Fast Multipole Method for Accelerating the Evaluation of Splines

We consider the problem of interpolating scattered data using spline methods and present a general framework of using the multipole method to accelerate the evaluation of splines. The method depends on a tree-data structure and two hierarchical approximations: an upward multipole expansion approximation and a downward local Taylor series approximation. We also illustrate the performance of the eeciency and the accuracy of the fast multipole algorithm for 2D vector spline. In comparison with the CPU time of direct calculation, which increases at a quadratic rate with the number of points, the fast algorithm achieves a higher speed in evaluation at a linear rate.