Analysis of Linear Difference Schemes in the Sparse Grid Combination Technique

TL;DR

This paper derives error expansions for linear difference schemes used in sparse grid combination techniques, analyzing convergence and error behavior in high-dimensional problems with numerical examples up to eight dimensions.

Contribution

It introduces new error expansion formulas for linear difference schemes within sparse grid methods, enabling better understanding of convergence in high-dimensional settings.

Findings

Error formulas of the form ε = O(h^p |log h|^{d-1}) derived

Convergence analysis depends on dimension and smoothness

Numerical illustrations confirm theoretical results up to 8 dimensions

Abstract

Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a $d$-dimensional tensor product space, the number of grid points is $N = \mathcal O(h^{-1} |\log h|^{d-1})$, where $h$ is a mesh parameter. The so-called combination technique, based on hierarchical decomposition and extrapolation, requires specific multivariate error expansions of the discretisation error on Cartesian grids to hold. We derive such error expansions for linear difference schemes through an error correction technique of semi-discretisations. We obtain overall error formulae of the type $\epsilon = \mathcal{O} (h^p |\log h|^{d-1})$ and analyse the convergence, with its dependence on dimension and smoothness, by examples of linear elliptic and parabolic problems, with numerical illustrations in up to eight dimensions.