Related papers: Cubature Formulas for Symmetric Measures in Higher…
Cubature formulas, asymptotically optimal with respect to accuracy, are derived for calculating multidimensional weakly singular integrals. They are used for developing a universal code for calculating capacitances of conductors of…
Cubature formulas, asymptotically optimal with respect to accuracy, are derived for calculating multidimensional weakly singular integrals. They are used for developing a universal code for calculating capacitances of conductors of…
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…
This paper is a second part of our study of the Discrete Polyharmonic Cubature Formulas on the disc. It completes our study and provides a satisfactory cubature formula in terms of precision and number of evaluation points (coefficient of…
Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with…
Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods are classical approaches for the numerical integration of functions $f$ over $[0,1]^d$. While QMC methods can achieve faster convergence rates than MC in moderate dimensions, their…
Building on techniques developed by Lyons and Victoir, we present the first explicit construction of a degree-7 cubature formula for Wiener space over $\mathbb{R}^3$. We then examine and compare two approaches for computing cubature…
We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by…
Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the $s$-dimensional unit cube. Since no explicit constructions of such quadrature methods are known for dimensions $s > 2$, one usually has to…
Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the…
We derive new formulas for the high dimensional biharmonic potential acting on Gaussians or Gaussians times special polynomials. These formulas can be used to construct accurate cubature formulas of an arbitrary high order which are fast…
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…
We construct an interpolatory high-order cubature rule to compute integrals of smooth functions over self-affine sets with respect to an invariant measure. The main difficulty is the computation of the cubature weights, which we…
Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way,…
Symmetric polynomial quadrature rules for triangles are commonly used to efficiently integrate two-dimensional domains in finite-element-type problems. While the development of such rules focuses on the maximum degree a given number of…
As a complement to \cite{X12}, minimal cubature rules of degree $4m+1$ for the weight functions $$ \mathcal{W}_{\alpha,\beta ,\pm \frac12}(x,y) = |x+y|^{2\alpha+1} |x-y|^{2\beta+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are shown…
An important question in the theory of approximate integration is to study the conditions on the nodes $x_{k,n}$ and weights $w_{k,n}$ that allow an estimate of the form $$ \sup_{f\in \mathcal{B}_\gamma}|\sum_k…
We prove lower bounds for the error of optimal cubature formulae for $d$-variate functions from Besov spaces of mixed smoothness $B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 < p, \theta \le \infty$ and $\alpha > 1/p$, where…
We present novel fully-symmetric quadrature rules with positive weights and strictly interior nodes of degrees up to 84 on triangles and 40 on tetrahedra. Initial guesses for solving the nonlinear systems of equations needed to derive…
The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons. In the numerical section we…