Construction and application of provable positive and exact cubature formulas

TL;DR

This paper introduces a general linear algebra-based method using least squares to construct provably positive and exact cubature formulas for multi-dimensional numerical integration, ensuring stability and accuracy.

Contribution

It presents a novel, broadly applicable approach to derive positive, exact cubature formulas using least squares, expanding beyond standard cases.

Findings

Least squares method guarantees positivity and exactness with enough equidistributed points.

Application to nested high-order rules and positive interpolatory formulas.

Provides insights into conditions for success of existing multivariate integration methods.

Abstract

Many applications require multi-dimensional numerical integration, often in the form of a cubature formula. These cubature formulas are desired to be positive and exact for certain finite-dimensional function spaces (and weight functions). Although there are several efficient procedures to construct positive and exact cubature formulas for many standard cases, it remains a challenge to do so in a more general setting. Here, we show how the method of least squares can be used to derive provable positive and exact formulas in a general multi-dimensional setting. Thereby, the procedure only makes use of basic linear algebra operations, such as solving a least squares problem. In particular, it is proved that the resulting least squares cubature formulas are ensured to be positive and exact if a sufficiently large number of equidistributed data points is used. We also discuss the application of provable positive and exact least squares cubature formulas to construct nested stable high-order rules and positive interpolatory formulas. Finally, our findings shed new light on some existing methods for multivariate numerical integration and under which restrictions these are ensured to be successful.