H2Pack: High-performance H² Matrix Package for Kernel Matrices Using the Proxy Point Method

Dense kernel matrices represented in H² matrix format typically require less storage and have faster matrix-vector multiplications than when these matrices are represented in the standard dense format. In this article, we present H2Pack, a high-performance, shared-memory library for constructing and operating with H² matrix representations for kernel matrices defined by non-oscillatory, translationally invariant kernel functions. Using a hybrid analytic-algebraic compression method called the proxy point method, H2Pack can efficiently construct an H² matrix representation with linear computational complexity. Storage and matrix-vector multiplication also have linear complexity. H2Pack also introduces the concept of “partially admissible blocks” for H² matrices to make H² matrix-vector multiplication mathematically identical to the fast multipole method (FMM) if analytic expansions are used. We optimize H2Pack from both the algorithm and software perspectives. Compared to existing FMM libraries, H2Pack generally has much faster H² matrix-vector multiplications, since the proxy point method is more effective at producing block low-rank approximations than the analytic methods used in FMM. As a tradeoff, H² matrix construction in H2Pack is typically more expensive than the setup cost in FMM libraries. Thus, H2Pack is ideal for applications that need a large number of matrix-vector multiplications for a given configuration of data points.