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Let $D$ be the area bounded by a series of points $(x_i,y_i)_{i=1}^{N}$.(The area need not to be convex and the points are supposed to go along the boundary curve.)

Let $f$ be a function defined on $D$ but we only know its values on a given point set (finite and discrete), say $(x'_i,y'_i,f(x'_i,y'_i))_{i=1}^{N'}$.(The given data set need not to be "dense" in $D$.)

How can I do numeric integration of $f$ over $D$?

Here is what I think:
1) First we should approximate the boundary of $D$ by segments between those series of points.
2) Then we should do some interpolation on the given data set. However, interpolation in two-dimension is not always possible. Then I get stuck.

Can you please help? Thank you.

EDIT: The function value are only known on ($(x'_i,y'_i)$ in the interior of $D$. Its values on the boundary of $D$ (where the points are $(x_i,y_i)$, without prime) are not known.

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The values of $f$ are known only on the boundary, or are there "interior values" as well? – J. M. May 17 '11 at 16:30
@J.M.: My understanding was that no relationship is given between the points $(x'_i,y'_i)$ at which $f$ is known and the points $(x_i,y_i)$ on the boundary. – joriki May 17 '11 at 16:34
@J.M.: joriki is right. And I've edited the post for clarification, – Roun May 17 '11 at 16:36
up vote 2 down vote accepted

For point 2), you should create a triangle mesh using the interior and the boundary points. Then you can perform linear or higher order interpolation in each triangle. Mesh generation is a huge area and there are many algorithms. See for instance Chapter 14 of Computational Geometry: Algorithms and Applications. For software, see for instance Triangle.

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