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

## TL;DR

This paper presents PBBFMM3D, a parallel black-box algorithm that efficiently computes kernel matrix-vector products with linear complexity, suitable for large-scale scientific computations on multi-core systems.

## Contribution

It introduces a novel parallel black-box method for kernel matrix-vector multiplication with proven $O(N)$ complexity and practical implementation for shared-memory architectures.

## Key findings

- Achieves up to 19x speedup on 32 cores
- Demonstrates effectiveness in geostatistics PCA applications
- Provides a scalable solution for large kernel matrices

## 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.