Advancements in computer hardware have made it possible to utilize low- and mixed-precision arithmetic for enhanced computational efficiency. In practical predictive modeling, however, it is vital to quantify uncertainty due to rounding along other sources like measurement, sampling, and numerical discretization. Traditional deterministic rounding uncertainty analysis (DBEA) assumes that the rounding errors equal the unit roundoff $u$. However, despite providing strong guarantees, DBEA severely overestimates rounding uncertainty. This work presents a novel probabilistic rounding uncertainty analysis called VIBEA. By treating rounding errors as i.i.d. random variables and leveraging concentration inequalities, VIBEA provides high-confidence estimates for rounding uncertainty using higher-order rounding error statistics. The presented framework is valid for all problem sizes $n$, unlike DBEA, which necessitates $nu<1$. Further, it can account for the potential cancellation of rounding errors, resulting in rounding uncertainty estimates that grow slowly with $n$. We show that for $n>n_c(u)$, VIBEA produces tighter estimates for rounding uncertainty than DBEA. We also show that VIBEA improves existing probabilistic rounding uncertainty analysis techniques for $n\ge3$ by using higher-order rounding error statistics. We conduct numerical experiments on random vector dot products, a linear system solution, and a stochastic boundary value problem. We show that quantifying rounding uncertainty along with traditional sources (numerical discretization, sampling, parameters) enables a more efficient allocation of computational resources, thereby balancing computational efficiency with predictive accuracy. This study is a step towards a comprehensive mixed-precision approach that improves model reliability and enables budgeting of computational resources in predictive modeling and decision-making.

翻译：计算机硬件的进步使得利用低精度和混合精度算术提升计算效率成为可能。然而，在实际预测建模中，量化因舍入产生的不确定性（与其他来源如测量、采样和数值离散化的不确定性并列）至关重要。传统确定性舍入不确定性分析方法（DBEA）假设舍入误差等于单位舍入值$u$。尽管提供了强保证，DBEA严重高估了舍入不确定性。本研究提出一种名为VIBEA的新型概率性舍入不确定性分析方法。通过将舍入误差视为独立同分布随机变量并利用集中不等式，VIBEA利用高阶舍入误差统计量提供高置信度的舍入不确定性估计。该框架适用于所有问题规模$n$，而DBEA要求$nu<1$。此外，它能考虑舍入误差的潜在抵消效应，使得舍入不确定性估计随$n$缓慢增长。我们证明当$n>n_c(u)$时，VIBEA产生的舍入不确定性估计比DBEA更紧致。我们还证明VIBEA通过使用高阶舍入误差统计量，对$n\ge3$的情况改进了现有概率性舍入不确定性分析技术。我们在随机向量点积、线性系统求解和随机边值问题上进行了数值实验。结果表明，将舍入不确定性与传统不确定性来源（数值离散化、采样、参数）一并量化，能够更高效地分配计算资源，从而平衡计算效率与预测精度。本研究是迈向全面混合精度方法的一步，该方法可提升模型可靠性，并实现预测建模与决策中计算资源的预算化配置。