Sparse grids

Abstract

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 in Zenger (1991), 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.

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.

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.