cubature formula, on which the Nyström method is constructed, has an error that behaves like the best polynomial approximation of the integrand function. Consequently, it is also shown how the Nyström method inherits this property and, hence, the proposed numerical strategy is fast when the involved known functions are smooth. Some numerical examples illustrate the efficiency of the method, also in comparison with other methods known in the literature.

Search for the Best Cubature Formulas on the Sphere Invariant under the Icosahedral Rotation Group

Search for the Best Cubature Formulas on the Sphere Invariant under the Icosahedral Rotation Group

Compound Cubature Formulas on a Lattice

Compound Cubature Formulas on a Lattice

A Multi-Fidelity Integration Method for Reliability Analysis of Industrial Robots

Abstract A multi-fidelity integration method is proposed to analyze the reliability of multiple performance indicators (MPI) for industrial robots. In order to high-fidelity mapping the performance of industrial robots, a unified multi-domain model (UMDM) is first established. The contribution-degree analysis is then used to classify the input random variables into interacting and non-interacting ones. Thus, the high-dimensional integration of reliability analysis is separated into a low-dimensional integration and multiple one-dimensional integrations in an additive form. Here, the low-dimensional integration consisting of the interacting variables is calculated using the high-precision mixed-degree cubature formula (MDCF), and the computational results are treated as high-fidelity data. The one-dimensional integration consisting of non-interacting variables is then computed by the highly efficient five-point Gaussian Hermite quadrature (FGHQ), and the computational results are named low-fidelity data. A multi-fidelity integration method is constructed by fusing the high-fidelity data and the low-fidelity data to obtain the statistical moments of the MPI. Subsequently, the probability density function and the failure probability of the MPI are estimated using the saddlepoint approximation method. Finally, some representative methods are performed to verify the superiority of the proposed method.