A Moment Matrix Approach to Multivariable Cubature

Abstract

Abstract. We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure µ, the bound is based on estimating ρ(C) := inf{rank (T −C) : T Toeplitz and T ≥ C}, where C := C [µ] is a positive matrix naturally associated with the moments of µ. We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag (c1, . . . , cn) (including the case when µ is planar measure on the unit disk), ρ(C) is at least as large as the number of gaps c k > c k+1 .

Algorithms for computing cubatures based on moment theory

“…The link between cubatures and flat extensions was first expanded in Refs. and , and then provided a criterion for the existence of a Gaussian cubature, while Bucero et al showed how to compute some cubatures as a global optimization problem. The connection between Hankel operators and multiplication maps, which allows one to compute the cubature nodes as an eigenvalue problem, was, for instance, used in Refs.…”

“…More recently, a characterization of cubatures based on moment matrices was initiated in Ref. . Additional contributions in this direction were made regarding Gaussian cubatures or on a numerical method to compute some of them .…”

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

“…There is a natural indexing of the multivariate h-polynomials: each monomial α, in the variates ω, determines a vector of integer exponents 〈a ω 〉 ω∊ω defined by (20) and we define the corresponding multivariate h-polynomial by (21) so α is the pure monomial part of the unique highest degree term in h α . The natural indexing is a notational convenience: for coding purposes, it is useful to maintain a table of the various exponent vectors ⟨a ω ⟩ ω∊ω .…”

“…On small domains, where relatively few eigenvalues are important, one might expect the coupling among the first nine variates to be more important the coupling between the first nine and the last eleven, while the latter coupling effects become relatively more significant with increasing domain size. To test this, we replaced H α computed by PC for the (20, 2)-truncation by the corresponding H α computed for the (20, 2)-truncation; the replacements were made for the H α with index α in the set of 55 indices τ(9, 3) ∩ τ (20,2) common to both truncations. Acceleration factors were computed as before by MC, using the (20, 2)-table with the indicated replacements.…”

“…KLME offers an adaptive method, which is competitive in the low-variance case with PC expansions, when not too many terms are included in the expansion. Both cubature rules and interpolatory methods deserve further examination as approaches to stochastic problems; for more information, see [7,10,12,15,20,33,48,71,81]). …”