Sparse grids and related approximation schemes for higher dimensional
In L. Pardo, A. Pinkus, E. Suli, and M. Todd, editors,
Foundations of Computational Mathematics (FoCM05), Santander, pages
106-161. Cambridge University Press, 2006.
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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 constant.