This looks pretty good from a quick skim (it's 2007 so I probably saw it 10+ years ago, but don't remember). I did my PhD on radial basis functions, and they seem surprisingly unknown for something so useful. RBFs provide a really easy way to interpolate through scattered data of any number of dimensions. I interpolated through point data, integral data, and derivative data (but didn't use them to solve DEs like this).
That fits a multiquadric RBF (p2) (with linear terms, because it's conditionally positive definite; p5-6) to some carefully chosen data, then evaluates it (at the same points again, because I was in a hurry...) and converts to characters.
Fitting an RBF naively involves inverting a matrix as big as the number of points, but the FMM techniques use approximation to make the task practical.
Yes, I'm familiar with that paper. It's very complicated to implement - back when I was researching I didn't encounter any papers by others who had replicated it, and citations often mentioned its complexity. There are simpler ways to scale RBFs. I did it piecewise, using overlapping subdomains (each with ~200 points, so a small linear system) blended with partition of unity, and it scaled to millions of points in an embarrassingly parallelisable way. Here's a result I've linked on HN before, filling a hole in a point cloud (a small one, ~24000 points): https://imgur.com/a/k6AQi
Those course notes have some interesting stuff in them, thanks!
They also made a good well-obfuscated "peak hello world" attempt: https://news.ycombinator.com/item?id=23211972
That fits a multiquadric RBF (p2) (with linear terms, because it's conditionally positive definite; p5-6) to some carefully chosen data, then evaluates it (at the same points again, because I was in a hurry...) and converts to characters.