From ICERM: challenges faced by RBF methods

I recently gave a couple of talks at the ICERM workshop on localized kernel methods for PDEs. The first talk was about my research, and the second talk was about a software package I'm developing called KernelPack. There will be more about KernelPack in this space as I continue developing the software. I plan to document my efforts (and difficulties) as I develop it.

Towards the end of the workshop, we had a panel discussion on the challenges faced by Radial Basis Function (RBF) methods, both in terms of research challenges faced by our community, and in the broader challenges faced by our community in getting the word out there about the strengths of these methods. As a result of this fruitful and rather friendly discussion, several ideas emerged:

1. The need for theoretical results to help relate eigenvalues of global differentiation matrices to the RBF-finite difference weights generated by the local methods. This plays more generally into the theme of robustness: RBF-FD methods have become very robust and efficient over the past 3 years, but a lot more work needs to be done along these lines for a variety of problems.

2. The need to find interesting problems that specifically showcase the strength of RBF-FD methods: they are meshfree, high-order, extremely easy to generate and compute, and require little to no tuning.

3. The need to tackle missing features of RBF-FD methods: meshfree conservation, monotonicity, the ability to handle shocks, and the development of software packages that both budding and seasoned researchers can use.

4. The relative isolation of the RBF community from the wider world of meshfree methods.

Much like writing a grant proposal, this panel discussion has solidified in my mind the types of challenges I'd like to tackle in my research. This is why workshops and conferences can be awesome!