Motivated by the advent of machine learning, the last few years saw the return of hardware-supported low-precision computing. Computations with fewer digits are faster and more memory and energy efficient, but can be extremely susceptible to rounding errors. An application that can largely benefit from the advantages of low-precision computing is the numerical solution of partial differential equations (PDEs), but a careful implementation and rounding error analysis are required to ensure that sensible results can still be obtained. In this paper we study the accumulation of rounding errors in the solution of the heat equation, a proxy for parabolic PDEs, via Runge-Kutta finite difference methods using round-to-nearest (RtN) and stochastic rounding (SR). We demonstrate how to implement the scheme to reduce rounding errors and we derive \emph{a priori} estimates for local and global rounding errors. Let $u$ be the roundoff unit. While the worst-case local errors are $O(u)$ with respect to the discretization parameters, the RtN and SR error behavior is substantially different. We prove that the RtN solution is discretization, initial condition and precision dependent, and always stagnates for small enough $\Delta t$. Until stagnation, the global error grows like $O(u\Delta t^{-1})$. In contrast, we show that the leading order errors introduced by SR are zero-mean, independent in space and mean-independent in time, making SR resilient to stagnation and rounding error accumulation. In fact, we prove that for SR the global rounding errors are only $O(u\Delta t^{-1/4})$ in 1D and are essentially bounded (up to logarithmic factors) in higher dimensions.

中文翻译：

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四舍五入和随机四舍五入对低精度热方程数值求解的影响

在机器学习出现的推动下，过去几年见证了硬件支持的低精度计算的回归。位数较少的计算速度更快，内存和能源效率更高，但极易出现舍入错误。偏微分方程 (PDE) 的数值解是一个可以在很大程度上受益于低精度计算优势的应用程序，但需要仔细实施和舍入误差分析，以确保仍然可以获得合理的结果。在本文中，我们通过使用舍入到最近 (RtN) 和随机舍入 (SR) 的 Runge-Kutta 有限差分方法研究了热方程解中舍入误差的累积，热方程是抛物线偏微分方程的代理。我们演示了如何实施该方案以减少舍入误差，并推导出局部和全局舍入误差的 \emph{a priori} 估计。令 $u$ 为舍入单位。尽管相对于离散化参数而言，最坏情况下的局部误差为 $O(u)$，但 RtN 和 SR 误差行为却大不相同。我们证明了 RtN 解是离散化的，依赖于初始条件和精度，并且总是在足够小的 $\Delta t$ 时停滞。直到停滞，全局误差会像 $O(u\Delta t^{-1})$ 一样增长。相比之下，我们表明 SR 引入的前导阶误差是零均值、空间独立和时间均值无关的，使 SR 对停滞和舍入误差累积具有弹性。实际上，