Matrix-free construction of HSS representation using adaptive randomized sampling

TL;DR

This paper introduces adaptive randomized algorithms for constructing HSS matrices efficiently without full matrix access, improving accuracy control and scalability for large applications.

Contribution

It develops a partially matrix-free, adaptive randomized projection scheme with new stopping criteria for HSS construction, enhancing robustness and efficiency.

Findings

Effective in boundary element method matrices

Scalable parallel implementation demonstrated

Robust accuracy control in large applications

Abstract

We present new algorithms for the randomized construction of hierarchically semi-separable matrices, addressing several practical issues. The HSS construction algorithms use a partially matrix-free, adaptive randomized projection scheme to determine the maximum off-diagonal block rank. We develop both relative and absolute stopping criteria to determine the minimum dimension of the random projection matrix that is sufficient for the desired accuracy. Two strategies are discussed to adaptively enlarge the random sample matrix: repeated doubling of the number of random vectors, and iteratively incrementing the number of random vectors by a fixed number. The relative and absolute stopping criteria are based on probabilistic bounds for the Frobenius norm of the random projection of the Hankel blocks of the input matrix. We discuss parallel implementation and computation and communication cost of both variants. Parallel numerical results for a range of applications, including boundary element method matrices and quantum chemistry Toeplitz matrices, show the effectiveness, scalability and numerical robustness of the proposed algorithms.