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 HOLDR 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 HOLDR matrix. Our analyses further give insight on how one must choose the working precision in HOLDR matrix computations relative to the approximation error in order to not observe the effects of finite precision. Intuitively, when a HOLDR 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.

翻译：层次矩阵计算在科学与工程领域引起了广泛关注，因为利用数据稀疏结构能显著降低许多重要核心算法的计算复杂度。在此类结构中，层次非对角低秩（HODLR）格式是一种特别受欢迎的选项。本文证明，HODLR矩阵中由低秩矩阵近似的非对角块可以用低精度表示，而不会降低整体近似质量（误差增长受限于$2$倍因子）。我们还提出了一种用于构建和存储HODLR矩阵的自适应精度方案，并证明混合精度的使用不会损害所得HOLDR矩阵-向量乘积和LU分解的数值稳定性。也就是说，这些计算中的最终误差不会显著大于使用单一精度（例如双精度）构建和存储HOLDR矩阵的情况。我们的分析进一步揭示了在HOLDR矩阵计算中，如何根据近似误差选择工作精度以避免有限精度效应的影响。直观而言，当HOLDR矩阵的近似误差较大时，后续计算可以在较低精度下执行而不产生负面影响。我们通过一系列数值实验验证了理论结果的有效性。