Efficient Cubature Rules

TL;DR

This paper introduces 73 new efficient cubature rules for multidimensional integrals with symmetric regions and weights, significantly reducing the number of points needed compared to existing rules, and including high-degree formulas for specific cases.

Contribution

The paper presents 73 novel cubature rules that improve efficiency and reduce points needed for multidimensional integrals, including high-degree formulas for sphere and Gaussian-weighted integrals.

Findings

Most new rules have fewer points than existing ones of the same degree.

Twenty rules are within three points of Möller's lower bound.

Most rules have positive coefficients and some exhibit symmetry.

Abstract

73 new cubature rules are found for three standard multidimensional integrals with spherically symmetric regions and weights, using direct search with a numerical zero-finder. All but four of the new rules have fewer integration points than known rules of the same degree, and twenty are within three points of M{\"o}ller's lower bound. Most have all positive coefficients and most have some symmetry, including some supported by one or two concentric spheres. They include degree 7 formulas for integration over the sphere and Gaussian-weighted integrals over all space, each in 6 and 7 dimensions, with 127 and 183 points, respectively.