Zooming from Global to Local: A Multiscale RBF Approach

TL;DR

This paper introduces a multiscale radial basis function approach that allows for global approximation and local refinement on Earth's surface, with proven stability and an illustrative numerical example.

Contribution

It develops a multiscale RBF method enabling local zoom-in refinements on spherical data, with theoretical stability guarantees and practical implementation.

Findings

Stable multiscale refinement process on the sphere.

Bounded condition numbers across scales.

Numerical example demonstrating local zoom-in capability.

Abstract

Because physical phenomena on Earth's surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can "zoom in" on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in $\mathbb{R}^d$ from Wendland, Numer. Math. 116 (2012). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process.