Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions

Riener, Cordian; Schweighofer, Markus

doi:10.1016/j.jco.2017.10.002

Cited by 9 publications

(11 citation statements)

References 29 publications

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“…(Section 3.3) It also provides a bound on the number of nodes of minimal cubatures, namely,

((), \binom{d + n}{n})

, where d is the degree of the cubature. Analogous results were later proved with less constraint on the measure μ, and recently, this upper bound was revisited for cubatures in the plane . However, the proof of Tchakaloff's theorem is not constructive.…”

Section: Introductionmentioning

confidence: 91%

Algorithms for computing cubatures based on moment theory

Collowald

Hubert

2018

Stud Appl Math

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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.

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“…(Section 3.3) It also provides a bound on the number of nodes of minimal cubatures, namely,

((), \binom{d + n}{n})

Section: Introductionmentioning

confidence: 91%

Algorithms for computing cubatures based on moment theory

Collowald

Hubert

2018

Stud Appl Math

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“…The theory and application of moments is rich, see e.g. [KS53], [Ric57], [Rog58], [AK62], [Akh65], [Kem68], [KN77], [Sch91], [Mat92], [Rez92], [CF96a], [CF96b], [Sim98], [CF00], [Sch03], [FP05], [CF05], [PS08], [Mar08], [Lau09], [FN10], [CF13], [Las15], [Sch15], [Sto16], [Fia17], [IKLS17], [SdD17], [Sch17], [RS18], [dDS18a], [dDS18b], and references therein. But in the present section we only present definitions and results needed in the following sections, especially from [dDS18a] and [dDS18b] with extensions to mixtures as presented in the introduction.…”

Section: Preliminariesmentioning

confidence: 99%

“…In important cases, e.g., Gaussian and log-normal mixtures (Examples 1 and 2), the moment cone has no boundary points despite 0. So "standard" methods to bound C M A in [dDS18b] and [RS18] can not be applied. These "standard" methods are, e.g., "taking an inner point, removing an atom to get to the boundary and describe the boundary" or "close the moment cone by going from R n to projective space P n and to homogeneous polynomials".…”

Section: The Carathéodory Number C Mmentioning

confidence: 99%

“…In all these cases, a boundary point s = 0 would imply that there is an f ∈ lin A, f ≥ 0, such that supp µ ⊆ W(s) ⊆ Z(f ) := {x ∈ X | f (x) = 0} = X for all representing measures µ of s. But this is not possible as long as conditions 1), 2), and 3) shall hold and int supp δ x,σ = ∅, see Theorem 25 iii). So recent methods in [dDS18b] and [RS18] do not apply. Theorem 22 fills the gap.…”

Section: The Carathéodory Number C Mmentioning

confidence: 99%

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The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

Dio

2019

Proc. Amer. Math. Soc.

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We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carathéodory numbers. For A = {a 1 , . . . , am} continuous (sufficiently differentiable) functions on R n we give a general upper bound of m − 1 and a general lower bound of 2m (n+1)(n+2) . For polynomials of degree at most d in n variables we find that the number of Gaussian and log-normal mixtures is bounded by the Carathéodory numbers in [dDS18b]. Therefore, for univariate polynomials {1, x, . . . , x d } at most d+1 2 distributions are needed. For bivariate polynomials of degree at most 2d − 1 we find that 3d(d−1) 2 + 1 Gaussian distributions are sufficient. We also treat polynomial systems with gaps and find, e.g., that for {1, x 2 , x 3 , x 5 , x 6 } 3 Gaussian distributions are enough for almost all truncated moment sequences. For log-normal distributions the number is bounded by half of the moment number. We give an example of continuous functions where more Gaussian distributions are needed than Dirac delta measures. We show that any inner truncated moment sequence has a mixture which contains any given distribution.

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“…The Carathéodory number is the minimal number N such that every truncated moment sequence (with fixed truncation) is a sum of N atoms, i.e., Dirac measures. It has been studied in several contexts but in most cases the precise value of the Carathéodory number is not known [15,16,32,39,42,43,46,53].…”

Section: Introductionmentioning

confidence: 99%

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Dio

Kummer

2021

Math. Ann.

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In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

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Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions

Cited by 9 publications

References 29 publications

Algorithms for computing cubatures based on moment theory

Algorithms for computing cubatures based on moment theory

The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

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