Stable high-order cubature formulas for experimental data

TL;DR

This paper develops stable high-order cubature formulas suitable for experimental data, allowing integration over scattered points with nonnegative weights, and introduces two novel classes of such formulas.

Contribution

It introduces a new approach to construct stable high-order cubature formulas from experimental data, accommodating more data points than basis functions, and proposes two novel classes of these formulas.

Findings

Two novel classes of stable high-order cubature formulas are proposed.

The methods ensure stability by selecting weights that minimize specific norms.

The formulas are applicable to scattered and equidistant data points.

Abstract

In many applications, it is impractical -- if not even impossible -- to obtain data to fit a known cubature formula (CF). Instead, experimental data is often acquired at equidistant or even scattered locations. In this work, stable (in the sense of nonnegative only cubature weights) high-order CFs are developed for this purpose. These are based on the approach to allow the number of data points N to be larger than the number of basis functions K which are integrated exactly by the CF. This yields an (N-K)-dimensional affine linear subspace from which cubature weights are selected that minimize certain norms corresponding to stability of the CF. In the process, two novel classes of stable high-order CFs are proposed and carefully investigated.