Interpolation lattices in several variables

Abstract

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices. We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. (2000): 41A05 · 41A63 · 65D05 Mathematics Subject Classification

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

“…(1), for example, we obtain (8) (9) where (10) Similarly, Eqs. (2) and (3) produce (11) (12) where (13) Similarly, rewriting Eq. (1) in the form (14) multiplying by h ϕ and computing expectations yields (15) To proceed further, it is necessary to specify the field K as a function of the variables ω and their joint density ρ; specify the polynomials h α ; and compute explicitly certain integrals such as G βα .…”

“…The corresponding interpolatory polynomials are defined by (59) If instead, one dilates the standard simplex by the factor d, the knots coincide with the vectors ζ and the polynomials become p ζ (ω/d). Carnicer et al [12] have obtained error estimates for such interpolations. A simple algorithm thus results by solving Eq.…”

“…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]). …”

Three-pencil lattices on triangulations

“…The first step to determine the points T n−k−j,k,j is the following. Lemma 1 Let T n−k−j,k,j , k, j ≥ 0, k + j ≤ n, be points of a three-pencil lattice, generated by centers C i given in (1). Then…”

“…In [5], these lattices have been generalized to the case of not necessarily parallel hyperplanes intersecting in so called centers. These lattices are known as (d + 1)-pencil lattices of order n. Further generalizations can be found in [1]. It is well-known that all these lattices admit correct interpolation in d n since they satisfy the GC condition (cf.…”