我们研究具有张量积形式的任意对称权函数的 d 维积分的立方体公式。我们提出一种构造，该构造具有高多项式精确度：对于固定次数 l=5 或 l=7 且维数大时，节点数仅略大于 Möller 下界，远小于已知构造。我们还证明，对于任意奇数次 l=2k+1，最小点数几乎与权函数无关。这对于（欧几里得）球面上的积分也成立。
英文摘要：
We study cubature formulas for d-dimensional integrals with an arbitrary symmetric weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree l=5 or l=7 and large dimension, the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions.
We also show, for any odd degree l=2k+1, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.
We propose a new extension of the proof of the Korevaar-Meyers conjecture by Bondarenko, Radchenko and Viazovska for cubature formulas with prefixed weights (we fix different weights for different points) on algebraic and Riemannian manifolds.
We propose and analyse randomized cubature formulae for the numerical integration of functions with respect to a given probability measure $μ$ defined on a domain $Γ\subseteq \mathbb{R}^d$, in any dimension $d$. Each cubature formula is exact on a given finite-dimensional subspace $V_n\subset L^2(Γ,μ)$ of dimension $n$, and uses pointwise evaluations of the integrand function $ϕ: Γ\to \mathbb{R}$ at $m>n$ independent random points. These points are drawn from a suitable auxiliary probability measure that depends on $V_n$. We show that, up to a logarithmic factor, a linear proportionality betwe…
Many applications require multi-dimensional numerical integration, often in the form of a cubature formula. These cubature formulas are desired to be positive and exact for certain finite-dimensional function spaces (and weight functions). Although there are several efficient procedures to construct positive and exact cubature formulas for many standard cases, it remains a challenge to do so in a more general setting. Here, we show how the method of least squares can be used to derive provable positive and exact formulas in a general multi-dimensional setting. Thereby, the procedure only makes…
Cubature formulas (CFs) based on radial basis functions (RBFs) have become an important tool for multivariate numerical integration of scattered data. Although numerous works have been published on such RBF-CFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for RBF-CFs. In particular, we prove stability for RBF-CFs based on compactly supported RBFs under certain conditions on the shape parameter and the data points. Moreover, it is shown that asymptotic stability of many RBF-CFs is independent of polynom…
In many applications, it is impractical -- if not even impossible -- to obtain data to fit a known cubature formula (CF). Instead, experimental data is often acquired at equidistant or even scattered locations. In this work, stable (in the sense of nonnegative only cubature weights) high-order CFs are developed for this purpose. These are based on the approach to allow the number of data points N to be larger than the number of basis functions K which are integrated exactly by the CF. This yields an (N-K)-dimensional affine linear subspace from which cubature weights are selected that minimize…
Positive interpolatory cubature formulas (CFs) are constructed for quite general integration domains and weight functions. These CFs are exact for general vector spaces of continuous real-valued functions that contain constants. At the same time, the number of data points -- all of which lie inside the domain of integration -- and cubature weights -- all positive -- is less or equal to the dimension of that vector space. The existence of such CFs has been ensured by Tchakaloff in 1957. Yet, to the best of the author's knowledge, this work is the first to provide a procedure to successfully con…
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…
Optimal with respect to accuracy algorithms for calculation of multidimensional weakly singular integrals and applications to calculation of capacitances of conductors of arbitrary shapes
Optimal with respect to accuracy algorithms for calculation of multidimensional weakly singular integrals and applications to calculations of capacitances of conductors of arbitrary shapes
Computation of volume potentials over bounded domains via approximate approximations
Sapienza University of Rome（罗马第一大学）；University of Linköping（林雪平大学）；University of Liverpool（利物浦大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）
In this paper the numerical solution of potential problems defined on 3D unbounded domains is addressed with Boundary Element Methods (BEMs), since in this way the problem is studied only on the boundary, and thus any finite approximation of the infinite domain can be avoided. The isogeometric analysis (IGA) setting is considered and in particular B-splines and NURBS functions are taken into account. In order to exploit all the possible benefits from using spline spaces, an important point is the development of specific cubature formulas for weakly and nearly singular integrals. Our proposal f…
