Higher-Dimensional Integration with Gaussian Weight for Applications in Probabilistic Design

Higher-dimensional Gaussian weighted integration is of interest in probabilistic simulations. Motivated by the need for variance calculations with functions being at least quadratic, the family of degree 5 formulae is considered. Using an existing formula for the integration over the surface of an n-sphere, an efficient, new formula for Gaussian weighted integration is obtained. Several other formulae that have appeared in the numerical integration literature are also given. The number of function evaluations required by the formulae is compared to a minimal bound result. The degree 5 formulae are applied to simple test problems and the relative errors are compared.