Re: Integrating scattered data

[Thomas Shores]

On Tuesday 21 August 2007 02:17:57 pm Jordi Gutiérrez Hermoso wrote:
> On 21/08/07, Thomas Shores <address@hidden> wrote:
> > 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.
...
>
> I don't think my problem is ugly enough to warrant Monte Carlo
> methods. Or do you disagree?
>
> - Jordi G. H.

I should have added that an important advantage of Monte Carlo 
methods is that they handle uncertainty more gracefully than 
deterministic methods.  However, given the additional information 
you have added in your last posting, it seems that you do not have 
that degree of uncertainty, either in sampling point placement or 
the value of the sample, that would call for Monte Carlo.  The 
nicely detailed response by David Bateman to your query should be 
more than adequate. 

This is assuming that you have reasonably full control over the 
placement of your sampling points, so that you can have a 
sufficiently dense (not necessarily regular) placement of points.  
You need this for two reasons:
1. If your integrand is highly oscillatory, you need to worry about 
an "aliasing" effect that is typically reduced by sufficiently fine 
grid construction.
2. An appropriate definition of h for convergence purposes is the 
maximum of the widths of triangles in your triangularization of the 
domain. 

It's also helpful for error estimates which would require running the 
experiment with smaller values of the mesh size h and extrapolating.  
The calculations should be interesting.  Good luck with them.

Thomas Shores