Gaussian quadrature for splines via homotopy continuation: rules for $C^2$ cubic splines

TL;DR

This paper presents a novel method for deriving optimal quadrature rules for $C^2$ cubic splines by transferring known rules from a source space using polynomial homotopy continuation, validated through numerical experiments.

Contribution

It introduces a homotopy continuation approach to generate optimal quadrature rules for spline spaces, including $C^2$ cubic splines, which were previously only conjectured.

Findings

Successfully derived optimal rules for $C^2$ cubic splines

Quadrature rules are independent of the deformation path

Method confirms conjectured rules for specific spline spaces

Abstract

We introduce a new concept for generating optimal quadrature rules for splines. Given a target spline space where we aim to generate an optimal quadrature rule, we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one, preserving the number of quadrature points and therefore optimality. The quadrature nodes and weights are, considered as a higher-dimensional point, a zero of a particular system of polynomial equations. As the space is continuously deformed by modifying the source knot vector, the quadrature rule gets updated using polynomial homotopy continuation. For example, starting with $C^1$ cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform $C^2$ cubic spline spaces where the rule was only conjectured heretofore. We validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen.