Higher order quadrature on sparse grids

Abstract

Sparse grids have turned out to be a very efficient discretization scheme that, to some extent, breaks the curse of dimensionality and, therefore, is especially well-suited for higher dimensional scenarios. Besides the classical sparse grid application, the numerical solution of partial differential equations, sparse grids have been used for various topics such as Fourier transform, image compression, numerical quadrature, or data mining, so far. In this paper, we summarize and assess recent results concerning the application of sparse grids to integrate functions of higher dimensionality, the focus being on the explicit and adaptive use of higher order basis polynomials. © Springer-Verlag Berlin Heidelberg 2004.