# Point distributions for RBF interpolation

## Background

Radial basis functions can be used to approximate a function by representing it as weighted sum of basis functions that are radial in the sense that they only depend on the distance to a center point. The accuracy of such interpolants depend on the distribution of the center points where the basis functions are located.

The image above shows some simple examples of point distributions. The three first on a square, and the last one on an irregular geometry. A random distribution as in the third image is easily seen problematic as points may coincide, but also distributions with too much structure as the uniform distribution in the second image can cause troubles.

## Project Description

The aim of this project is to investigate different methods for generating point distributions and to evaluate the quality of the resulting interpolants. Some desirable properties are:

- Generating point distributions on arbitrary geometries.
- Distribute points more densely in interesting areas.
- Efficient generation.
- Quality of generated points.

Possible methods to evaluate include treating the points as charged particles and do a molecular dynamics simulation to find the distribution, and using linear algebra methods as described in [BOS].

Contact: Elisabeth Larsson

## References

[BOS] L. Bos, S. De Marchi, A. Sommariva, and M. Vianello, *Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra*, SIAM J. Numer. Anal. 48 (2010), 1984-1999