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<p>We study polynomial approximation on a <i>d</i>-cube, where <i>d</i> is large, and compare interpolation on sparse grids, aka&#xa0;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 [<CitationRef CitationID="CR4">4</CitationRef>]. We repeat and extend their theoretical analysis and numerical experiments for SA and compare them 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.</p>

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

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年6月2日

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  • Jakob Eggl,
  • Elias Mindlberger,
  • Mario Ullrich

摘要

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 [4]. We repeat and extend their theoretical analysis and numerical experiments for SA and compare them 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.