Efficient Spherical Designs with Good Geometric Properties

Abstract

Spherical $t$-designs on ${\mathbb{S}}^d\subset {\mathbb{R}}^{d+1}$ provide $N$ nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most $t$. This paper considers the generation of efficient, where $N$ is comparable to $(1+t{)}^d/d$, spherical $t$-designs with good geometric properties as measured by their mesh ratio, the ratio of the covering radius to the packing radius. Results for ${\mathbb{S}}^2$ include computed spherical $t$-designs for $t=1,...,180$ and symmetric (antipodal) $t$-designs for degrees up to $325$, all with low mesh ratios. These point sets provide excellent points for numerical integration on the sphere. The methods can also be used to computationally explore spherical $t$-designs for $d=3$ and higher.

to appear in Festschrift for the 80th Birthday of Ian H. Sloan

Cited by in corpus (1)

• Efficient Evaluation of Arbitrary Static Fields For Symplectic Particle Tracking