Generation and application of multivariate polynomial quadrature rules

TL;DR

This paper introduces new mathematical bounds and an efficient algorithm for generating minimal multivariate polynomial quadrature rules with positive weights, outperforming existing methods in high-dimensional applications.

Contribution

It provides a novel lower bound analysis for quadrature node counts and a new algorithm for constructing optimal rules on complex domains.

Findings

The algorithm successfully generates rules in up to 20 dimensions.

Generated rules outperform sparse grids, Monte Carlo, and Stroud rules in tests.

The method applies to dimension reduction and chemical kinetics problems.

Abstract

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second main contribution comes in the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.