Anisotropic Kernel-based Multilevel Interpolation of High-Dimensional Functions on Sparse Grids

This paper presents the Tensor Product Multilevel (TPML) method for approximating high-dimensional functions from scattered data on general bounded domains. The approach combines Smolyak's sparse grid construction with kernel-based multilevel interpolation, where the kernel scale is adaptively coupled to the fill distance at each level to maintain uniformly bounded condition numbers. The original formulation relied implicitly on the target function, limiting numerical evaluation; this work addresses that limitation.