Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

• Authors: Robert Krasny;Lei Wang
• Venue: SIAM Journal on Scientific Computing
• Year: 2011

A treecode is presented for evaluating sums defined in terms of the multiquadric radial basis function (RBF), $\phi({\bf x}) = (|{\bf x}|^2+c^2)^{1/2}$, where ${\bf x} \in \mathbb{R}^3$ and $c \ge 0$. Given a set of $N$ nodes, evaluating an RBF sum directly requires CPU time that scales like $O(N^2)$. For a given level of accuracy, the treecode reduces the CPU time to $O(N\log N)$ using a far-field expansion of $\phi({\bf x})$. We consider two options for the far-field expansion: (1) a Laurent series previously used in applications of the Fast Multipole Method to multiquadric RBFs, and (2) a certain Taylor series previously used in treecode particle simulations, but not yet in the context of multiquadric RBFs. It is known that the Laurent series converges when the RBF parameter $c$ lies in an interval $0 \le c \le \bar{c}$, where $\bar{c}$ is proportional to the minimum node spacing, but here we show that the Taylor series converges uniformly for $c \ge 0$. We implement the treecode in Cartesian coordinates and use a recurrence relation to compute the Taylor coefficients. Numerical results exhibit the treecode error, CPU time, and memory usage in two test cases, random nodes in a cube and on the surface of a sphere. The treecode approach presented here is applicable to generalized multiquadrics in any dimension.