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Zbl 1122.65405
Bungartz, Hans-Joachim; Griebel, Michael
Sparse grids.
(English)
[J] Acta Numerica 13, 147-269 (2004). ISSN 0962-4929; ISSN 1474-0508/e

Summary: We present a survey of the fundamentals and the applications of sparse grids, with a focus on the solution of partial differential equations (PDEs). The sparse grid approach, introduced by {\it C. Zenger} [Notes Numer. Fluid Mech. 31, 241--251 (1991; Zbl 0763.65091)], is based on a higher-dimensional multiscale basis, which is derived from a one-dimensional multi-scale basis by a tensor product construction. Discretizations on sparse grids involve $O(N \cdot (\log N)^{d-1})$ degrees of freedom only, where $d$ denotes the underlying problem's dimensionality and where $N$ is the number of grid points in one coordinate direction at the boundary. The accuracy obtained with piecewise linear basis functions, for example, is $O(N^{-2} \cdot (\log N)^{d-1})$ with respect to the $L_{2^-}$ and $L_{\infty}$-norm, if the solution has bounded second mixed derivatives. This way, the curse of dimensionality, i.e., the exponential dependence $O(N^d)$ of conventional approaches, is overcome to some extent. For the energy norm, only $O(N)$ degrees of freedom are needed to give an accuracy of $O(N^{-1})$. That is why sparse grids are especially well-suited for problems of very high dimensionality. \par The sparse grid approach can be extended to nonsmooth solutions by adaptive refinement methods. Furthermore, it can be generalized from piecewise linear to higher-order polynomials. Also, more sophisticated basis functions like interpolets, prewavelets, or wavelets can be used in a straightforward way. \par We describe the basic features of sparse grids and report the results of various numerical experiments for the solution of elliptic PDEs as well as for other selected problems such as numerical quadrature and data mining.
MSC 2000:
*65N50 Mesh generation and refinement (BVP of PDE)
65-02 Research monographs (numerical analysis)

Keywords: survey article; sparse grids; nonsmooth solutions; adaptive refinement; numerical experiments

Citations: Zbl 0763.65091

Cited in: Zbl 1231.65010 Zbl 1128.82017

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