Multivariate piecewise linear interpolation of a random field

TL;DR

This paper analyzes the approximation accuracy of multivariate piecewise linear interpolation for continuous random fields, focusing on locally stationary fields and providing asymptotic and upper bound error estimates.

Contribution

It introduces asymptotic error analysis for locally stationary random fields and bounds for smooth fields, advancing understanding of interpolation performance in high dimensions.

Findings

Asymptotic approximation accuracy for locally stationary fields derived.

Upper bounds for approximation error in smooth fields established.

Performance characterized for large sample sizes and various field classes.

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.