On Quasi-Localized Dual Pairs in Reproducing Kernel Hilbert Spaces

TL;DR

This paper explores various basis choices in scattered data approximation within reproducing kernel Hilbert spaces, emphasizing localized and orthogonal bases for improved computational properties and demonstrating their effectiveness through numerical experiments.

Contribution

It introduces and compares localized Lagrange, orthogonal, and multiresolution bases, highlighting their advantages and providing benchmark results for scattered data approximation.

Findings

Orthogonal bases lead to symmetric preconditioners.

Localized bases are feasible for scattered data approximation.

Numerical experiments validate the effectiveness of the proposed bases.

Abstract

In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics.