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

  author = {M.~Griebel},
  title = {Sparse grids and related approximation schemes for higher
		  dimensional problems},
  booktitle = {Foundations of Computational Mathematics (FoCM05),
  editor = {L. Pardo and A. Pinkus and E. Suli and M.J. Todd},
  year = {2006},
  pages = {106-161},
  publisher = {Cambridge University Press},
  annote = {series,C2,ALM,1145},
  abstract = {The efficient numerical treatment of high dimensional
		  problems is hampered by the curse of dimensionality. We
		  review approximation techniques which overcome this problem
		  to some extent. Here, we focus on methods stemming from
		  Kolmogorov's theorem, the ANOVA decomposition and the
		  sparse grid approach and discuss their prerequisites and
		  properties. Moreover, we present energy-norm based sparse
		  grids and demonstrate that, for functions with bounded
		  mixed derivatives on the unit hypercube, the associated
		  approximation rate in terms of the involved degrees of
		  freedom shows no dependence on the dimension at all,
		  neither in the approximation order nor in the order
  pdf = { 1}