Preprint ANL/MCS-P4001-1212 A FAST SUMMATION TREE CODE FOR MATÉRN KERNEL

The Matérn family of functions is a widely used covariance kernel in spatial statistics for Gaussian process modeling, which in many instances requires calculation with a covariance matrix. In this paper, we design a fast summation algorithm for the Matérn kernel in order to efficiently perform matrix-vector multiplications. This algorithm is based on the Barnes–Hut tree code framework, and several important aspects are addressed: the partitioning of the point set, the computation of the Taylor approximation with error estimates, and the handling of multiple sets of weights originating from multiple matrix-vector multiplications with the same matrix. The computational cost of the derived algorithm scales as O(n logn) for n points. Comprehensive numerical experiments are shown to demonstrate the practicality of the design. The development of a similar algorithm based on the multipole expansion framework is also discussed.