Towards stability of radial basis function based cubature formulas

TL;DR

This paper advances the stability theory of radial basis function-based cubature formulas, establishing conditions for their stability and independence from polynomial terms, thus improving their reliability for multivariate numerical integration.

Contribution

It provides new stability conditions for RBF-CFs with compactly supported RBFs and shows asymptotic stability independence from polynomial terms.

Findings

Proved stability for RBF-CFs under specific conditions.

Demonstrated asymptotic stability is independent of polynomial terms.

Identified gaps for future research in RBF-CF stability theory.

Abstract

Cubature formulas (CFs) based on radial basis functions (RBFs) have become an important tool for multivariate numerical integration of scattered data. Although numerous works have been published on such RBF-CFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for RBF-CFs. In particular, we prove stability for RBF-CFs based on compactly supported RBFs under certain conditions on the shape parameter and the data points. Moreover, it is shown that asymptotic stability of many RBF-CFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of RBF-CFs, the present work also demonstrates that there are still many gaps to fill in future investigations.