# Adaptive scattered data fitting by extension of local approximations to hierarchical splines

## TL;DR

This paper presents an adaptive method for fitting scattered data using hierarchical splines, which adjusts local polynomial degrees based on data density and matrix singular values, improving approximation accuracy.

## Contribution

It extends local least squares approximations to hierarchical spline spaces with an adaptive scheme that considers data distribution and matrix properties for better fitting.

## Key findings

- Effective approximation of real scattered terrain data
- Adaptive local polynomial degree selection improves fit quality
- Hierarchical splines enable efficient data approximation

## Abstract

We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of (variable degree) polynomial approximations according not only to the number of data points locally available, but also to the smallest singular value of the local collocation matrices. These local approximations are subsequently combined without the need of additional computations with the construction of hierarchical quasi-interpolants described in terms of truncated hierarchical B-splines. A selection of numerical experiments shows the effectivity of our approach for the approximation of real scattered data sets describing different terrain configurations.