Piecewise-Multilinear Interpolation of a Random Field

Abstract

We consider a multivariate piecewise linear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured by the integrated mean square error. Multivariate piecewise linear interpolator is defined by N field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field in mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large N . Moreover, for certain classes of continuous and continuously differentiable fields we provide the upper bound for the approximation accuracy in the uniform mean square norm.

Extremes of 𝛼(𝑡)-locally stationary Gaussian random fields

“…Moreover, the analysis of the extremes of GRF's leads to technical difficulties, see e.g., the excellent monographs Piterbarg (1996) and Adler and Taylor (2007). Recently, Abramowicz and Seleznjev (2011) deal with multivariate piecewise linear interpolation of locally stationary random fields, whereas Hashorva et al (2012) investigates the piece-wise approximation of α(t)locally stationary processes. With motivation from the aforementioned papers and Dȩbicki and Kisowski (2008), we consider, in this paper, extremes of α(t)-locally stationary GRF {X(t), t ∈ [0, T ] k } (to be defined below).…”

Stratified Monte Carlo Quadrature for Continuous Random Fields

“…Let the hypercube D be partitioned into hyperrectangular strata by design points T N , for N ≥ 1. We consider cross regular sequences of grid designs (see, e.g., Abramowicz and Seleznjev, 2011a). The designs T N :…”

“…where by the positiveness and uniform continuity of local stationarity functions, we have that ε N = max{|q N,i |, i ∈ I} = o(1) as N → ∞ (cf. Abramowicz and Seleznjev, 2011a). Recall that the hyperrectangle D i is determined by the vertex t i = (t 1,i1 , .…”

“…The proof is based on the inequality for the arithmetic and geometric means (cf. Abramowicz and Seleznjev, 2011a), i.e.,…”

“…We use cross regular sequences of designs, generalizing the well known regular sequences pioneered by Sacks and Ylvisaker (1966). We focus on random fields satisfying a local stationarity condition proposed for stochastic processes by Berman (1974) and extended for random fields in Abramowicz and Seleznjev (2011a). Approximation of random functions from this class is studied in, e.g., Abramowicz and Seleznjev (2011a,b); Hüsler et al (2003); Seleznjev (2000).…”