On the construction of general cubature formula by flat extensions

TL;DR

This paper introduces a novel method for constructing general cubature formulas using flat extensions of Hankel operators, involving algebraic analysis, optimization, and algorithms to minimize points and compute formulas efficiently.

Contribution

It presents a new approach combining algebraic properties and convex optimization to compute minimal cubature formulas via flat extensions of Hankel operators.

Findings

Successfully computes cubature formulas with minimal points

Develops an algorithm to test flat extension property

Demonstrates the method on various examples

Abstract

We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyse the algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated Hankel operator. We next present an algorithm to test the flat extension property and to additionally compute the decomposition. To generate cubature formulae with a minimal number of points, we propose a new relaxation hierarchy of convex optimization problems minimizing the nuclear norm of the Hankel operators. For a suitably high order of convex relaxation, the minimizer of the optimization problem corresponds to a cubature formula. Furthermore cubature formulae with a minimal number of points are associated to faces of the convex sets. We illustrate our method on some examples, and for each we obtain a new minimal cubature formula.