A Fast Summation Tree Code for Matérn Kernel

Jie Chen, Lei Wang, Mihai Anitescu

Abstract: The Matérn family of functions is a widely used covariance kernel in spatial statistics for Gaussian process modeling, which in many instances requires calculations 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 addresses several practical issues: the anisotropy of the kernel, the nonuniform distribution of the point set, and a tight error estimate of the approximation. Even though the algorithmic details differ from the standard tree code in several aspects, empirically the computational cost of our algorithm scales as $O(n\log n)$ for $n$ points. Comprehensive numerical experiments are shown to demonstrate the practicality of the design.