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A Radial Basis Function Generated Finite-Differences (RBF-FD) inspiredtechnique for evaluating definite integrals over bounded volumes that havesmooth boundaries in three dimensions is described. A key aspect of thisapproach is that it allows the user to approximate the value of the integralwithout explicit knowledge of an expression for the boundary surface. Instead,a tesselation of the node set is utilized to inform the algorithm of the domaingeometry. Further, the method applies to node sets featuring spatially varyingdensity, facilitating its use in Applied Mathematics, Mathematical Physics andmyriad other application areas, where the locations of the nodes might be fixedby experiment or previous simulation. By using the RBF-FD-like approach, theproposed algorithm computes quadrature weights for $N$ arbitrarily scatterednodes in only $O(N\mbox{ log}N)$ operations with tunable orders of accuracy.

Approximate Integrals Over Bounded Volumes with Smooth Boundaries