Sparse grids (Quantitative Finance)

Abstract

One coordinate direction at the boundary. The accuracy obtained with piece- wise linear basis functions, for example, is O(N2 (logN)d1) with respect to the L2-and L-norm, if the solution has bounded second mixed derivatives. This way, the curse of dimensionality, i.e., the exponential dependence O(Nd) 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(N1). 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.