Error bounds for overdetermined and underdetermined generalized centred simplex gradients

TL;DR

This paper extends the centered simplex gradient method using Moore--Penrose pseudoinverse to handle arbitrary sample set sizes, providing error bounds and calculus rules, with potential benefits demonstrated through examples.

Contribution

It introduces the generalized centred simplex gradient, broadening the original method to any sample set size and establishing error bounds with calculus rules.

Findings

Error bounds of order O(Δ^2) under full-rank condition

Introduction of calculus rules for generalized gradients

Illustrative examples demonstrating method benefits

Abstract

Using the Moore--Penrose pseudoinverse, this work generalizes the gradient approximation technique called centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the \emph{generalized centred simplex gradient}. We develop error bounds and, under a full-rank condition, show that the error bounds have order $O(\Delta^2)$, where $\Delta$ is the radius of the sample set of points used. We establish calculus rules for generalized centred simplex gradients, introduce a calculus-based generalized centred simplex gradient and confirm that error bounds for this new approach are also order $O(\Delta^2)$. We provide several examples to illustrate the results and some benefits of these new methods.