Dimension-adaptive sparse grid quadrature for integrals with boundary singularities

Classical Gaussian quadrature rules achieve exponential convergence for univariate functions that are infinitely smooth and where all derivatives are uniformly bounded. The aim of this paper is to construct generalized Gaussian quadrature rules based on non-polynomial basis functions, which yield exponential convergence even for integrands with (integrable) boundary singularities whose exact type is not a-priori known. Moreover, we use sparse tensor-products of these new formulae to compute d-dimensional integrands with boundary singularities by means of a dimension-adaptive approach. As application, we consider, besides standard model problems, the approximation of multivariate normal probabilities using the Genz-algorithm.