@Article{ Gerstner.Griebel:1998,
author = {Gerstner, T. and Griebel, M.},
title = {Numerical Integration using Sparse Grids},
journal = {Numer. Algorithms},
volume = {18},
note = {(also as SFB 256 preprint 553, Univ. Bonn, 1998)},
pages = {209--232},
ps = {http://wissrech.ins.uni-bonn.de/research/pub/gerstner/quad.ps.gz}
,
abstract = { We present new and review existing algorithms for the
numerical integration of multivariate functions defined
over $d$--dimensional cubes using several variants of the
sparse grid method first introduced by Smolyak. In this
approach, multivariate quadrature formulas are constructed
using combinations of tensor products of suited
one--dimensional formulas. The computing cost is almost
independent of the dimension of the problem if the function
under consideration has bounded mixed derivatives. We
suggest the usage of extended Gauss (Patterson) quadrature
formulas as the one--dimensional basis of the construction
and show their superiority in comparison to previously used
sparse grid approaches based on the trapezoidal,
Clenshaw--Curtis and Gauss rules in several numerical
experiments and applications. For the computation of path
integrals further improvements can be obtained by combining
generalized Smolyak quadrature with the Brownian bridge
construction. },
year = {1998},
annote = {article}
}