Title: Robustness and uncertainty of direct numerical simulation under the influence of rounding and noise Show Chinese title

Title: 直接数值模拟在舍入和噪声影响下的稳健性和不确定性

Abstract: Numerical precision in large-scale scientific computations has become an emerging topic due to recent developments in computer hardware. Lower floating point precision offers the potential for significant performance improvements, but the uncertainty added from reducing the numerical precision is a major obstacle for it to reach prevalence in high-fidelity simulations of turbulence. In the present work, the impact of reducing the numerical precision under different rounding schemes is investigated and compared to the presence of white noise in the simulation data to obtain statistical averages of different quantities in the flow. To investigate how this impacts the simulation, an experimental methodology to assess the impact of these sources of uncertainty is proposed, in which each realization $u^i$ at time $t_i$ is perturbed, either by constraining the flow to a coarser discretization of the phase space (corresponding to low precision formats rounded with deterministic and stochastic rounding) or by perturbing the flow with white noise with a uniform distribution. The purpose of this approach is to assess the limiting factors for precision, and how robust a direct numerical simulation (DNS) is to noise and numerical precision. Our results indicate that for low-Re turbulent channel flow, stochastic rounding and noise impacts the results significantly less than deterministic rounding, indicating potential benefits of stochastic rounding over conventional round-to-nearest. We find that to capture the probability density function of the velocity change in time, the floating point precision is especially important in regions with small relative velocity changes and low turbulence intensity, but less important in regions with large velocity gradients and variations such as in the near-wall region. Show Chinese abstract

Abstract: 数值精度在大规模科学计算中已成为一个新兴话题，这得益于计算机硬件的最新发展。较低的浮点精度提供了显著提高性能的潜力，但通过减少数值精度所增加的不确定性是其在高保真湍流模拟中普及的主要障碍。在目前的工作中，研究了不同舍入方案下降低数值精度的影响，并将其与仿真数据中的白噪声进行比较，以获得流中不同量的统计平均值。为了调查这对仿真有何影响，提出了一种实验方法来评估这些不确定性的来源，其中每个时间点$u^i$的实现$t_i$被扰动，要么通过将流限制到相空间的较粗离散化（对应于用确定性和随机舍入舍入的低精度格式），要么通过用具有均匀分布的白噪声扰动流。这种方法的目的是评估精度的限制因素，以及直接数值仿真（DNS）对噪声和数值精度的鲁棒性。我们的结果显示，对于低雷诺数湍流通道流，随机舍入和噪声对结果的影响远小于确定性舍入，表明随机舍入相对于传统的最近舍入可能具有的潜在优势。我们发现，为了捕捉速度随时间变化的概率密度函数，在相对速度变化小且湍流强度低的区域，浮点精度尤为重要，但在速度梯度大且变化大的区域，如靠近壁面的区域，则不那么重要。