Rounding Research Articles

Half precision wave simulation

In recent years, half precision floating-point arithmetic has gained wide support in hardware and software stack thanks to the advance of artificial intelligence and machine learning applications. Operating at half precision can significantly reduce the memory footprint comparing to operating at single or double precision. For memory bound applications such as time domain wave simulations, this is an attractive feature. However, the narrower width of the half precision data format can lead to degradation of the solution quality due to larger roundoff errors. In this work, we illustrate with carefully designed numerical experiments the negative impact caused by the accumulation of roundoff errors in wave simulations. Specifically, the energy-conserving property of the wave equations is employed as a convenient diagnosis tool. The corresponding remedy in the form of compensated sum is then provided, with its efficacy demonstrated using numerical examples involving both acoustic and elastic wave equations on hardware that natively support half precision arithmetic.

Computational Approaches to Compressible Micropolar Fluid Flow in Moving Parallel Plate Configurations

In this paper, we consider the unsteady flow of a compressible micropolar fluid between two moving, thermally isolated parallel plates. The fluid is characterized as viscous and thermally conductive, with polytropic thermodynamic properties. Although the mathematical model is inherently three-dimensional, we assume that the variables depend on only a single spatial dimension, reducing the problem to a one-dimensional formulation. The non-homogeneous boundary conditions representing the movement of the plates lead to moving domain boundaries. The model is formulated in mass Lagrangian coordinates, which leads to a time-invariant domain. This work focuses on numerical simulations of the fluid flow for different configurations. Two computational approaches are used and compared. The first is based on the finite difference method and the second is based on the Faedo–Galerkin method. To apply the Faedo–Galerkin method, the boundary conditions must first be homogenized and the model equations reformulated. On the other hand, in the finite difference method, the non-homogeneous boundary conditions are implemented directly, which reduces the computational complexity of the numerical scheme. In the performed numerical experiments, it was observed that, for the same accuracy, the Faedo–Galerkin method was approximately 40 times more computationally expensive compared to the finite difference method. However, on a dense numerical grid, the finite difference method required a very small time step, which could lead to an accumulation of round-off errors. On the other hand, the Faedo–Galerkin method showed the convergence of the solutions as the number of expansion terms increased, despite the higher computational cost. Comparisons of the obtained results show good agreement between the two approaches, which confirms the consistency and validity of the numerical solutions.

Influence of tool micro-texturing, cooling conditions and cutting conditions on workpiece surface integrity and tool condition

Improving cutting tool life is very crucial in manufacturing industry. The quality of the finished product is equally important. Investigations are being done to identify different methods to minimize tool wear and improve surface integrity. Micro-texturing of tools is one of the possible methods to improve life of the cutting tool. This work investigates the influence of type of tool: micro-textured, untextured tool; cooling conditions: dry, flood machining; process parameters like cutting speed: low and high, feed: low and high and depth of cut: low and high on surface integrity of machined low alloy steel. Micro-textured tools showed better performance over untextured tools by decreasing the tool crater wear area, roundness error and surface roughness of the finished part. Type of tool, cutting condition, feed and depth of cut influenced roundness error the most. Cutting condition influenced crater wear and feed influenced surface roughness the most.

Accurate Sum and Dot Product with New Instruction for High-Precision Computing on ARMv8 Processor

The accumulation of rounding errors can lead to unreliable results. Therefore, accurate and efficient algorithms are required. A processor from the ARMv8 architecture has introduced new instructions for high-precision computation. We have redesigned and implemented accurate summation and the accurate dot product. The number of floating-point operations has been reduced from 7n−5 and 10n−5 to 4n−2 and 7n−2, compared with the classic compensated precision algorithms. It has been proven that our accurate summation and dot algorithms’ error bounds are γn−1γncond+u and γnγn+1cond+u, where ’cond’ denotes the condition number, γn=n·u/(1−n·u), and u denotes the relative rounding error unit. Our accurate summation and dot product achieved a 1.69× speedup and a 1.14× speedup, respectively, on a simulation platform. Numerical experiments also illustrate that, under round-towards-zero mode, our algorithms are as accurate as the classic compensated precision algorithms.

Investigation of the corner rounding effect near the diffraction limit in advanced projection lithography with a rigorous imaging model.

The corner rounding effect in lithography refers to the phenomenon where the corners or angles of a pattern created by lithography are rounded off, rather than remaining square and sharp. This occurs mainly due to the diffraction of light. In addition, mask pattern design, numerical aperture, and the limited resolution of the lithographic process also influence it. The rounding of corners can severely degrade the performance and reliability of microelectronic devices due to incomplete pattern transfer. Therefore, minimizing the corner radius in mask correction is crucial for enhancing pattern fidelity. We propose a methodology that relates the corner radius to the diffraction-limited spot width. By employing aerial image simulation, we delve into the factors that significantly impact the corner radius in projection lithography. By combining a rigorous computational imaging model with the optimization of lithography parameters, the limit of the corner radius is determined in immersion lithography. Our comprehensive investigation reveals the nature of the corner rounding effect, analyzes the contributing factors for corner rounding, and ultimately leads to improvements in imaging quality in photolithography.

MainModelingStr: A Structural Dynamic Nonlinear Analysis Framework with Adaptive Beam-Column Finite Elements

This study introduces a new method for nonlinear structural analysis of buildings using high-order Hermitian interpolation function matrices for Timoshenko elements. This method aims to improve the accuracy of traditional beam-column finite element analyses by using higher-order shape function polynomials. Thus, a p-adaptive procedure is employed to address the potential issues of increased complexity and round-off error associated with higher-order functions. The study uses a technique for removing outliers in the p-adaptive method, which is more consistent with the theoretical approach. Moreover, a novel strategy based on the generalized alpha method is applied to reduce even more convergence problems, allowing its parameters to adapt in each instant of the time-history analysis. These adaptive parameters permit setting a low tolerance error and avoiding changing these parameters manually in time-consuming analyses. The efficiency of using fourth-order beam-column elements is also investigated. Finally, the importance of the proposed method is demonstrated through examples of building analyses, which is the base of a proposed software called MainModelingStr.

Building a BBB Pseudorandom Permutation using Lai-Massey Networks

In spite of being a popular technique for designing block ciphers, Lai-Massey networks have received considerably less attention from a security analysis point of view than Feistel networks and Substitution-Permutation networks. In this paper we study the beyond-birthday-bound (BBB) security of Lai-Massey networks with independent random round functions against chosen-plaintext adversaries. Concretely, we show that five rounds are necessary and sufficient to achieve BBB security.

On-machine measurement system for roundness and axis straightness errors in deep hole boring

On-machine measurement system for roundness and axis straightness errors in deep hole boring

A novel distinguishing attack on Rocca

AbstractThe security of an encryption algorithm often hinges on the indistinguishability between ciphertext and random numbers. In block ciphers, pseudorandomness and super‐pseudorandomness are commonly used to depict the indistinguishability between ciphertext and random numbers. Whereas, stream cipher algorithms can be viewed as functions of variables such as the key K, initialization vector IV, and plaintext M. For an ideal stream cipher algorithm, the ciphertext sequences for any two different plaintexts should exhibit good independence across various K‐IV pairs. Consequently, the probability of the two ciphertext sequences being equal or having sufficiently long matching segments should be negligible. This study examines a new type of attack that stems from key commitment attacks. By exploiting the independence between the ciphertext sequences, a novel distinguishing attack against the stream cipher is constructed and applied to the encryption of Rocca. Focusing on the authenticated encryption algorithm Rocca, a guess‐and‐determine method is employed to demonstrate that for any plaintext and two different sets of , another plaintext can be found with a time complexity , resulting in identical ciphertext sequences except for the initial bits. Furthermore, it is proved that under chosen plaintext conditions, the time complexity of this distinguishing attack is . These findings imply that Rocca does not offer 256‐bit security against such distinguishing attacks, providing valuable insights into the design of round update functions for stream ciphers like Rocca.

A High-Precision Roundness Error Separation Method Based on Time–Frequency-Domain Transformation for Parameter Optimization

A High-Precision Roundness Error Separation Method Based on Time–Frequency-Domain Transformation for Parameter Optimization

Epidemiology of Rounding Error.

This work represents a significant contribution to understanding the importance of appropriately rounding numbers with minimal error. That is, to reduce inexact rounding and data truncation error and simultaneously eliminate unintentional misleading findings in epidemiological studies. The rounding of numbers represents a compromise solution that attempts to find a balance between the loss of information from reporting too few significant digits versus retaining more digits than necessary. Substituting a rounded number for its original value may be acceptable and practical in many applied situations if an adequate degree of accuracy is retained. On the other hand, numeric error may result from improper rounding or data truncation which, in effect, compromises the credibility of study findings and may lead to a false sense of discovery. Performing complex computations on such values, especially when sequential or composite operations are involved, can lead to error propagation and inaccurate results. Having an overall awareness of the nature and impact of rounding error, including preventive actions, can contribute greatly to the integrity of research, yielding more reliable and accurate conclusions. Heuristic examples are provided to illustrate the consequences of rounding and data truncation error in epidemiology studies, specifically those pertaining to relative effect estimation.

Rotor Position Estimation Method for Permanent Magnet Synchronous Motor Based on High-Order Extended Kalman Filter

To address the issue of decreased rotor position estimation accuracy in permanent magnet synchronous motors (PMSMs) caused by linearization rounding errors in the extended Kalman filter (EKF), this paper proposes a rotor position estimation method for PMSMs based on higher-order extended Kalman filtering. This method relies on the state-space equations of a PMSM in a stationary coordinate system and establishes a higher-order Taylor series expansion based on the least squares approach. It constructs a prediction and update model for the state variables using the higher-order Taylor series expansion and designs an algorithm for estimating the rotor position of PMSMs based on higher-order extended Kalman filtering. The simulation results indicate that, compared to the EKF, the proposed method reduces the root-mean-square error by 10%.

Symmetric Twin Column Parity Mixers and Their Applications

The circulant twin column parity mixer (TCPM) is a type of mixing layer for the round function of cryptographic permutations designed by Hirch et al. at CRYPTO 2023. It has a bitwise differential branch number of 12 and a bitwise linear branch number of 4, which makes it competitive in applications where differential security is required. Hirch et al. gave a concrete instantiation of a permutation using such a mixing layer, named Gaston, and showed the best 3-round differential and linear trails of Gaston have much higher weights than those of Ascon. In this paper, we first prove why the TCPM has linear branch number 4 and then show that Gaston’s linear behavior is worse than Ascon for more than 3 rounds. Motivated by these facts, we aim to enhance the linear security of the TCPM. We show that adding a specific set of row cyclic shifts to the TCPM can make its differential and linear branch numbers both 12. Notably, by setting a special relationship between the row shift parameters of the modified TCPM, we obtain a special kind of mixlayer called the symmetric circulant twin column parity mixer. The symmetric TCPM has a unique design property that its differential and linear branch histograms are the same, which makes the parameter selection process and the security analysis convenient. Using the symmetric TCPM, we present two new 320-bit cryptographic permutations, namely (1) Gaston-S where we replace the mixing layer in Gaston with the symmetric TCPM and (2) SBD which uses a low-latency degree-4 S-box as the non-linear layer and the symmetric TCPM as the mixing layer. We evaluate the security of these permutations considering differential, linear and algebraic analysis, and then provide the performance comparison with Gaston in both hardware and software. Our results indicate that Gaston-S and SBD are competitive with Gaston in both security and performance.

Comparison of the combinations of Fourier pseudospectral method, finite volume method, Euler method and fourth order Runge-Kutta method used to solve Burgers equation

The Burgers equation is a mathematical model frequently used in Computational Fluid Dynamics. It is often employed to test and calibrate numerical methods, as it is one of the few nonlinear transport equations with an exact analytical solution. In this paper, numerical solutions are obtained using the Finite Difference Method (FDM) and the Fourier Pseudospectral Method (FPSM) for spatial discretization, combined with the Euler Method and the Fourth-Order Runge-Kutta Method (FRKM) for time discretization. The results are compared with the exact analytical solution in terms of accuracy, convergence rate, and computational cost. The findings indicate that the combination of the FDM and Euler Method achieves excellent computational efficiency when compared to the other approaches. Meanwhile, the combination of FPSM and FRKM demonstrates superior accuracy (achieving round-off errors) and a high order of convergence (spectral convergence order). Thus, combining methods with similar convergence rates and accuracy is the optimal strategy for obtaining efficient numerical solutions of partial differential equations (PDEs).

An On-Machine Measuring Apparatus for Dimension and Form Errors of Deep-Hole Parts.

The precise measurement of inner dimensions and contour accuracy is required for deep-hole parts, particularly during the manufacturing process, to monitor quality and obtain real-time error parameters. However, on-machine measurement is challenging due to the limited inner space of deep holes. This study proposes an automatic on-machine measuring apparatus for assessing inner diameter, straightness, and roundness errors. Based on the axial-section measurement principle, an integrated measuring module was designed, including a self-centering mechanism, a diameter measuring sensor, and a positioning reference sensor, all embedded within a control system. On this basis, calculations of the inner diameter, and evaluations of the straightness and roundness errors are presented. Experimental verification is conducted on a blind deep hole with a nominal 100 mm inner diameter and 700 mm depth. Compared with measurements performed on a coordinate measuring machine (CMM), which is limited to a maximum hole depth of 300 mm, the proposed apparatus achieved full-depth on-machine measurements. Meanwhile, the measurement results were consistent with the data obtained by the CMM. The straightness error is considered less than 0.05 mm, and the roundness error is considered less than 0.015 mm. Ultimately, without requiring any additional reference platform, the proposed apparatus shows promise for measuring deep-hole parts on various machine tools, with diameters of no less than 80 mm and theoretically unlimited hole depth.

An adaptive data-driven algorithm with machine learning for modeling high-order nonlinear beam-column finite elements

Structural analyses of buildings using beam-column finite elements are typically performed using procedures based on rotation polynomials truncated to the second order. This truncation will lead to inaccurate results that must be reduced. For this purpose, this study develops straightforward high-order Hermitian interpolation function matrices for Timoshenko elements. Moreover, a novel distributed plasticity hinge is devised. However, the structure complexity and the round-off error will increase with the interpolation function order. Nonetheless, a p-adaptive procedure can alleviate this problem. In contrast to other research, a technique for removing outliers for the maximum permissible error in the p-adaptive method is used to obtain results in a discrete space instead of using integrals in the used norm, which is more consistent with the theory. Moreover, as with other studies, it has been noticed that naïve p-adaptive processes diverge because of oscillations in the iterations. The proposed algorithm groups elements to be refined in different stages, employing the clustering function of the K-means method. Additionally, it is demonstrated that using fourth-order beam-column elements is also inefficient in elements that must be refined. After showing the theoretical base, this investigation's importance is proved with examples of buildings of different complexities.

A generalized maximum correntropy based constraint adaptive filtering: Constraint-forcing and performance analyses.

The quadratic cost functions, exemplified by mean-square-error, often exhibit limited robustness and flexibility when confronted with impulsive noise contamination. In contrast, the generalized maximum correntropy (GMC) criterion, serving as a robust nonlinear similarity measure, offers superior performance in such scenarios. In this paper, we develop a recursive constrained adaptive filtering algorithm named recursive generalized maximum correntropy with a forgetting factor (FF-RCGMC). This algorithm integrates the exponential weighted GMC criterion with a linear constraint framework based on least-squares. However, the lack of constraint information during the learning process may lead to divergence or malfunctioning of FF-RCGMC after a certain number of iterations because of round-off errors. To rectify this deficiency, we introduce a constraint-forcing strategy into FF-RCGMC, resulting in a more stable variant termed robust type constraint-forcing FF-RCGMC (CFFF-RCGMC). In the context of CFFF-RCGMC, we embark on a thorough examination of its computational burden, encompassing both mean and mean-square stability analyses, along with an in-depth exploration of its transient and steady-state filtering characteristics under a set of plausible assumptions. Our simulation-based evaluations, specifically tailored for system identification tasks within non-Gaussian noisy environments, unequivocally underscore the excellent performance of CFFF-RCGMC when against its relevant algorithmic counterparts.

Quantization via Distillation and Contrastive Learning.

Quantization is a critical technique employed across various research fields for compressing deep neural networks (DNNs) to facilitate deployment within resource-limited environments. This process necessitates a delicate balance between model size and performance. In this work, we explore knowledge distillation (KD) as a promising approach for improving quantization performance by transferring knowledge from high-precision networks to low-precision counterparts. We specifically investigate feature-level information loss during distillation and emphasize the importance of feature-level network quantization perception. We propose a novel quantization method that combines feature-level distillation and contrastive learning to extract and preserve more valuable information during the quantization process. Furthermore, we utilize the hyperbolic tangent function to estimate gradients with respect to the rounding function, which smoothens the training procedure. Our extensive experimental results demonstrate that the proposed approach achieves competitive model performance with the quantized network compared to its full-precision counterpart, thus validating its efficacy and potential for real-world applications.

Automated Roundoff Error Analysis of Probabilistic Floating-Point Computations

Automated Roundoff Error Analysis of Probabilistic Floating-Point Computations

High-precision roundness measuring system for tungsten ball tips with the diameter of less than 100 µm

Abstract Tungsten balls with a diameter of less than 100 μm can function as probe tips for micro-coordinate measuring machines to measure ultraprecise workpieces with complex features. A high-precision measuring system was developed to evaluate the roundness of self-made tungsten ball tips. The system, based on the two-point method, is composed of two oppositely placed interferometers and a turntable. The turntable rotates the tested ball, and the interferometers measure the variation in radius. A new model based on the minimum zone method is proposed to evaluate roundness. The fabrication principle of tungsten balls is introduced and an analysis is conducted on the maximum permissible contact force and bending stress of tungsten ball tips. The elastic mechanism is designed, based on the analysis. A ruby sphere with a sphericity of 130 nm is measured to verify the system’s effectiveness. Repeated experiments are performed for two tungsten ball tips. The results show that the roundness error of tungsten ball-A is between 550.9 and 586.2 nm with a standard deviation of 11.0 nm while tungsten ball-B’s roundness error is between 898.5 and 959.9 nm with a standard deviation of 21.7 nm. The uncertainty is calculated as 144.6 nm. The developed system can stably measure the roundness of tungsten ball tips.