2016/11/09 by An, Congpei, Chen, Siyong
#65D30 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1611.02785
This paper studies numerical integration over the unit sphere \mathbbS2 ⊂ ℝ3 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 ℍs(\mathbbS2) . 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.