A higher order sparse grid combination technique

TL;DR

This paper introduces a generalized sparse grid combination technique that enhances the accuracy of finite difference solutions from second to fourth order in multiple dimensions, improving computational efficiency for high-dimensional problems.

Contribution

It develops a new sparse grid method combining multi-variate extrapolation with standard techniques, extending accuracy and applicability to higher dimensions.

Findings

Achieves fourth order accuracy on sparse grids

Validates the method with Poisson problem in up to seven dimensions

Provides a theoretical error expansion and practical convergence results

Abstract

We show that a generalised sparse grid combination technique which combines multi-variate extrapolation of finite difference solutions with the standard combination formula lifts a second order accurate scheme on regular meshes to a fourth order combined sparse grid solution. In the analysis, working in a general dimension, we characterise all terms in a multivariate error expansion of the scheme as solutions of a sequence of semi-discrete problems. This is first carried out formally under suitable assumptions on the truncation error of the scheme, stability and regularity of solutions. We then verify the assumptions on the example of the Poisson problem with smooth data, and illustrate the practical convergence in up to seven dimensions.