Fast evaluation of vector splines in three dimensions

Vector spline techniques have been developed as general-purpose methods for vector field reconstruction. However, such vector splines involve high computational complexity, which precludes applications of this technique to many problems using large data sets. In this paper, we develop a fast multipole method for the rapid evaluation of the vector spline in three dimensions. The algorithm depends on a tree-data structure and two hierarchical approximations: an upward multipole expansion approximation and a downward local Taylor series approximation. In comparison with the CPU time of direct calculation, which increases at a quadratic rate with the number of points, the presented fast algorithm achieves a higher speed in evaluation at a linear rate. The theoretical error bounds are derived to ensure that the fast method works well with a specific accuracy. Numerical simulations are performed in order to demonstrate the speed and the accuracy of the proposed fast method.