Matrix-free construction of HSS representation using adaptive randomized sampling
This work addresses the practical challenge of efficiently constructing HSS representations for large-scale matrices, which is important for scientific computing and data analysis.
The paper introduces new algorithms for constructing hierarchically semi-separable (HSS) matrices using adaptive randomized sampling, which determine the maximum off-diagonal block rank with relative and absolute stopping criteria. The algorithms demonstrate effectiveness, scalability, and numerical robustness across applications like boundary element method and quantum chemistry Toeplitz matrices.
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.