Year: 2025
Type: article
Abstract: . Hierarchical matrix computations have attracted significant attention in the science and engineering community, as exploiting data-sparse structures can significantly reduce the computational complexity of many important kernels. One particularly popular option within this class is the hierarchical off-diagonal low-rank (HODLR) format. In this paper, we show that the off-diagonal blocks of HODLR matrices that are approximated by low-rank matrices can be represented in low precision without degenerating the quality of the overall approximation (with the error growth bounded by a factor of 2). We also present an adaptive-precision scheme for constructing and storing HODLR matrices, and we prove that the use of mixed precision does not compromise the numerical stability of the resulting HODLR matrix–vector product and LU factorization. That is, the resulting error in these computations is not significantly greater than the case where we use one precision (say, double) for constructing and storing the HODLR matrix. Our analyses further give insight into how one must choose the working precision in HODLR matrix computations relative to the approximation error in order to not observe the effects of finite precision. Intuitively, when a HODLR matrix is subject to a high degree of approximation error, subsequent computations can be performed in a lower precision without detriment. We demonstrate the validity of our theoretical results through a range of numerical experiments. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/chenxinye/mhodlr and https://github.com/inEXASCALE/mphodlr_exp in the supplementary material ( M168392_SM.zip [124MB]). [Formula: see text]
DOI: 10.1137/24m1683925
Authors: Erin Carson, Xinye Chen, Xiaobo Liu
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4410045135