Fast Evaluation of Vector Splines in Two Dimensions

This paper presents an algorithm for the rapid evaluation of the vector spline in two dimensions. The algorithm is based on the fast multipole method and has an asymptotic CPU time estimate of O(N), where N is the number of data points. The main idea of the algorithm is to combine a cluster of evaluation kernels at diierent data sources into a single kernel in the form of a multipole expansion at the center of this cluster, so that the contributions come from this cluster of points can be approximated by a truncated multipole expansion. In this paper, we establish the series approximation theory and present the implementation details of the algorithm. The associated theoretic error bounds are derived for numerical computation to any speciic accuracy. In addition, numerical simulation experiments are carried out to demonstrate the eeciency and the accuracy of the algorithm.