Multidimensional Hierarchical Interpolation Method on Sparse Grids for the Absorption Problem

The numerical integration of multidimensional functions using some variables of the sparse grid method for the absorption problem is presented in this paper. The multivariate quadrature expressions are constructed by combining tensor of suited one dimensional formula. We develop a multidimensional adaptive quadrature algorithm for the implementation of sparse grid based on a hierarchical basis. Furthermore, we obtain a new error bound at each sparse grid point. The numerical examples are shown to demonstrate the efficiency of our algorithm for the absorption problem and confirm the theoretical estimates.