# Optimal selection of local approximants in RBF-PU interpolation

## TL;DR

This paper introduces a method for optimally selecting local data set sizes and shape parameters in RBF-PU interpolation, enhancing accuracy for large scattered data problems with non-uniform density.

## Contribution

It proposes a novel approach to adaptively choose local data points and shape parameters by minimizing an a priori error estimate, improving interpolation accuracy.

## Key findings

- High accuracy achieved with non-homogeneous data density
- Effective selection of local data set size and shape parameter
- Numerical experiments validate the method's robustness

## Abstract

The Partition of Unity (PU) method, performed with local Radial Basis Function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.