Sparse grids vs. random points for high-dimensional polynomial approximation

TL;DR

This paper compares sparse grid interpolation and random point least squares methods for high-dimensional polynomial approximation, showing that least squares often outperforms sparse grids in very high dimensions.

Contribution

It extends previous theoretical and numerical analysis of sparse grids and demonstrates the superiority of least squares methods in high-dimensional settings.

Findings

Least squares matches sparse grid accuracy in low dimensions.

In high dimensions, least squares significantly outperforms sparse grids.

Extensive experiments up to 100 dimensions support these conclusions.

Abstract

We study polynomial approximation on a $d$-cube, where $d$ is large, and compare interpolation on sparse grids, aka Smolyak's algorithm (SA), with a simple least squares method based on randomly generated points (LS) using standard benchmark functions. Our main motivation is the influential paper [Barthelmann, Novak, Ritter: High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12, 2000]. We repeat and extend their theoretical analysis and numerical experiments for SA and compare to LS in dimensions up to 100. Our extensive experiments demonstrate that LS, even with only slight oversampling, consistently matches the accuracy of SA in low dimensions. In high dimensions, however, LS shows clear superiority.