EFFICIENT INTEGRATION OF INTENSITY FUNCTIONS ON THE UNIT-SPHERE

To integrate peaking intensity functions over all ray directions, comm only occuring in radiative transfer calculations, we present efficient quadrature formulae by calculating appropriate nodes and weights. Ins tead of product formulae using univariate quadrature rules we construc t multivariate quadrature formulae for the sphere. Due to the fact tha t there is no Gaussian quadrature for the unit sphere for grid point n umbers of interest, approximate grids and corresponding weights have t o 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 configurat ions. We find that the vertices of the cube and pentagon dodecahedron are not the optimal distribution, although they have as Platonian bodi es equally distributed vertices. The algorithm finds even high-resolvi ng grids (N similar to 1000) with moderate computational effort (4 h o n a 30 MFlop workstation). The corresponding weights of the quadrature rule are obtained by evaluating special Gegenbauer polynomials at pro ducts of the nodes and inverting the resulting symmetric matrix by Cho lesky-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.