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

TL;DR

This paper enhances the tensor product multilevel method (TPML) for high-dimensional function approximation on sparse grids, reducing computational costs and demonstrating its effectiveness through numerical examples.

Contribution

It introduces two improvements to TPML that lower evaluation costs and provides numerical evidence of its effectiveness and innovation.

Findings

Reduced computational cost of point evaluations in TPML

Demonstrated effectiveness of improved TPML through numerical examples

Validated the method's applicability to high-dimensional, kernel-based reconstruction

Abstract

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.