Thematische Bibliographien / Stochastic rounding

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Auswahl der wissenschaftlichen Literatur zum Thema „Stochastic rounding“

Autor: Grafiati

Veröffentlicht am 25. Mai 2024

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Zeitschriftenartikel zum Thema "Stochastic rounding"

Paxton, E. Adam, Matthew Chantry, Milan Klöwer, Leo Saffin, and Tim Palmer. "Climate Modeling in Low Precision: Effects of Both Deterministic and Stochastic Rounding." Journal of Climate 35, no. 4 (2022): 1215–29. http://dx.doi.org/10.1175/jcli-d-21-0343.1.

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Abstract Motivated by recent advances in operational weather forecasting, we study the efficacy of low-precision arithmetic for climate simulations. We develop a framework to measure rounding error in a climate model, which provides a stress test for a low-precision version of the model, and we apply our method to a variety of models including the Lorenz system, a shallow water approximation for flow over a ridge, and a coarse-resolution spectral global atmospheric model with simplified parameterizations (SPEEDY). Although double precision [52 significant bits (sbits)] is standard across operational climate models, in our experiments we find that single precision (23 sbits) is more than enough and that as low as half precision (10 sbits) is often sufficient. For example, SPEEDY can be run with 12 sbits across the code with negligible rounding error, and with 10 sbits if minor errors are accepted, amounting to less than 0.1 mm (6 h)−1 for average gridpoint precipitation, for example. Our test is based on the Wasserstein metric and this provides stringent nonparametric bounds on rounding error accounting for annual means as well as extreme weather events. In addition, by testing models using both round-to-nearest (RN) and stochastic rounding (SR) we find that SR can mitigate rounding error across a range of applications, and thus our results also provide some evidence that SR could be relevant to next-generation climate models. Further research is needed to test if our results can be generalized to higher resolutions and alternative numerical schemes. However, the results open a promising avenue toward the use of low-precision hardware for improved climate modeling. Significance Statement Weather and climate models provide vital information for decision-making, and will become ever more important in the future with a changed climate and more extreme weather. A central limitation to improved models are computational resources, which is why some weather forecasters have recently shifted from conventional 64-bit to more efficient 32-bit computations, which can provide equally accurate forecasts. Climate models, however, still compute in 64 bits, and adapting to lower precision requires a detailed analysis of rounding errors. We develop methods to quantify rounding error in a climate model, and find similar precision acceptable across weather and climate models, with even 16 bits often sufficient for an accurate climate. This opens a promising avenue for computational efficiency gains in climate modeling.

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Connolly, Michael P., Nicholas J. Higham, and Theo Mary. "Stochastic Rounding and Its Probabilistic Backward Error Analysis." SIAM Journal on Scientific Computing 43, no. 1 (2021): A566—A585. http://dx.doi.org/10.1137/20m1334796.

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Gupta, Anupam, R. Ravi, and Amitabh Sinha. "LP Rounding Approximation Algorithms for Stochastic Network Design." Mathematics of Operations Research 32, no. 2 (2007): 345–64. http://dx.doi.org/10.1287/moor.1060.0237.

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Arciniega, Armando, and Edward Allen. "Rounding Error in Numerical Solution of Stochastic Differential Equations." Stochastic Analysis and Applications 21, no. 2 (2003): 281–300. http://dx.doi.org/10.1081/sap-120019286.

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Arar, El-Mehdi El, Devan Sohier, Pablo de Oliveira Castro, and Eric Petit. "Stochastic Rounding Variance and Probabilistic Bounds: A New Approach." SIAM Journal on Scientific Computing 45, no. 5 (2023): C255—C275. http://dx.doi.org/10.1137/22m1510819.

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McCarl, Bruce A. "Generalized Stochastic Dominance: An Empirical Examination." Journal of Agricultural and Applied Economics 22, no. 2 (1990): 49–55. http://dx.doi.org/10.1017/s1074070800001796.

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Abstract Use of generalized stochastic dominance (GSD) requires one to place lower and upper bounds on the risk aversion coefficient. This study showed that breakeven risk aversion coefficients found assuming the exponential utility function delineate the places where GSD preferences switch between prospects. However, between these break points, multiple, overlapping GSD intervals can be found. Consequently, when one does not have risk aversion coefficient information, discovery of breakeven coefficients instead of GSD use is recommended. The investigation also showed GSD results are insensitive to wealth and data scaling but are sensitive to rounding.

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Hopkins, Michael, Mantas Mikaitis, Dave R. Lester, and Steve Furber. "Stochastic rounding and reduced-precision fixed-point arithmetic for solving neural ordinary differential equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2166 (2020): 20190052. http://dx.doi.org/10.1098/rsta.2019.0052.

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Although double-precision floating-point arithmetic currently dominates high-performance computing, there is increasing interest in smaller and simpler arithmetic types. The main reasons are potential improvements in energy efficiency and memory footprint and bandwidth. However, simply switching to lower-precision types typically results in increased numerical errors. We investigate approaches to improving the accuracy of reduced-precision fixed-point arithmetic types, using examples in an important domain for numerical computation in neuroscience: the solution of ordinary differential equations (ODEs). The Izhikevich neuron model is used to demonstrate that rounding has an important role in producing accurate spike timings from explicit ODE solution algorithms. In particular, fixed-point arithmetic with stochastic rounding consistently results in smaller errors compared to single-precision floating-point and fixed-point arithmetic with round-to-nearest across a range of neuron behaviours and ODE solvers. A computationally much cheaper alternative is also investigated, inspired by the concept of dither that is a widely understood mechanism for providing resolution below the least significant bit in digital signal processing. These results will have implications for the solution of ODEs in other subject areas, and should also be directly relevant to the huge range of practical problems that are represented by partial differential equations. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

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Ji, Sai, Dachuan Xu, Donglei Du, and Yijing Wang. "LP-rounding approximation algorithms for two-stage stochastic fault-tolerant facility location problem." Applied Mathematical Modelling 58 (June 2018): 76–85. http://dx.doi.org/10.1016/j.apm.2017.12.009.

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Tovissodé, Chénangnon Frédéric, Sèwanou Hermann Honfo, Jonas Têlé Doumatè, and Romain Glèlè Kakaï. "On the Discretization of Continuous Probability Distributions Using a Probabilistic Rounding Mechanism." Mathematics 9, no. 5 (2021): 555. http://dx.doi.org/10.3390/math9050555.

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Most existing flexible count distributions allow only approximate inference when used in a regression context. This work proposes a new framework to provide an exact and flexible alternative for modeling and simulating count data with various types of dispersion (equi-, under-, and over-dispersion). The new method, referred to as “balanced discretization”, consists of discretizing continuous probability distributions while preserving expectations. It is easy to generate pseudo random variates from the resulting balanced discrete distribution since it has a simple stochastic representation (probabilistic rounding) in terms of the continuous distribution. For illustrative purposes, we develop the family of balanced discrete gamma distributions that can model equi-, under-, and over-dispersed count data. This family of count distributions is appropriate for building flexible count regression models because the expectation of the distribution has a simple expression in terms of the parameters of the distribution. Using the Jensen–Shannon divergence measure, we show that under the equidispersion restriction, the family of balanced discrete gamma distributions is similar to the Poisson distribution. Based on this, we conjecture that while covering all types of dispersions, a count regression model based on the balanced discrete gamma distribution will allow recovering a near Poisson distribution model fit when the data are Poisson distributed.

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Чубич, Владимир Михайлович, and Светлана Олеговна Кулабухова. "Square-root algorithms for robust modifications of the continuous-discrete cubature Kalman filter." Вычислительные технологии, no. 3 (July 15, 2020): 88–98. http://dx.doi.org/10.25743/ict.2020.25.3.010.

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Предложены две устойчивые к ошибкам машинного округления и к аномальным данным квадратно-корневые модификации непрерывно-дискретного кубатурного фильтра Калмана, основанные на вариационном байесовском и коррентропийном подходах. Апробация разработанных алгоритмов на модельной задаче со случайным характером расположения аномальных наблюдений показала их работоспособность при сопоставимом качестве фильтрации. Подтверждена алгебраическая эквивалентность представленных квадратно-корневых и стандартных версий Rounding errors due to the finite length of machine word can significantly affect the quality of estimation and filtering when solving the corresponding problems in various subject areas. In this regard, to improve the reliability of the obtained results, it is advisable to develop and then apply square-root modifications of the used algorithms. Purpose: developing the square-root modifications of the continuous-discrete cubature Kalman filter on the basis of variational Bayesian and correntropy approaches. Methodology: matrix orthogonal QR decomposition. Findings: two robust (resistant to the possible presence of anomalous data and to machine rounding errors) modifications of the continuous-discrete cubature Kalman filter have been developed. The first (variational Bayesian) algorithm is obtained by extending the known discrete equations of the extrapolation stage to the continuous-discrete case. The second algorithm, based on the maximum correntropy criterion, is proposed in this paper for the first time. The developed square-root algorithms for nonlinear filtering are validated on the example of one stochastic dynamical system model with the random location of anomalous observations. In doing so, the filtering quality, estimated by the value of the accumulated mean square error, was quite comparable for both modifications during equivalent results obtained for the corresponding root-free analogues. Value: the proposed square-root versions of robust modifications of the continuous-discrete cubature Kalman filter are algebraically equivalent to their standard analogues. Meanwhile, positive definiteness and symmetry of covariance matrices of the state vector estimates at the extrapolation and the filtration stages are provided. The developed algorithms will be used to develop software and mathematical support for parametric identification of stochastic nonlinear continuous-discrete systems in the presence of anomalous observations in the measurement data

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Dissertationen zum Thema "Stochastic rounding"

El, Arar El-Mehdi. "Stochastic models for the evaluation of numerical errors." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPASG104.

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L'idée de considérer les erreurs d'arrondi comme des variables aléatoires n'est pas nouvelle. Basées sur des outils tels que l'indépendance des variables aléatoires ou le théorème central limite, plusieurs propositions ont démontré des bornes d'erreur en O(√n). Cette thèse est dédiée à l'étude de l'arrondi stochastique (SR) en tant que remplaçant du mode d'arrondi déterministe par défaut. Tout d'abord, nous introduisons une nouvelle approche pour dériver une borne probabiliste de l'erreur en O(√n), basée sur le calcul de la variance et l'inégalité de Bienaymé-Chebyshev. Ensuite, nous développons un cadre général permettant l'analyse probabiliste des erreurs des algorithmes sous SR. Dans ce contexte, nous décomposons l'erreur en une martingale plus un biais. Nous montrons que le biais est nul pour les algorithmes présentant des erreurs multilinéaires, tandis que l'analyse probabiliste de la martingale conduit à des bornes probabilistes de l'erreur en O(√n). Pour le calcul de la variance, nous montrons que le biais est négligeable au premier ordre par rapport à la martingale, et nous prouvons des bornes probabilistes de l'erreur en O(√n)<br>The idea of assuming rounding errors as random variables is not new. Based on tools such as independent random variables or the Central Limit Theorem, various propositions have demonstrated error bounds in O(√n). This thesis is dedicated to studying stochastic rounding (SR) as a replacement for the default deterministic rounding mode. First, we introduce a new approach to derive a probabilistic error bound in O(√n) based on variance calculation and Bienaymé-Chebyshev inequality. Second, we demonstrate a general framework that allows the probabilistic error analysis of algorithms under SR. In this context, we decompose the error into a martingale plus a drift. We show that the drift is zero for algorithms with multi-linear errors, while the probabilistic analysis of the martingale term leads to probabilistic error bounds in O(√n). We show that the drift is negligible at the first order compared to the martingale term for the variance computation, and we prove probabilistic error bounds in O(√n)

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Picot, Romain. "Amélioration de la fiabilité numérique de codes de calcul industriels." Electronic Thesis or Diss., Sorbonne université, 2018. http://www.theses.fr/2018SORUS242.

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De nombreux travaux sont consacrés à la performance des simulations numériques, or il est important de tenir compte aussi de l'impact des erreurs d'arrondi sur les résultats produits. Ces erreurs d'arrondi peuvent être estimées grâce à l'Arithmétique Stochastique Discrète (ASD), implantée dans la bibliothèque CADNA. Les algorithmes compensés permettent d'améliorer la précision des résultats, sans changer le type numérique utilisé. Ils ont été conçus pour être généralement exécutés en arrondi au plus près. Nous avons établi des bornes d'erreur pour ces algorithmes en arrondi dirigé et montré qu'ils peuvent être utilisés avec succès avec le mode d'arrondi aléatoire de l'ASD. Nous avons aussi étudié l’impact d’une précision cible des résultats sur les types numériques des différentes variables. Nous avons développé l'outil PROMISE qui effectue automatiquement ces modifications de types tout en validant les résultats grâce à l’ASD. L'outil PROMISE a ainsi fourni de nouvelles configurations de types mêlant simple et double précision dans divers programmes numériques et en particulier dans le code MICADO développé à EDF. Nous avons montré comment estimer avec l'ASD les erreurs d'arrondi générées en quadruple précision. Nous avons proposé une version de CADNA qui intègre la quadruple précision et qui nous a permis notamment de valider le calcul de racines multiples de polynômes. Enfin nous avons utilisé cette nouvelle version de CADNA dans l'outil PROMISE afin qu'il puisse fournir des configurations à trois types (simple, double et quadruple précision)<br>Many studies are devoted to performance of numerical simulations. However it is also important to take into account the impact of rounding errors on the results produced. These rounding errors can be estimated with Discrete Stochastic Arithmetic (DSA), implemented in the CADNA library. Compensated algorithms improve the accuracy of results, without changing the numerical types used. They have been designed to be generally executed with rounding to nearest. We have established error bounds for these algorithms with directed rounding and shown that they can be used successfully with the random rounding mode of DSA. We have also studied the impact of a target precision of the results on the numerical types of the different variables. We have developed the PROMISE tool which automatically performs these type changes while validating the results thanks to DSA. The PROMISE tool has thus provided new configurations of types combining single and double precision in various programs and in particular in the MICADO code developed at EDF. We have shown how to estimate with DSA rounding errors generated in quadruple precision. We have proposed a version of CADNA that integrates quadruple precision and that allowed us in particular to validate the computation of multiple roots of polynomials. Finally we have used this new version of CADNA in the PROMISE tool so that it can provide configurations with three types (single, double and quadruple precision)

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Huber, Anna [Verfasser]. "Randomized rounding and rumor spreading with stochastic dependencies / vorgelegt von Anna Huber." 2010. http://d-nb.info/1008296163/34.

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Buchteile zum Thema "Stochastic rounding"

Giessing, Sarah. "Flexible Rounding Based on Consistent Post-tabular Stochastic Noise." In Privacy in Statistical Databases. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33627-0_3.

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Yuan, Geng, Sung-En Chang, Qing Jin, et al. "You Already Have It: A Generator-Free Low-Precision DNN Training Framework Using Stochastic Rounding." In Lecture Notes in Computer Science. Springer Nature Switzerland, 2022. http://dx.doi.org/10.1007/978-3-031-19775-8_3.

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Konferenzberichte zum Thema "Stochastic rounding"

Mikaitis, Mantas. "Stochastic Rounding: Algorithms and Hardware Accelerator." In 2021 International Joint Conference on Neural Networks (IJCNN). IEEE, 2021. http://dx.doi.org/10.1109/ijcnn52387.2021.9533756.

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Chang, Sung-En, Geng Yuan, Alec Lu, et al. "Hardware-efficient stochastic rounding unit design for DNN training." In DAC '22: 59th ACM/IEEE Design Automation Conference. ACM, 2022. http://dx.doi.org/10.1145/3489517.3530619.

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Arar, El-Mehdi El, Devan Sohier, Pablo de Oliveira Castro, and Eric Petit. "The Positive Effects of Stochastic Rounding in Numerical Algorithms." In 2022 IEEE 29th Symposium on Computer Arithmetic (ARITH). IEEE, 2022. http://dx.doi.org/10.1109/arith54963.2022.00018.

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Qian Zhang, Sai, Bradley McDanel, and H. T. Kung. "FAST: DNN Training Under Variable Precision Block Floating Point with Stochastic Rounding." In 2022 IEEE International Symposium on High-Performance Computer Architecture (HPCA). IEEE, 2022. http://dx.doi.org/10.1109/hpca53966.2022.00067.

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Essam, Mohaned, Tong Boon Tang, Eric Tatt Wei Ho, and Hsin Chen. "Dynamic point stochastic rounding algorithm for limited precision arithmetic in Deep Belief Network training." In 2017 8th International IEEE/EMBS Conference on Neural Engineering (NER). IEEE, 2017. http://dx.doi.org/10.1109/ner.2017.8008430.

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Chang, Sung-En, Geng Yuan, Alec Lu, et al. "ESRU: Extremely Low-Bit and Hardware-Efficient Stochastic Rounding Unit Design for Low-Bit DNN Training." In 2023 Design, Automation & Test in Europe Conference & Exhibition (DATE). IEEE, 2023. http://dx.doi.org/10.23919/date56975.2023.10137222.

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Ghenaiet, Adel. "Study of Sand Particle Trajectories and Erosion Into the First Fan Stage of a Turbofan." In ASME Turbo Expo 2010: Power for Land, Sea, and Air. ASMEDC, 2010. http://dx.doi.org/10.1115/gt2010-22415.

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Ingestion of dust particles by aero-engines or stationary gas turbines is inevitable when operating in extremely polluted environments. The impingements of particles on the surfaces of blades cause erosion damage and permanent losses in engine performance. This paper presents a study of the particle dynamics and erosion in the first stage of a turbofan. The steady flow field through the turbomachinery components was solved separately from the solid phase. The particle trajectories computations used a stochastic Lagrangian tracking code that implements probabilistic modeling for particle size rebound and fragmentation, and considers the eddy-lifetime concept for turbulence and the complex flow features near walls. The equations of a particle motion were solved in a stepwise manner using the seventh order RK-Fehlberg technique, whereas particle tracking in different cells of the computational domain used the finite element method. Computations of particle trajectories were carried out for sand particles MIL-E5007E (0–1000 microns) at low, mid and high concentrations. As the locations of impacts were predicted, erosion contours were estimated and the subsequent blade deteriorations were assessed. The rotor blade shows a noticeable erosion of the blade leading and trailing edges almost from root to tip and a rounding of blade tip. Erosion patterns in the diffuser depict high erosion at blade leading and trailing edges and the erosion of pressure side is spreading almost from root to tip, in addition to erosion over the suction side. The actual findings may serve in improving erosion resistance of the blades in this fan stage.

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Dodson, C. T. J., and W. W. Sampson. "Effect of Correlated Free Fibre Lengths on Pore Size Distribution in Fibrous Mats." In Advances in Paper Science and Technology, edited by S. J. I’Anson. Fundamental Research Committee (FRC), Manchester, 2005. http://dx.doi.org/10.15376/frc.2005.2.943.

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We provide a simulator for a range of bivariate stochastic processes of various application in the physics of stochastic fibrous networks. We illustrate the effects of local correlation on the statistics of voids in the bulk and the surface of fibre mats in general and paper in particular. The reference case of random isotropy has an inherent ‘ground-state’ correlation of adjacent free-fibre-lengths; this explains the classical observation of Corte that pores seem mainly ‘roundish’ in real paper samples. In the isotropic case, the mean pore radius can be reduced from that in a random network by 20% through structural changes associated with increased flocculation. The mean eccentricity of pores seems to give a measure of the variability in free-fibre-length distributions that is not due to local correlation. We find a uniform effect of local correlation on mean pore eccentricity over a range of stochastic network structures; at a given correlation, increased flocculation increases mean eccentricity slightly.

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