TITLE: When is 2d sparse numerical integration by Voronoi regions better than using triangular mesh elements
QUESTION [10 upvotes]: I've inherited some numerical analysis code that integrates a 2D function that is only
known at a large set of unstructured points.
The way it does this is by Delauney triangulating the domain using the sample points, and then calculating the area of the Voronoi cell at each point, and using that as the weight associated with each point. Neglecting edge effects the sum of the areas will total up to the area of the domain, and so it seems a valid approach - but not how I'd approach it.
I'd use the Delauney triangulation and derive the weights from the finite element style basis function at each vertex. (1/3 the area of the sum of the triangles with the same vertex.). Again the functions weights sum to the area of the domain.
It seems in 1D the two approaches would be the same, but I don't think this is the case for 2D. 
So my question is are there known advantages/disadvantages for each style of numerical integration?

REPLY [4 votes]: The PhD thesis of Willem Schaap ("DTFE: the Delaunay Tessellation Field Estimator" University of Groningen, Kapteyn Astronomical Institute) discusses Voronoi versus Delaunay weighting schemes for re-sampling irregular point sets on to a regular grid: http://irs.ub.rug.nl/ppn/298831376.  
Once re-sampled you can integrate, take FFT's and so on.  Schaap gives cogent reasons as to why the Delaunay weighting (DTFE) is to be preferred over Voronoi weighting (VTFE).  This has been used widely in the analysis of astronomical data sets.  There are generalisations to get higher levels of continuity, if necessary, using "Natural Neighbours" on a Delaunay tessellation  (http://arxiv.org/abs/1107.1488).