A Node Elimination Algorithm for Cubature of High-Dimensional Polytopes

TL;DR

This paper presents a novel node elimination algorithm for constructing efficient high-dimensional cubature rules for polytopes, improving upon existing methods by introducing new selection criteria and optimization strategies.

Contribution

It introduces a new node elimination criterion based on linearized moment equations and a penalized solver to ensure positive weights and valid nodes within the domain.

Findings

High efficiency cubature rules for 2-4 dimensional polytopes

Improved node selection method enhances rule quality

New rules outperform tensor product and analytical rules

Abstract

Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. This paper introduces a new criterion for selecting which nodes to eliminate that is based on a linearization of the moment equation. In addition, a penalized iterative solver is introduced, that ensures that weights are positive and nodes are inside the integration domain. A strategy for constructing an initial quadrature rule for various polytopes in several space dimensions is described. High efficiency rules are presented for two, three and four dimensional polytopes. The new rules are compared with rules that are obtained by combining tensor products of one dimensional quadrature rules and domain transformations, as well as with known analytically constructed cubature rules.