Approximate Integrals Over Bounded Volumes with Smooth Boundaries

A Radial Basis Function Generated Finite-Differences (RBF-FD) inspired technique for evaluating definite integrals over bounded volumes that have smooth boundaries in three dimensions is described. Such methods are necessary in many areas of Applied Mathematics, Mathematical Physics and myriad other application areas. Previous approaches needed restrictive uniformity in the node set, which the algorithm presented here does not require. By using RBF-FD approach, the proposed algorithm computes quadrature weights for $N$ arbitrarily scattered nodes in only $O(N\mbox{ log}N)$ operations with high orders of accuracy.