Limiting behaviour of the generalized simplex gradient as the number of points tends to infinity on a fixed shape in R^n

TL;DR

This paper analyzes the asymptotic behavior of the generalized simplex gradient method, demonstrating that with proper construction, its error bounds remain finite even as the number of sample points increases infinitely.

Contribution

It introduces new error bounds for the GSG method that do not depend on the number of sample points, clarifying its limiting behavior.

Findings

Error bounds for GSG remain finite as points tend to infinity.

Proper construction of sample points ensures bounded approximation error.

The results apply to functions in finite-dimensional spaces.

Abstract

This work investigates the asymptotic behaviour of the gradient approximation method called the generalized simplex gradient (GSG). This method has an error bound that at first glance seems to tend to infinity as the number of sample points increases, but with some careful construction, we show that this is not the case. For functions in finite dimensions, we present two new error bounds ad infinitum depending on the position of the reference point. The error bounds are not a function of the number of sample points and thus remain finite.