PBBFMM3D: a parallel black-box algorithm for kernel matrix-vector multiplication

Abstract: Kernel matrix-vector product is ubiquitous in many science and engineering applications. However, a naive method requires $O(N^2)$ operations, which becomes prohibitive for large-scale problems. We introduce a parallel method that provably requires $O(N)$ operations to reduce the computation cost. The distinct feature of our method is that it requires only the ability to evaluate the kernel function, offering a black-box interface to users. Our parallel approach targets multi-core shared-memory machines and is implemented using OpenMP. Numerical results demonstrate up to $19\times$ speedup on 32 cores. We also present a real-world application in geostatistics, where our parallel method was used to deliver fast principle component analysis of covariance matrices.