Algorithms for computing cubatures based on moment theory

Abstract

Quadrature is an approximation of the definite integral of a function by a weighted sum of function values at specified points, or nodes, within the domain of integration. Gaussian quadratures are constructed to yield exact results for any polynomials of degree 2r−1 or less by a suitable choice of r nodes and weights. Cubature is a generalization of quadrature in higher dimension. In this article, we elaborate algorithms to compute all minimal cubatures for a given domain and a given degree. We propose first algorithm in symbolic computation to characterize all cubatures of a given degree with a fixed number of nodes. The determination of the nodes and weights is then left to the computation of the eigenvectors of the matrix identified at the characterization stage and can be performed numerically. The characterization of cubatures on which our algorithms are based stems from moment theory. We formulate the results there in a basis‐independent way: Rather than considering the moment matrix, the central object in moment problems, we introduce the underlying linear map from the polynomial ring to its dual, the Hankel operator. This makes natural the use of bases of polynomials other than the monomial basis, and proves to be computationally relevant, either for numerical properties or to exploit symmetry.

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

“…Identifying the support of a linear form on a polynomial ring K[x] already has applications in optimization, tensor decomposition and cubature [1,2,9,13,15,36,37]. How to take advantage of symmetry in some of these applications appears in [14,21,52].…”

“…The usual Hankel or mixed Hankel-Toepliz matrices that appeared in the literature on sparse interpolation [8,35] are actually the matrices of the Hankel operator mentioned above in the different univariate polynomial bases considered. The recovery of the support of a linear form with this type of technique also appears in optimization, tensor decomposition and cubature [2,9,13,15,36,37]. We present new developments to take advantage of the invariance or semi-invariance of the linear form.…”

Waring, tangential and cactus decompositions

“…Symmetric Tensor Decomposition (SymTD) is one of the most active research topic of the last decades and it has received many attentions both from the pure mathematical community and applied ones (signal processing [31], phylogenetics [2], quantum information [38,41,17], computational complexity [43], geometric modeling of shapes [29]). The push towards the generation of algorithms that efficiently compute a specific type of decomposition of given symmetric tensors has not only a practical interest but also extremely deep theoretical facets.…”

Preface: Orthogonal polynomials, special functions, and applications

“…This special issue features papers by plenary speakers at OPSFA14. These reflect the varied nature of talks at the conference with papers discussing aspects of Askey–Wilson polynomials, Sonine formulas for Bessel functions, cubatures, exceptional orthogonal polynomials, rational solutions of Painlevé equations, and multipeakon solutions of partial differential equations …”

“…Collowald and Hubert give a symbolic computation algorithm to characterize all cubatures, which are generalizations of quadratures to numerically compute multiple integrals.…”