1. Matrix-free construction of HSS representation using adaptive randomized sampling
- Author
-
Gorman, Christopher, Chávez, Gustavo, Ghysels, Pieter, Mary, Théo, Rouet, François-Henry, and Li, Xiaoye Sherry
- Subjects
FOS: Computer and information sciences ,Computer Science - Numerical Analysis ,FOS: Mathematics ,Computer Science - Mathematical Software ,Numerical Analysis (math.NA) ,Mathematical Software (cs.MS) - 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., Comment: 24 pages, 4 figures, 20 references
- Published
- 2018
- Full Text
- View/download PDF