Optimal compression of kernel matrices by interpolets with application to high-dimensional approximation

Abstract

We consider the compression of kernel matrices on the unit interval $[0,1]$ by interpolets that have sufficiently many vanishing moments. We define a compression rule which discards most matrix coefficients without compromising the accuracy offered by the underlying discretization. Since interpolets can be scaled such that the compressed kernel matrices are well conditioned, we derive a fully discrete scheme that solves a kernel interpolation problem under consideration in linear overall complexity. We finally generalize this approach to the unit $n$-cube $[0,1{]}^n$ by means of the sparse grid combination technique. Numerical experiments are carried out to validate the theoretical findings.