Error bounds for overdetermined and underdetermined generalized centred simplex gradients. (11th December 2020)

• Record Type: Journal Article
• Authors: Hare, Warren; Jarry–Bolduc, Gabriel; Planiden, Chayne
• Is Part Of: IMA journal of numerical analysis. Volume 42:Number 1(2022)
• Page Start: 744
• Page End: 770
• Publication Date: 2020-12-11
• Subjects: Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/draa089
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: Using the Moore–Penrose pseudoinverse this work generalizes the gradient approximation technique called the centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the generalized centred simplex gradient . We develop error bounds and, under a full-rank condition, show that the error bounds have ${\mathcal O}(\varDelta ^2)$, where $\varDelta $ 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 ${\mathcal O}(\varDelta ^2)$ . We provide several examples to illustrate the results and some benefits of these new methods.