Average Case $(s, t)$-weak tractability of non-homogenous tensor product problems

TL;DR

This paper investigates the average case complexity of multivariate problems with Gaussian measures, establishing conditions for $(s,t)$-weak tractability, applicable to various covariance kernels like Euler, Wiener, and Korobov.

Contribution

It provides a necessary and sufficient condition for $(s,t)$-weak tractability in multivariate Gaussian problems with product kernels, extending understanding to several important kernel types.

Findings

Derived a precise condition for $(s,t)$-weak tractability.

Applicable to Euler, Wiener, and Korobov kernels.

Enhances understanding of multivariate Gaussian problem complexity.

Abstract

We study $d$-variate problem in the average case setting with respect to a zero-mean Gaussian measure. The covariance kernel of this Gaussian measure is a product of univariate kernels and satisfies some special properties. We study $(s, t)$-weak tractability of this multivariate problem, and obtain a necessary and sufficient condition for $s>0$ and $t\in(0,1)$. Our result can apply to the problems with covariance kernels corresponding to Euler and Wiener integrated processes, Korobov kernels, and analytic Korobov kernels.