Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

Abstract

A set $\mathcal{X}_{N}$ of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over $\mathcal{X}_{N}$ is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical t-designs on the unit sphere $\mathbb{S}^2\subset\mathbb{R}^3$ when $N\geq(t+1)^2$, the dimension of the space $\mathbb{P}_t$ of spherical polynomials of degree at most t. We show how to construct well conditioned spherical designs with $N\geq(t+1)^2$ points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints. Interval methods are then used to prove the existence of a true spherical t-design very close to the calculated points and to provide a guaranteed interval containing the determinant. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example.