Mixed precision HODLR matrices

TL;DR

This paper demonstrates that using mixed precision in HODLR matrix computations can reduce computational costs without significantly affecting accuracy or stability, supported by theoretical analysis and numerical experiments.

Contribution

It introduces an adaptive-precision scheme for HODLR matrices and proves that mixed precision does not compromise numerical stability or accuracy.

Findings

Low precision representation of off-diagonal blocks maintains approximation quality.

Mixed precision does not significantly increase error in matrix-vector and LU computations.

Theoretical insights guide the choice of working precision relative to approximation error.

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 on 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.