Re: Integrating scattered data

[Thomas Shores]

On Monday 20 August 2007 11:27:04 am Jordi Gutiérrez Hermoso wrote:
> I have a surface in some irregular domain of R^2 that I'm sampling
> at scattered, unstructured points. I'd like to find the volume
> under this surface.
>
> Ideally, I'd like to not interpolate to points not in my data set,
> since my surface is rather wiggly and I don't think interpolation
> could be very accurate. A Delaunay triangulation is an obvious
> first step, and then I could compose my surface of triangles to
> get some sort of O(h) integration.
>
> Can I do better? Are there routines already in Octave to help me
> do this? Is it possible to do some analogue of Simpson's rule to
> get O(h^2) precision?
>
> Thanks,
> - Jordi G. H.

Unless there is additional information about the data that you are 
not providing, it seems to me that deterministic methods don't make 
a whole lot of sense in this context.  You mention O(h), which I 
presume is something like the maximum distance between sample domain 
points.  If your sampling is really unstructured, then you have 
little control over h, and thus the usual order estimates (which are 
premised on h and factors such as reasonably small higher 
derivatives and regularity of the domain, which you suggest you may 
not have)  Is the domain of function generating the surface even 
completely known?  Moreover, by use of the word "sampling" you 
suggest that there might be error in your sampled values.  Is this 
so?

  I'm assuming that what you are after the integral of a function 
f(x,y) over a recognizable bounded domain D in R^2.  What would make 
more sense is to me in this situation is a Monte Carlo method.  
These have the disadvantages of relatively low accuracy and 
probabilistic estimates of error to boot, but they have advantages 
of being relatively less sensitive to higher dimensional integration 
and ease of implementation.   Do a google on something 
like "multivariate Monte Carlo methods"  and you'll get a lot of 
information.

T. Shores