Efficient integration of intensity functions on the unit sphere

To integrate peaking intensity functions over all ray directions, commonly occuring in radiative transfer calculations, we present efficient quadrature formulae by calculating appropriate nodes and weights. Instead of product formulae using univariate quadrature rules we construct multivariate quadrature formulae for the sphere. Due to the fact that there is no Gaussian quadrature for the unit sphere for grid point numbers of interest, approximate grids and corresponding weights have to be calculated. Using a special Metropolis algorithm, we minimize the potential energy of an N-charged particle distribution on the sphere and discuss the resulting, nearly isotropically distributed configurations. We find that the vertices of the cube and pentagon dodecahedron are not the optimal distribution, although they have as Platonian bodies equally distributed vertices. The algorithm finds even high-resolving grids (N similar to 1000) with moderate computational effort (4 h on a 30 MFlop workstation). The corresponding weights of the quadrature rule are obtained by evaluating special Gegenbauer polynomials at products of the nodes and inverting the resulting symmetric matrix by Cholesky-decomposition. Thus we get very precise quadrature rules (with a relative error of the order 10(-12)) though the weights are not equal. Copyright (C) 1996 Elsevier Science Ltd.