Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  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 = { 1},
  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}