Multilevel Sparse Kernel-Based Interpolation

TL;DR

The paper introduces MLSKI, a multilevel sparse kernel-based interpolation method that efficiently reconstructs large high-dimensional datasets, outperforming classical RBF methods in stability, speed, and convergence.

Contribution

It extends sparse grid ideas to kernel-based functions within a hierarchical multilevel framework, enabling efficient high-dimensional interpolation.

Findings

Numerical experiments show stability and efficiency for large datasets in 2-4 dimensions.

MLSKI outperforms classical RBF methods in complexity, runtime, and convergence.

Effective for datasets with tens to hundreds of thousands of points.

Abstract

A multilevel kernel-based interpolation method, suitable for moderately high-dimensional function interpolation problems, is proposed. The method, termed multilevel sparse kernel-based interpolation (MLSKI, for short), uses both level-wise and direction-wise multilevel decomposition of structured (or mildly unstructured) interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites, whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level. On each level, anisotropic radial basis functions are used for solving a number of small interpolation problems, which are subsequently linearly combined to produce the interpolant. MLSKI can be viewed as an extension of $d$-boolean interpolation (which is closely related to ideas in sparse grid and hyperbolic crosses literature) to kernel-based functions, within the hierarchical multilevel framework to achieve accelerated convergence. Numerical experiments suggest that the new algorithm is numerically stable and efficient for the reconstruction of large data in $\mathbb{R}^{d}\times \mathbb{R}$, for $d = 2, 3, 4$, with tens or even hundreds of thousands data points. Also, MLSKI appears to be generally superior over classical radial basis function methods in terms of complexity, run time and convergence at least for large data sets.