Asymptotic analysis of average case approximation complexity of additive random fields

TL;DR

This paper analyzes how the complexity of approximating additive random fields grows with dimension, providing asymptotic results and applying them to fields with Korobov kernel-based processes.

Contribution

It introduces asymptotic analysis of approximation complexity for additive random fields as dimension increases, under natural assumptions, with applications to Korobov kernel processes.

Findings

Derived general asymptotic formulas for approximation complexity growth.

Established results for additive fields with Korobov kernel marginals.

Provided insights into high-dimensional approximation challenges.

Abstract

We study approximation properties of sequences of centered additive random fields $Y_d$, $d\in\mathbb{N}$. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate $Y_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Under natural assumptions we obtain general results concerning asymptotics of $n^{Y_d}(\varepsilon)$. We apply our results to additive random fields with marginal random processes corresponding to the Korobov kernels.