Numerical Integration over the Unit Sphere by using spherical t-design
For researchers in numerical analysis and approximation theory, this work provides a comparative study of spherical t-designs for integration, though it is incremental as it builds on existing designs and transformations.
This paper investigates numerical integration over the unit sphere using spherical t-designs, comparing well-conditioned (WSTD) and efficient (ESTD) designs up to t=160. Results show WSTD outperforms ESTD in worst-case error in Sobolev space and is effective for integrands with point singularities, achieving better uniform errors than bivariate trapezoidal rule and equal area points.
This paper studies numerical integration over the unit sphere $ \mathbb{S}^2 \subset \mathbb{R}^{3} $ by using spherical $t$-design, which is an equal positive weights quadrature rule with polynomial precision $t$. We investigate two kinds of spherical $t$-designs with $t$ up to 160. One is well conditioned spherical $t$-design(WSTD), which was proposed by [1] with $ N=(t+1)^{2} $. The other is efficient spherical $t$-design(ESTD), given by Womersley [2], which is made of roughly of half cardinality of WSTD. Consequently, a series of persuasive numerical evidences indicates that WSTD is better than ESTD in the sense of worst-case error in Sobolev space $ \mathbb{H}^{s}(\mathbb{S}^2) $. Furthermore, WSTD is employed to approximate integrals of various of functions, especially including integrand has a point singularity over the unit sphere and a given ellipsoid. In particular, to deal with singularity of integrand, Atkinson's transformation [3] and Sidi's transformation [4] are implemented with the choices of `grading parameters' to obtain new integrand which is much smoother. Finally, the paper presents numerical results on uniform errors for approximating representive integrals over sphere with three quadrature rules: Bivariate trapezoidal rule, Equal area points and WSTD.