Robustness and uncertainty of direct numerical simulation under the influence of rounding and noise

Martin Karp; Niclas Jansson; Philipp Schlatter; Saleh Rezaeiravesh; Stefano Markidis

REVIEW 4 major objections 4 minor 12 references

Robustness and uncertainty of direct numerical simulation under the influence of rounding and noise

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Reducing precision with stochastic rounding preserves turbulent channel flow statistics far better than round-to-nearest.

desk verdict A clean, honest empirical study of rounding schemes in a turbulent channel DNS; the SR-over-RTN result is plausible but rests on single runs and a state-only perturbation model. read the letter →

arxiv 2505.01140 v1 pith:JC5D7H47 submitted 2025-05-02 physics.flu-dyn cs.MS

Martin Karp , Niclas Jansson , Saleh Rezaeiravesh , Stefano Markidis , Philipp Schlatter This is my paper · ORCID

classification physics.flu-dyncs.MS

keywords directnumericalsimulationturbulentchannelflowreducedprecisionstochasticroundinground-to-nearestfloating-pointarithmeticuncertaintyquantificationvelocityincrementPDF

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how much numerical precision a direct numerical simulation of turbulence really needs, and whether the answer depends on how numbers are rounded. To isolate rounding from the rest of the simulation, the authors insert a perturbation after each time step: either round the state vector to a lower floating-point format using round-to-nearest (RTN) or stochastic rounding, or multiply it by a uniform white-noise factor. For low-Reynolds-number turbulent channel flow, stochastic rounding and noise change the first- and second-order flow statistics far less than RTN does, and FP16 with stochastic rounding behaves like much higher precision. The results identify the probability density function of the velocity change between time steps as the quantity that sets the required precision, with the channel center, where relative velocity changes and turbulence intensity are small, being the most fragile region.

What carries the argument

The carrying device is the scalar map $G$ applied to every component of the state vector after each application of the time-stepping map $F$, so the perturbed simulation is $\hat{u}^{i+1}=G(F(\hat{u}^i))$. Three versions of $g$ are compared: deterministic round-to-nearest to a floating-point format, stochastic rounding (rounding up or down with probability proportional to the distance to the nearest representable values), and multiplication by $(1+\varepsilon\delta)$ with $\delta$ uniform in $[-1,1]$. The paper also distinguishes rounding the integrated state from rounding only sampled observations, and uses the probability density function of the velocity increment $\Delta u = u_i-u_{i-1}$ as the central diagnostic, motivated by the fact that once $\Delta u$ falls below the representable spacing the state stagnates. This construction lets the authors attribute statistical degradation either to bias introduced by deterministic rounding or to the inability to resolve small increments.

What would settle it

Run the same channel flow at $\mathrm{Re}_\tau=180$ with genuine FP16 arithmetic applied inside the time-stepping and spatial discretization, not just to the state, and compare round-to-nearest with stochastic rounding; if RTN no longer produces the larger statistical deviations seen here, the state-only rounding model is the reason for the ranking.

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Extended reading notes

Core claim

The paper's central claim is that for a statistically stationary, chaotic turbulent channel flow at $\mathrm{Re}_\tau=180$, the way a state vector is rounded, not just the number of bits, controls how much precision loss degrades the simulation. With deterministic round-to-nearest, low precision (FP16 or below) introduces bias in the mean and visibly distorts the Reynolds stresses and the joint PDF of velocity and its time change; with stochastic rounding, the same precision behaves almost like FP64 and FP32, with the main effect being a longer time needed for the statistics to converge. The same close-to-reference behavior appears when the state is perturbed by uniform white noise, so the authors interpret stochastic rounding as a controlled noise injection rather than a systematic bias. They also separate observation-only rounding, where the state is rounded after the integration for sampling purposes, from true state perturbation, showing that even coarse observation precision preserves means under SR and noise but distorts second moments. Precision matters most where the relative velocity change per time step is small and turbulence intensity is low, which is why the channel center is the hardest region to represent at low precision.

Load-bearing premise

The ranking rests on modeling low precision as a rounding of the state vector at each time step rather than rounding the arithmetic operations inside the time integrator and spatial discretization.

Editorial extensions

If this is right

If the ranking holds, simulations can run in FP16 with stochastic rounding and retain FP64-level first and second moments, cutting memory and bandwidth costs.
The PDF criterion gives a location-dependent precision requirement: the channel center demands the most precision for resolving velocity changes, not the near-wall region.
Observation-only rounding under SR and noise preserves mean profiles even at quarter precision, suggesting sampled simulation data can be stored or transmitted in much lower precision than the integration itself.
A fixed-point format with spacing tuned to the near-wall velocity standard deviation might outperform floating point for turbulent state vectors, since floating point wastes resolution on large mean values.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: the paper rounds only the state vector, not the arithmetic inside the time integrator, so a real reduced-precision hardware implementation that also rounds intermediate sums and products may accumulate RTN and SR errors differently.
Extension: because stochastic rounding behaves like injected white noise, its errors may be statistically indistinguishable from physical noise and could be absorbed into stochastic turbulence models or uncertainty-quantification frameworks.
Extension: a testable prediction is that the location-dependent precision map, with the channel center as the bottleneck, should shift as the Reynolds number changes the ratio of near-wall to centerline turbulence intensity.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: this paper is a legitimate step forward for the low-precision DNS discussion. It compares stochastic rounding, round-to-nearest, and uniform noise in a unified 'perturb the state after each time step' framework, and it produces a genuinely new diagnostic: the wall-normal PDF of velocity change as a function of precision. That makes it more useful than the typical 'here's a reduced-precision run that worked' paper.

The good: the methodology is explicit, the perturbation map G is clearly defined for RTN, SR, and noise, they use CPFloat for faithful format emulation, and they compare against Moser's reference data. The distinction between perturbing the observation (tilde U) and perturbing the state (hat U) is smart and clarifies that sampling statistics in FP16 is fine while evolving the state in FP16 with RTN is not. The observation that SR and noise behave similarly, and both do better than RTN at low precision, is coherent with the known bias of RTN in chaotic systems. Their point that floating point may not be the ideal format for DNS state vectors is worth taking seriously.

The soft spots are real but not fatal. First, every conclusion comes from a single realization per case. For a chaotic system, one run cannot statistically separate the effect of the rounding scheme from the butterfly effect; SR is stochastic, so you'd want at least a few seeds to show the ranking is stable. Second, the model perturbs only the state vector G(u), not the arithmetic inside F. Real reduced-precision hardware would round sums, products, and derivatives, and that could change the error accumulation, especially for RTN. The authors are upfront about this constraint, but it does limit how far you can push the claim that 'SR is better for low-precision DNS' without testing actual mixed-precision arithmetic. Third, they mention a few runs that did not complete (bfloat16 RTN, quarter-precision SR/noise for hat U); that's an important caveat and it gets only a passing mention. I'd like to see those failures explained, since they hint that the ranking is not universal.

Net: the paper is worth reading for anyone working on low-precision CFD or stochastic rounding. It doesn't fully prove the transferability of the SR advantage to real arithmetic, but it poses the question well and provides a useful diagnostic. I'd send it out for peer review, with the expectation that the authors add ensemble runs and address the state-only perturbation limitation more prominently.

Referee Report

4 major / 4 minor

Summary. This manuscript proposes a perturbation framework for assessing the sensitivity of a chaotic direct numerical simulation (DNS) to reduced numerical precision and to noise. The baseline is an FP64 spectral-element simulation of turbulent channel flow at Re_tau=180, and at each time step the state vector is mapped through G(u), where G applies deterministic round-to-nearest, stochastic rounding, or uniform multiplicative noise with amplitude equal to the machine epsilon of the target format. Two protocols are compared: perturbing the state that is fed back into the time integration (denoted hat-U) and perturbing only the recorded samples used for statistics (denoted tilde-U). The authors report mean and Reynolds-stress profiles compared with Moser et al., and joint PDFs of the streamwise velocity u and its time increment at three wall-normal locations. They conclude that stochastic rounding and noise affect the results significantly less than deterministic rounding, and that floating-point precision matters most in regions of small relative velocity changes and low turbulence intensity, such as the channel center.

Significance. The perturbation framework is clearly defined, the numerical experiments are reproducible through the CPFloat emulation library, and the comparison with Moser et al. is a sensible external reference. The paper also identifies a practically relevant observation: the channel center, where relative velocity changes are small, is the limiting region for low-precision representation, and this suggests considering alternative number formats. However, the central ranking claim is currently supported only by single-realization visual comparisons, and the perturbation model excludes rounding inside the arithmetic operations of the time integrator. If the authors add ensemble estimates and address the transferability of the state-only model, the proposed methodology would be a useful contribution to the mixed-precision DNS community.

major comments (4)

[Section 2, Figure 1] The statement in the abstract and in Section 2 that stochastic rounding and noise impact the results 'significantly less' than deterministic rounding is based on a single realization per configuration. Because the system is chaotic and SR/noise are stochastic, the differences between schemes could be smaller than run-to-run variability. Please provide ensemble averages, bootstrap confidence intervals, or at least multiple independent realizations for each combination before making a ranking claim.
[Section 1, system definition] Reduced precision is modeled solely as quantization of the state vector through G applied to u, while all arithmetic inside F is performed in FP64. The authors acknowledge this limitation, but the abstract and conclusions are stated in terms of 'DNS under the influence of rounding.' In an actual mixed-precision DNS, rounding occurs in every arithmetic operation, producing error accumulation inside F that behaves differently for RTN (systematic bias) than for SR (random walk). The ranking of rounding schemes obtained here may therefore not transfer. Please either revise the central claim to refer to state-discretization perturbations or add a comparison with an implementation that rounds the arithmetic operations.
[Section 2, paragraph after Figure 1] The manuscript reports that three configurations 'did not complete' (bfloat16 RTN and quarter-precision SR/noise for hat-U) but does not describe the failure mode or include these cases in the quantitative comparison. In a robustness study, failed runs are an outcome and should be reported explicitly (stagnation versus blow-up, time step of failure, and affected quantities). Omitting them could bias the qualitative ranking, especially for the lowest precisions.
[Section 2, noise model definition] The noise perturbation is defined as (1+epsilon*delta)u with delta uniform in [-1,1], so its amplitude equals the machine epsilon of the target precision. This makes the noise comparable in magnitude to the rounding unit, but its structure (continuous uniform relative perturbation versus discrete rounding) is markedly different. The paper does not justify why equal epsilon is the meaningful comparison for evaluating robustness to noise. Please include a sensitivity study in noise amplitude or an explicit argument that epsilon is the appropriate scale for this reference perturbation.

minor comments (4)

[Figure 2] The joint PDFs are displayed with color maps whose normalization the authors state depends on the time-series length; please add a quantitative distance metric, such as the Hellinger distance or KL divergence, so that the claimed similarity between cases is supported quantitatively rather than visually.
[Figure 1 caption] The caption states that 'only isotropic components of the Reynolds stress tensor are plotted' but does not specify which components these are; please define the plotted quantity in the caption or in the text.
[Section 2, numerical setup] The numerical setup is incompletely described: no grid resolution, time step, time-series length, number of samples, or CPFloat emulation details are provided; please add a methods paragraph so that the experiments can be reproduced.
[Section 1, paragraph on stagnation] The sentence stating that 'the change in time Δu never completely stagnates' is contradicted by the later report that low-precision cases stagnate; please qualify this statement as applying to the baseline FP64 system only.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the perturbation framework is independently defined and compared to external DNS data, with only minor, non-load-bearing self-citations.

full rationale

The paper's central comparison is self-contained rather than circular. The perturbation operators are defined independently of the target statistics: roundRTN,FP, roundSR,FP, and noise g(u) = (1 + epsilon delta)u with delta uniform over [-1,1], and these are applied to a baseline FP64 DNS state through G(F(u)). The outputs are then compared to the baseline U and to the external reference data of Moser et al. [7]. No parameter is fitted to any quantity that is later presented as a prediction, and no uniqueness theorem or ansatz is imported from prior work to force the conclusions. The self-citations that appear, such as the reference to Karp et al. [4] for the observation that RTN time series diverge around b = 10, and the citation to the authors' own Neko code [1], are contextual and not load-bearing for the central claim that stochastic rounding and noise affect the simulated statistics less than deterministic rounding. The paper's explicitly stated scope limitation, that only the state vector is perturbed while the time integration F is performed in FP64, is a modeling choice and a caveat for transferability, not a circular step. The finding that precision matters most where relative velocity changes are small is an observed consequence of applying relative-error perturbations to the state, but it is presented as an empirical result rather than as a derivation whose conclusion is embedded in its premises. The matching of the noise amplitude epsilon to the machine epsilon of each format is a normalization choice, not a fitted parameter. Consequently, there is no circular derivation chain; at most there is a minor self-citation that does not carry the argument.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on four unproven premises: ergodicity and positive Lyapunov exponent for the channel flow; the adequacy of state-only rounding as a model of reduced precision; the comparability of uniform noise of amplitude epsilon to rounding to a format with machine epsilon; and the reliability of the FP64 baseline. No parameters are fitted to data, and no new entities are invented.

assumptions (4)

domain assumption The discretized Navier-Stokes system for turbulent channel flow at Re_tau=180 is ergodic and has a positive Lyapunov exponent, so time-averaged statistics converge to the attractor distribution.
Invoked in Section 1 where the authors state they 'focus on systems F that are ergodic and chaotic' and 'assume a positive Lyapunov exponent'; no direct evidence for this specific simulation is given.
ad hoc to paper Rounding the state vector u at each time step (G applied to u) adequately represents the effect of running a DNS in lower precision.
Section 1: 'we constrain our focus to only evaluate the effect of the discretization of the state of the flow'; this is the central modeling choice and is not validated against a full low-precision run.
ad hoc to paper Uniform white noise with amplitude equal to machine epsilon is a meaningful reference perturbation comparable to rounding to that precision.
Section 1 defines noise as g(u)=(1+epsilon delta)u; the equivalence between additive noise and rounding is asserted, not derived.
domain assumption The FP64 baseline simulation is an adequate proxy for the true solution.
The paper compares profiles to Moser et al. [7] and treats FP64 as reference, but does not report a grid-convergence or time-step-convergence study for the baseline.

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Cite this review

Pith. "Pith review of Robustness and uncertainty of direct numerical simulation under the influence of rounding and noise." pith.science (2026). https://pith.science/paper/JC5D7H47

@misc{pith2026250501140,
  author       = {Pith},
  title        = {Pith review of: Robustness and uncertainty of direct numerical simulation under the influence of rounding and noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JC5D7H47}},
  note         = {Machine review of arXiv:2505.01140}
}

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.

Discussion (0). Continue with ORCID to comment.

Reference graph

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[1] [1]

Computers & Fluids

Jansson, N., Karp, M., Podobas, A., Markidis, S., Schlatter, P.: Neko: A modern, portable, and scalable framework for high-fidelity computational fluid dynamics. Computers & Fluids. 275 (2024)

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ACM Transactions on Mathematical Software

Fasi, M., Mikaitis, M.: CPFloat: A C library for simulating low-precision arithmetic. ACM Transactions on Mathematical Software. 49(2), 1--32 (2023)

work page 2023

[3] [3]

D., Jameson, A.: Impact of number representation for high-order implicit large-eddy simulations

Witherden, F. D., Jameson, A.: Impact of number representation for high-order implicit large-eddy simulations. AIAA journal, 58(1), 184--197 (2020)

work page 2020

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et al.: Uncertainty Quantification of Reduced-Precision Time Series in Turbulent Channel Flow

Karp, M. et al.: Uncertainty Quantification of Reduced-Precision Time Series in Turbulent Channel Flow. In Proceedings of the SC'23 Workshops, 387--390. (2023)

work page 2023

[5] [5]

J., Mary, T., Mikaitis, M.: Stochastic rounding: implementation, error analysis and applications

Croci, M., Fasi, M., Higham, N. J., Mary, T., Mikaitis, M.: Stochastic rounding: implementation, error analysis and applications. Royal Society Open Science. 9(3) (2022)

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A.: Roundoff-induced periodicity and the correlation dimension of chaotic attractors

Grebogi, C., Ott, E., Yorke, J. A.: Roundoff-induced periodicity and the correlation dimension of chaotic attractors. Physical Review A, 38(7) (1988)

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D., Kim, J., Mansour, N

Moser, R. D., Kim, J., Mansour, N. N.: Direct numerical simulation of turbulent channel flow up to Re_ = 590. Physics of fluids, 11(4), 943--945 (1999)

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Broy, M.: Software engineering --- from auxiliary to key technologies. In: Broy, M., Dener, E. (eds.) Software Pioneers, pp. 10-13. Springer, Heidelberg (2002)

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