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. A key aspect of this approach is that it allows the user to approximate the value of the integral without explicit knowledge of an expression for the boundary surface. Instead, a tesselation of the node set is utilized to inform the algorithm of the domain geometry. Further, the method applies to node sets featuring spatially varying density, facilitating its use in Applied Mathematics, Mathematical Physics and myriad other application areas, where the locations of the nodes might be fixed by experiment or previous simulation. By using the RBF-FD-like approach, the proposed algorithm computes quadrature weights for $N$ arbitrarily scattered nodes in only $O(N\mbox{ log}N)$ operations with tunable orders of accuracy.