M7: The numerical linear algebra of approximation involving radial basis functions (RBFs)

Background

Radial basis functions (RBFs) provide an elegant scheme for multidimensional scattered data approximation, one foundation of mesh-free method and an emerging method for surface computing. They are applicable in many areas, such as surface reconstruction, machine learning, neural networks and artificial intelligence, and allow for the possibility of attractive mesh-free methods for PDEs on unstructured meshes with multi-physics features. Such methods are advantageous because they are mesh-free/mesh-less, easy to apply on high dimensional spaces and have good approximation properties. However, these methods can lead to some significant problems in linear algebra.

Techniques and Challenges

One challenge in using these methods is the conditioning issue – the linear systems can be highly ill-conditioned. The properties of the linear systems from this method are quite different to that of linear systems in a finite difference or finite elements method context, and since there is no diagonal-dominant and M-property, traditional preconditioning techniques like incomplete factorisation methods may fail. We shall use approximation theory, sampling techniques and numerical linear algebra techniques to deal with these challenges and develop new methods.

Results

We have developed a general Schur product theorem. This can be used to investigate the properties of interpolation matrices with RBFs with different scales, investigate practical error estimation formulae and find the distribution pattern of the eigenvalues of certain kernel matrices.

The Future

In the future, we expect to seek new techniques or find suitable techniques to deal with special ill-conditioned linear systems, and build up a foundation for constructing an effective solver with a preconditioner.

References

[12/80] Wathen A.J., Zhu S.: On the spectral distribution of kernel matrices related to radial basis functions

[12/57] Zhu S.: Compactly supported radial basis functions: How and why?, Technical report, SIAM

Wendland H.: Scatter Data Approximation, Cambridge University Press