Approximation of additive random fields based on standard information: average case and probabilistic settings

TL;DR

This paper investigates the effectiveness of standard information algorithms in approximating additive and tensor product random fields, showing that they incur at most a logarithmic error loss compared to general linear methods, using randomization techniques.

Contribution

It provides the first detailed analysis of the power of standard information algorithms for these fields, quantifying the approximation error loss compared to general linear information.

Findings

Standard information algorithms have at most a logarithmic error loss.

The results are obtained using randomization techniques.

In most cases, standard information is nearly as effective as general linear information.

Abstract

We consider approximation problems for tensor product and additive random fields based on standard information in the average case setting. We also study the probabilistic setting of the mentioned problem for tensor products. The main question we are concerned with in this paper is ``How much do we loose by considering standard information algorithms against those using general linear information?'' For both types of the fields, the error of linear algorithms has been studied in great detail. However, the power of standard information for them was not addressed so far, which we do here. Our main conclusion is that in most interesting cases there is no more than a logarithmic loss in approximation error when information is being restricted to the standard one. The results are obtained by randomization techniques.