Abstract Motivated by the advent of machine learning, the last few years have seen 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. As shown by recent studies into reduced-precision climate simulations, an application that can largely benefit from the advantages of low-precision computing is the numerical solution of partial differential equations (PDEs). However, 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 a priori estimates for local and global rounding errors. Let $u$ be the unit roundoff. While the worst-case local errors are $\mathcal{O}(u)$ with respect to the discretization parameters (mesh size and timestep), the RtN and SR error behaviour is substantially different. In fact, the RtN solution always stagnates for small enough $\varDelta t$, and until stagnation the global error grows like $\mathcal{O}(u\varDelta t^{-1})$. 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 $\mathcal{O}(u\varDelta t^{-1/4})$ in one dimension 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解的影响

摘要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，但极易受到舍入错误的影响。正如最近对降低精度气候模拟的研究所表明的那样，偏微分方程 (PDE) 的数值解法可以极大地受益于低精度计算的优势。但是，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，我们研究了热方程解中舍入误差的累积，热方程是抛物线 PDE 的代表，通过使用最近舍入 (RtN) 和随机舍入 (SR) 的 Runge-Kutta 有限差分方法。我们演示了如何实施该方案以减少舍入误差，并得出局部和全局舍入误差的先验估计。设 $u$ 为单位舍入。虽然就离散化参数（网格大小和时间步长）而言，最坏情况的局部误差是 $\mathcal{O}(u)$，但 RtN 和 SR 误差行为有很大不同。事实上，对于足够小的 $\varDelta t$，RtN 解总是停滞不前，并且在停滞之前，全局误差会像 $\mathcal{O}(u\varDelta t^{-1})$ 一样增长。相比之下，我们表明 SR 引入的前导误差是零均值、空间独立和时间均值独立的，这使得 SR 对停滞和舍入误差累积具有弹性。实际上，