The machine learning explosion has created a prominent trend in modern computer hardware towards low precision floating-point operations. In response, there have been growing efforts to use low and mixed precision in general scientific computing. One important area that has received limited exploration is time-integration methods, which are used for solving differential equations that are ubiquitous in science and engineering applications. In this work, we develop two new approaches for leveraging mixed precision in exponential time integration methods. The first approach is based on a reformulation of the exponential Rosenbrock--Euler method allowing for low precision computations in matrix exponentials independent of the particular algorithm for matrix exponentiation. The second approach is based on an inexact and incomplete Arnoldi procedure in Krylov approximation methods for computing matrix exponentials and is agnostic to the chosen integration method. We show that both approaches improve accuracy compared to using purely low precision and offer better efficiency than using only double precision when solving an advection-diffusion-reaction partial differential equation.

翻译：机器学习领域的爆发式增长推动了现代计算机硬件向低精度浮点运算的显著趋势。为此，通用科学计算领域越来越多地尝试采用低精度和混合精度技术。其中，时间积分方法作为解决科学与工程应用中普遍存在的微分方程的关键技术，其混合精度探索尚不充分。本研究提出了两种利用混合精度加速指数时间积分方法的新方案。第一种方案基于对指数Rosenbrock-Euler方法的重新表述，允许在矩阵指数计算中独立于具体算法实现低精度运算。第二种方案基于Krylov近似方法中用于计算矩阵指数的非精确不完全Arnoldi过程，且与所选积分方法无关。研究表明，在求解对流-扩散-反应偏微分方程时，这两种方案相比纯低精度计算能提升精度，同时较纯双精度计算具有更优的效率。