Kernel interpolation on generalized sparse grids

TL;DR

This paper introduces a scalable kernel interpolation method on sparse grids for high-dimensional scattered data, with improved error estimates and an efficient solver capable of handling billions of points.

Contribution

It develops a novel kernel interpolation approach on generalized sparse grids, featuring new error bounds and a sparse direct solver with matrix compression for large-scale problems.

Findings

Achieves accurate interpolation on large datasets with billions of points.

Provides improved error estimates for kernel interpolation on sparse grids.

Demonstrates the method's efficiency through numerical experiments.

Abstract

We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data interpolation on the product region. For this, we derive new improved error estimates for the respective kernel interpolation error by invoking duality arguments. An efficient algorithm to solve the underlying linear system of equations is proposed. The algorithm is based on the sparse grid combination technique, where a sparse direct solver is used for the elementary anisotropic tensor product kernel interpolation problems. The application of the sparse direct solver is facilitated by applying a samplet matrix compression to each univariate kernel matrix, resulting in an essentially sparse representation of the latter. In this way, we obtain a method that is able to deal with large problems up to billions of interpolation points, especially in case of reproducing kernels of nonlocal nature. Numerical results are presented to qualify and quantify the approach.