Related papers: Deterministic and Probabilistic Rounding Error Ana…

Many recent computational accelerators provide non-standard (e.g., reduced precision) arithmetic operations to enhance performance for floating-point matrix multiplication. Unfortunately, the properties of these accelerators are not widely…

Finite-precision floating point arithmetic unavoidably introduces rounding errors which are traditionally bounded using a worst-case analysis. However, worst-case analysis might be overly conservative because worst-case errors can be…

Although mixed precision arithmetic has recently garnered interest for training dense neural networks, many other applications could benefit from the speed-ups and lower storage cost if applied appropriately. The growing interest in…

The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for…

Modern graphics computing units (GPUs) are designed and optimized to perform highly parallel numerical calculations. This parallelism has enabled (and promises) significant advantages, both in terms of energy performance and calculation. In…

Finite precision computations using digital computers involve the following inherent errors: (1) Round-off error of finite precision computations (2) Binary computer arithmetic precludes exact number representation of traditional decimal…

Probabilistic model checking computes probabilities and expected values related to designated behaviours of interest in Markov models. As a formal verification approach, it is applied to critical systems; thus we trust that probabilistic…

In this paper we develop the first fine-grained rounding error analysis of finite element (FE) cell kernels and assembly. The theory includes mixed-precision implementations and accounts for hardware-acceleration via matrix multiplication…

We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale…

In recent years, a new kind of accelerated hardware has gained popularity in the Artificial Intelligence (AI) and Machine Learning (ML) communities which enables extremely high-performance tensor contractions in reduced precision for deep…

Renewed interest in mixed-precision algorithms has emerged due to growing data capacity and bandwidth concerns, as well as the advancement of GPUs, which enable significant speedup for low precision arithmetic. In light of this, we propose…

Modern lunar-planetary ephemerides are numerically integrated on the observational timespan of more than 100 years (with the last 20 years having very precise astrometrical data). On such long timespans, not only finite difference…

We propose a multi-precision extension of the Quadratic Regularization (R2) algorithm that enables it to take advantage of low-precision computations, and by extension to decrease energy consumption during the solve. The lower the precision…

Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one…

Due to the limited number of bits in floating-point or fixed-point arithmetic, rounding is a necessary step in many computations. Although rounding methods can be tailored for different applications, round-off errors are generally…

Classical probabilistic rounding error analysis is particularly well suited to stochastic rounding (SR), and it yields strong results when dealing with floating-point algorithms that rely heavily on summation. For many numerical linear…

We present a detailed study of roundoff errors in probabilistic floating-point computations. We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are…

Probabilistic rounding error analysis can yield much sharper bounds than classical worst-case theory, but existing results typically rely on zero-mean rounding errors and often leave the confidence parameter implicit. This work revisits…

Within the past years, hardware vendors have started designing low precision special function units in response to the demand of the Machine Learning community and their demand for high compute power in low precision formats. Also the…

We analyse the convergence of the proximal gradient algorithm for convex composite problems in the presence of gradient and proximal computational inaccuracies. We derive new tighter deterministic and probabilistic bounds that we use to…