[HN Gopher] A practical guide to radial basis functions (2007) [...
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       A practical guide to radial basis functions (2007) [pdf]
        
       Author : Phithagoras
       Score  : 53 points
       Date   : 2021-09-14 20:09 UTC (2 days ago)
        
 (HTM) web link (num.math.uni-goettingen.de)
 (TXT) w3m dump (num.math.uni-goettingen.de)
        
       | mkl wrote:
       | 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).
       | 
       | 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.
        
         | sdfwefwfe wrote:
         | I first heard about them at Siggraph 2001. This paper
         | https://dl.acm.org/doi/10.1145/383259.383266 by an acquaintance
         | described a way of fitting RBFs to very large point clouds. The
         | trick was to use Fast Multipole Methods, which are a bit like
         | treecodes for n-body problems:
         | https://en.wikipedia.org/wiki/Barnes%E2%80%93Hut_simulation
         | 
         | 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.
         | 
         | JP Lewis has some course notes which explain the basics of RBF
         | techniques well:
         | http://scribblethink.org/Courses/ScatteredInterpolation/scat...
         | 
         | For a slightly more technical account, try Sage Shaw's
         | excellent contribution to this Wikipedia page:
         | https://en.wikipedia.org/wiki/Radial_basis_function_interpol...
        
           | mkl wrote:
           | 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!
        
       | raphlinus wrote:
       | A Rust implementation, with visuals and pointers to other
       | resources: https://crates.io/crates/rbf-interp
        
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       (page generated 2021-09-16 23:02 UTC)