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<p>This paper investigates the approximation of functions with finite smoothness defined on domains with a Cartesian product structure. The recently proposed <i>tensor product multilevel method (TPML)</i> combines Smolyak’s sparse grid method with a kernel-based residual correction technique. The contributions of this paper are twofold. First, we present two improvements on the TPML that reduce the computational cost of point evaluations compared to a naive implementation. Second, we provide numerical examples that demonstrate the effectiveness and innovation of the TPML.</p>

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Numerical Aspects of the Tensor Product Multilevel Method for High-Dimensional, Kernel-Based Reconstruction on Sparse Grids

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年12月2日

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  • Markus Büttner,
  • Rüdiger Kempf,
  • Holger Wendland

摘要

This paper investigates the approximation of functions with finite smoothness defined on domains with a Cartesian product structure. The recently proposed tensor product multilevel method (TPML) combines Smolyak’s sparse grid method with a kernel-based residual correction technique. The contributions of this paper are twofold. First, we present two improvements on the TPML that reduce the computational cost of point evaluations compared to a naive implementation. Second, we provide numerical examples that demonstrate the effectiveness and innovation of the TPML.