The Matern 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 Matern 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 log n) 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.

%B SIAM Journal on Scientific Computing %V 36 %P A289-A309 %8 01/2014 %G eng %N 1 %1 http://www.mcs.anl.gov/papers/P4001-1212.pdf