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Journal articles on the topic 'Numerical error'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 December 2024

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1

Duyan, Yalçın Akın, and Fuat Balcı. "Numerical error monitoring." Psychonomic Bulletin & Review 25, no. 4 (2018): 1549–55. http://dx.doi.org/10.3758/s13423-018-1506-x.

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2

La Follette, Peter T., Adriaan J. Teuling, Nans Addor, Martyn Clark, Koen Jansen, and Lieke A. Melsen. "Numerical daemons of hydrological models are summoned by extreme precipitation." Hydrology and Earth System Sciences 25, no. 10 (2021): 5425–46. http://dx.doi.org/10.5194/hess-25-5425-2021.

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Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation such as that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation. In this experiment, a large number of hydrographs are generated with the modular modeling framework FUSE (Framework for Understanding Structural Errors), using eight numerical techniques across a variety of forcing data sets. All constructed models are conceptual and lumped. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational cost and numerical error associated with each hydrograph were recorded. Numerical error is assessed via root mean square error and normalized root mean square error. It was found that the root mean square error usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both a small numerical error and a low computational cost. A small literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be suboptimal. We conclude that relatively large numerical errors may be common in current models, highlighting the need for robust numerical techniques, in particular in the face of increasing precipitation extremes.

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3

Tzeng, Jengnan. "Linear Regression to Minimize the Total Error of the Numerical Differentiation." East Asian Journal on Applied Mathematics 7, no. 4 (2017): 810–26. http://dx.doi.org/10.4208/eajam.161016.300517a.

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AbstractIt is well known that numerical derivative contains two types of errors. One is truncation error and the other is rounding error. By evaluating variables with rounding error, together with step size and the unknown coefficient of the truncation error, the total error can be determined. We also know that the step size affects the truncation error very much, especially when the step size is large. On the other hand, rounding error will dominate numerical error when the step size is too small. Thus, to choose a suitable step size is an important task in computing the numerical differentiation. If we want to reach an accuracy result of the numerical difference, we had better estimate the best step size. We can use Taylor Expression to analyze the order of truncation error, which is usually expressed by the big O notation, that is, E(h) = Chk. Since the leading coefficient C contains the factor f(k)(ζ) for high order k and unknown ζ, the truncation error is often estimated by a roughly upper bound. If we try to estimate the high order difference f(k)(ζ), this term usually contains larger error. Hence, the uncertainty of ζ and the rounding errors hinder a possible accurate numerical derivative.We will introduce the statistical process into the traditional numerical difference. The new method estimates truncation error and rounding error at the same time for a given step size. When we estimate these two types of error successfully, we can reach much better modified results. We also propose a genetic approach to reach a confident numerical derivative.

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4

Cviklovič, V., D. Hrubý, M. Olejár, and O. Lukáč. "Comparison of numerical integration methods in strapdown inertial navigation algorithm ." Research in Agricultural Engineering 57, Special Issue (2011): S30—S34. http://dx.doi.org/10.17221/58/2010-rae.

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The numerical mathematical theory provides a few ways of numerical integration with different errors. It is necessary to make use of the most exact method with respect to the computing power for a majority of microprocessors, because errors are integrated within them due to the algorithm. In our contribution, trapezoidal rule and Romberg&rsquo;s method of numerical integration are compared in the velocity calculation algorithm of the strapdown inertial navigation. The sample frequency of acceleration and angular velocity measurement was 816.6599 Hz. Inertial navigation velocity was compared with precise incremental encoder data. Trapezoidal method velocity error in this example was 1.23 &times; 10<sup>&ndash;3</sup> m/s in the fifteenth-second measurement. Romberg&rsquo;s method velocity error was 0.16 &times; 10<sup>&ndash;3 </sup>m/s for the same input data. The numerical mathematical theory provides a few ways of numerical integration with different errors. It is necessary to make use of the most exact method with respect to the computing power for a majority of microprocessors, because errors are integrated within them due to the algorithm. In our contribution, trapezoidal rule and Romberg&rsquo;s method of numerical integration are compared in the velocity calculation algorithm of the strapdown inertial navigation. The sample frequency of acceleration and angular velocity measurement was 816.6599 Hz. Inertial navigation velocity was compared with precise incremental encoder data. Trapezoidal method velocity error in this example was 1.23 &times; 10<sup>&ndash;3</sup> m/s in the fifteenth-second measurement. Romberg&rsquo;s method velocity error was 0.16 &times; 10<sup>&ndash;3 </sup>m/s for the same input data.

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5

Cui, Jing Jun. "Analysis of Machining Error in Numerical Control Milling." Applied Mechanics and Materials 312 (February 2013): 710–13. http://dx.doi.org/10.4028/www.scientific.net/amm.312.710.

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Generally speaking, the error in machining is an important indicator measuring the accuracy of finished surface. The machining error often occurs in numerical control milling. Such error will be influenced by multiple factors, such as cutter wear, thermal deformation, machine tool deformation, vibration or positioning error. Nowadays, though our science and technology develops rapidly, machining error problem in numerical control milling occurs frequently. At present, several methods can be applied to forecast machining error problems in numerical control milling, including on the basis of machining theory, experimental study, design study and artificial intelligence. The analysis and forecast of machining error problems in numerical control milling can to some extent improve the degree of machining errors so as to promote the machining accuracy in milling. The author expresses the views on machining error problems according to current situations of numerical control milling.

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6

Ferreira, F., E. Gendron, G. Rousset, and D. Gratadour. "Numerical estimation of wavefront error breakdown in adaptive optics." Astronomy & Astrophysics 616 (August 2018): A102. http://dx.doi.org/10.1051/0004-6361/201832579.

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Aims. Adaptive optics (AO) system performance is improved using post-processing techniques, such as point spread function (PSF) deconvolution. The PSF estimation involves characterization of the different wavefront (WF) error sources in the AO system. We propose a numerical error breakdown estimation tool that allows studying AO error source behavior such as their correlations. We also propose a new analytical model for anisoplanatism and bandwidth errors that were validated with the error breakdown estimation tool. This model is the first step for a complete AO residual error model that is expressed in deformable mirror space, leading to practical usage such as PSF reconstruction or turbulent parameters identification. Methods. We have developed in the computing platform for adaptive optics systems (COMPASS) code, which is an end-to-end simulation code using graphics processing units (GPU) acceleration, an estimation tool that provides a comprehensive error breakdown by the outputs of a single simulation run. We derive the various contributors from the end-to-end simulator at each iteration step: this method provides temporal buffers of each contributor. Then, we use this tool to validate a new model of anisoplanatism and bandwidth errors including their correlation. This model is based on a statistical approach that computes the error covariance matrices using structure functions. Results. On a SPHERE-like system, the comparison between a PSF computed from the error breakdown with a PSF obtained from classical end-to-end simulation shows that the statistics convergence limits converge very well, with a sub-percent difference in terms of Strehl ratio and ensquared energy at 5λ/D separation. A correlation analysis shows significant correlations between some contributors, especially WF measurement deviation error and bandwidth error due to centroid gain, and the well-known correlation between bandwidth and anisoplanatism errors is also retrieved. The model we propose for the two latter errors shows an SR and EE difference of about one percent compared to the end-to-end simulation, even if some approximations exist.

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7

T., V., Solomon G. Mikhlin, and Reinhard Lehmann. "Error Analysis in Numerical Processes." Mathematics of Computation 60, no. 201 (1993): 431. http://dx.doi.org/10.2307/2153180.

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8

Kinsella, A. "Numerical methods for error evaluation." American Journal of Physics 54, no. 5 (1986): 464–66. http://dx.doi.org/10.1119/1.14588.

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9

Stollberger, Claudia. "Correction: Numerical Error in Abstract." Annals of Internal Medicine 120, no. 4 (1994): 347. http://dx.doi.org/10.7326/0003-4819-120-4-199402150-00031.

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10

Huang, Yu Bin, Wei Sun, Qing Chao Sun, Yue Ma, and Hong Fu Wang. "Numerical Analysis of Thermal Error for a 4-Axises Horizontal Machining Center." Applied Mechanics and Materials 868 (July 2017): 64–68. http://dx.doi.org/10.4028/www.scientific.net/amm.868.64.

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Thermal deformations of machine tool are among the most significant error source of machining errors. Most of current thermal error modeling researches is about 3-axies machine tool, highly reliant on collected date, which could not predict thermal errors in design stage. In This paper, in order to estimate the thermal error of a 4-axise horizontal machining center. A thermal error prediction method in machine tool design stage is proposed. Thermal errors in workspace in different working condition are illustrated through numerical simulation and volumetric error model. Verification experiments shows the outcomes of this prediction method are basically correct.

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11

Faber, Vance, and Thomas A. Manteuffel. "Orthogonal Error Methods." SIAM Journal on Numerical Analysis 24, no. 1 (1987): 170–87. http://dx.doi.org/10.1137/0724014.

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12

Ditkowski, Adi, Sigal Gottlieb, and Zachary J. Grant. "Two-Derivative Error Inhibiting Schemes and Enhanced Error Inhibiting Schemes." SIAM Journal on Numerical Analysis 58, no. 6 (2020): 3197–225. http://dx.doi.org/10.1137/19m1306129.

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13

Wang, Xiao Jun, and Xiao Guang Fu. "Geometric Error Compensation Methods of Numerical Control Machine Tool." Advanced Materials Research 426 (January 2012): 239–42. http://dx.doi.org/10.4028/www.scientific.net/amr.426.239.

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In this paper the characteristics of geometric errors is discussed in detail, error compensation methods used in productive practice and relevant examples are given. Finally, the application of error compensation in different situation is discussed according to the characteristics of machining center. The machine accuracy can be improved by error compensation. It has important practical reference value for reasonable use and maintaining of NC machine tool.

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14

Samrowski, Tatiana. "Combined Error Estimates in the Case of Dimension Reduction." Computational Methods in Applied Mathematics 14, no. 1 (2014): 113–34. http://dx.doi.org/10.1515/cmam-2013-0024.

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Abstract. We consider the stationary reaction-diffusion problem in a domain $\Omega \subset \mathbb {R}^3$ having the size along one coordinate direction essentially smaller than along the others. By an energy type argumentation, different simplified models of lower dimension can be deduced and solved numerically. For these models, we derive a guaranteed upper bound of the difference between the exact solution of the original problem and a three-dimensional reconstruction generated by the solution of a dimensionally reduced problem. This estimate of the total error is determined as the sum of discretization and modeling errors, which are both explicit and computable. The corresponding discretization errors are estimated by a posteriori estimates of the functional type. Modeling error majorants are also explicitly evaluated. Hence, a numerical strategy based on the balancing modeling and discretization errors can be derived in order to provide an economical way of getting an approximate solution with an a priori given accuracy. Numerical tests are presented and discussed.

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15

Journal, Baghdad Science. "A Note on the Perturbation of arithmetic expressions." Baghdad Science Journal 13, no. 1 (2016): 190–97. http://dx.doi.org/10.21123/bsj.13.1.190-197.

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In this paper we present the theoretical foundation of forward error analysis of numerical algorithms under;• Approximations in "built-in" functions.• Rounding errors in arithmetic floating-point operations.• Perturbations of data.The error analysis is based on linearization method. The fundamental tools of the forward error analysis are system of linear absolute and relative a prior and a posteriori error equations and associated condition numbers constituting optimal of possible cumulative round – off errors. The condition numbers enable simple general, quantitative bounds definitions of numerical stability. The theoretical results have been applied a Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.

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16

Yang, Hongtao, Mei Shen, Li Li, Yu Zhang, Qun Ma, and Mengyao Zhang. "New identification method for computer numerical control geometric errors." Measurement and Control 54, no. 5-6 (2021): 1055–67. http://dx.doi.org/10.1177/00202940211010835.

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To address the problems of the low accuracy of geometric error identification and incomplete identification results of the linear axis detection of computer numerical control (CNC) machine tools, a new 21-item geometric error identification method based on double ball-bar measurement was proposed. The model between the double ball-bar reading and the geometric error term in each plane was obtained according to the three-plane arc trajectory measurement. The mathematical model of geometric error components of CNC machine tools is established, and the error fitting coefficients are solved through the beetle antennae search particle swarm optimization (BAS–PSO) algorithm, in which 21 geometric errors, including roll angle errors, were identified. Experiments were performed to compare the optimization effect of the BAS–PSO and PSO and BAS and genetic particle swarm optimization (GA–PSO) algorithms. Experimental results show that the PSO algorithm is trapped in the local optimum, and the BAS–PSO is superior to the other three algorithms in terms of convergence speed and stability, has higher identification accuracy, has better optimization performance, and is suitable for identifying the geometric error coefficient of CNC machine tools. The accuracy and validity of the identification results are verified by the comparison with the results of the individual geometric errors detected through laser interferometer experiments. The identification accuracy of the double ball-bar is below 2.7 µm. The proposed identification method is inexpensive, has a short processing time, is easy to operate, and possesses a reference value for the identification and compensation of the linear axes of machine tools.

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17

McGuigan, S. M. "The use of Statistics in the British Journal of Psychiatry." British Journal of Psychiatry 167, no. 5 (1995): 683–88. http://dx.doi.org/10.1192/bjp.167.5.683.

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BackgroundStatistical error rates in the medical literature are generally high.MethodAll papers published in the British Journal of Psychiatry in 1993 which presented numerical results were reviewed by the author for statistical errors.ResultsA total of 248 papers were published, of which 164 (66%) presented numerical results.Sixty-five (40% of 164) papers contained statistical errors. Many errors were not serious in nature, but some were serious enough to cast doubt on conclusions. The error rates are similar to those found in an earlier study.ConclusionsThe statistical error rate is unacceptably high. There is no evidence of a change in the statistical error rate over time.

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18

Kellison, Ariel E., and Justin Hsu. "Numerical Fuzz: A Type System for Rounding Error Analysis." Proceedings of the ACM on Programming Languages 8, PLDI (2024): 1954–78. http://dx.doi.org/10.1145/3656456.

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Algorithms operating on real numbers are implemented as floating-point computations in practice, but floating-point operations introduce roundoff errors that can degrade the accuracy of the result. We propose Λ num , a functional programming language with a type system that can express quantitative bounds on roundoff error. Our type system combines a sensitivity analysis, enforced through a linear typing discipline, with a novel graded monad to track the accumulation of roundoff errors. We prove that our type system is sound by relating the denotational semantics of our language to the exact and floating-point operational semantics. To demonstrate our system, we instantiate Λ num with error metrics proposed in the numerical analysis literature and we show how to incorporate rounding operations that faithfully model aspects of the IEEE 754 floating-point standard. To show that Λ num can be a useful tool for automated error analysis, we develop a prototype implementation for Λ num that infers error bounds that are competitive with existing tools, while often running significantly faster. Finally, we consider semantic extensions of our graded monad to bound error under more complex rounding behaviors, such as non-deterministic and randomized rounding.

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19

Miao, Zhongzheng, and Jinhai Zhang. "Reducing error accumulation of optimized finite-difference scheme using the minimum norm." GEOPHYSICS 85, no. 5 (2020): T275—T291. http://dx.doi.org/10.1190/geo2019-0758.1.

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The finite-difference (FD) scheme is popular in the field of seismic exploration for numerical simulation of wave propagation; however, its accuracy and computational efficiency are restricted by the numerical dispersion caused by numerical discretization of spatial partial derivatives using coarse grids. The constant-coefficient optimization method is used widely for suppressing the numerical dispersion by tuning the FD weights. Although gaining a wider effective bandwidth under a given error tolerance, this method undoubtedly encounters larger errors at low wavenumbers and accumulates significant errors. We have developed an approach to reduce the error accumulation. First, we construct an objective function based on the [Formula: see text] norm, which can constrain the total error better than the [Formula: see text] and [Formula: see text] norms. Second, we translated our objective function into a constrained [Formula: see text]-norm minimization model, which can be solved by the alternating direction method of multipliers. Finally, we perform theoretical analyses and numerical experiments to illustrate the accuracy improvement. The proposed method is shown to be superior to the existing constant-coefficient optimization methods at the low-wavenumber region; thus, we can obtain higher accuracy with less error accumulation, particularly at longer simulation times. The widely used objective functions, defined by the [Formula: see text] and [Formula: see text] norms, could handle a relatively wider range of accurate wavenumbers, compared with our objective function defined by the [Formula: see text] norm, but their actual errors would be much larger than the given error tolerance at some azimuths rather than axis directions (e.g., about twice at 45°), which greatly degrade the overall numerical accuracy. In contrast, our scheme can obtain a relatively even 2D error distribution at various azimuths, with an apparently smaller error. The peak error of the proposed method is only 40%–65% that of the [Formula: see text] norm under the same error tolerance, or only 60%–80% that of the [Formula: see text] norm under the same effective bandwidth.

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20

Bharathi R and Selvarani R. "Software Reliability Assessment of Safety Critical System Using Computational Intelligence." International Journal of Software Science and Computational Intelligence 11, no. 3 (2019): 1–25. http://dx.doi.org/10.4018/ijssci.2019070101.

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In the recent past, automotive industries are concentrating on software controlled automatic functions for its safety operations. The automotive safety and reliability lie in its design, construction, and software implementation. To assess the software reliability, the hidden design errors are classified and quantified. The temporal characteristic of numerical error is analyzed and its probabilistic behavior is explored using a novel framework called software failure estimation with numerical error (SFENE). Here, a model is devised to assess the probability of occurrence of the numerical error and its propagations from the initial to various other states using a Hidden Markov Model. It is seen that the framework SFENE supports classifying and quantifying the behavior of numerical errors while interacting across its system components and aids in the assessment on software reliability at design stage. The sensitivity and precision are found to be satisfactory. This attempt will support in the development of cost effective and error free safety critical software system.

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21

Radaikin, O., L. Sabitov, Sergey Klyuev, L. Akhtyamova, T. Arakcheev, and A. Darvish. "ACCURACY OF THE NUMERICAL DIAGRAM METHOD FOR CALCULATING BAR REINFORCED CONCRETE ELEMENTS." Bulletin of Belgorod State Technological University named after. V. G. Shukhov 7, no. 6 (2022): 25–34. http://dx.doi.org/10.34031/2071-7318-2022-7-6-25-34.

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The sources available for analysis do not explain why they adopted exactly such an accuracy (error) of the numerical diagram method for calculating core reinforced concrete elements (calculated and maximum permissible error values), such a control parameter for its evaluation. There is no definitive approach to estimating the error of the method under consideration yet. The available literature does not have a strict theoretical basis. The article is intended to try to correct this situation. For this purpose, the mathematical theory of numerical methods, metrology and the theory of reinforced concrete are involved. The classification of errors arising in determining the true value of the control parameter that integrally characterizes the stress-strain state of the element has been developed (unavoidable errors (≈12, %) – errors of the discrete nonlinear deformation model and inaccuracies in the initial data, errors of the numerical diagram method (≈5 %), computational errors (≈0 %)). The curvature of the axis of the reinforced concrete rod is taken as such a parameter. It is concluded that the maximum value of the permissible error of the numerical diagram method, which characterizes the accuracy, should not exceed 5 % and can be adjusted to decrease (increase accuracy) by clarifying the errors of the computational model and experimental base.

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22

Chowdhary, Sangeeta, and Santosh Nagarakatte. "Fast shadow execution for debugging numerical errors using error free transformations." Proceedings of the ACM on Programming Languages 6, OOPSLA2 (2022): 1845–72. http://dx.doi.org/10.1145/3563353.

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This paper proposes, EFTSanitizer, a fast shadow execution framework for detecting and debugging numerical errors during late stages of testing especially for long-running applications. Any shadow execution framework needs an oracle to compare against the floating point (FP) execution. This paper makes a case for using error free transformations, which is a sequence of operations to compute the error of a primitive operation with existing hardware supported FP operations, as an oracle for shadow execution. Although the error of a single correctly rounded FP operation is bounded, the accumulation of errors across operations can result in exceptions, slow convergences, and even crashes. To ease the job of debugging such errors, EFTSanitizer provides a directed acyclic graph (DAG) that highlights the propagation of errors, which results in exceptions or crashes. Unlike prior work, DAGs produced by EFTSanitizer include operations that span various function calls while keeping the memory usage bounded. To enable the use of such shadow execution tools with long-running applications, EFTSanitizer also supports starting the shadow execution at an arbitrary point in the dynamic execution, which we call selective shadow execution. EFTSanitizer is an order of magnitude faster than prior state-of-art shadow execution tools such as FPSanitizer and Herbgrind. We have discovered new numerical errors and debugged them using EFTSanitizer.

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23

Li, Long, Binyang Chen, and Chengjun Wang. "Positioning Accuracy and Numerical Analysis of the Main Casting Mechanism of the Hybrid Casting Robot." Mathematical Problems in Engineering 2022 (April 13, 2022): 1–14. http://dx.doi.org/10.1155/2022/6140729.

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The positioning accuracy is a key index to measure the performance of the robot. This paper studies the positioning accuracy of the main pouring mechanism of the hybrid truss pouring robot and analyzes that the main error sources affecting the positioning accuracy are machining error, assembly error, and thermal deformation error. Error transfer matrix is constructed to describe the influence of machining errors and assembly errors on the position and pose of the terminal, and the error parameters have physical significance. The probability distribution of sensitive errors is discussed. A joint regression prediction model based on sensitive error sets is established to determine the thermal deformation error on the basis of fully considering the contribution rate of component error. The results show that the position error has a wide range of influences on the end pose, but the angle error is more sensitive, and the probability distribution of the sensitive error is concentrated. The reliable data can be obtained without reorganizing the measurement in the calibration process. The joint regression model considering the contribution rate of component error can effectively eliminate the collinearity problem in the prediction of thermal deformation from a single heat source. Compared with the single regression model, it has better prediction accuracy and effect.

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24

Mercer, Peter R. "Error Estimates for Numerical Integration Rules." College Mathematics Journal 36, no. 1 (2005): 27. http://dx.doi.org/10.2307/30044815.

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25

Reich, Sebastian. "Backward Error Analysis for Numerical Integrators." SIAM Journal on Numerical Analysis 36, no. 5 (1999): 1549–70. http://dx.doi.org/10.1137/s0036142997329797.

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26

Yu, Q., S. Günter, Y. Kikuchi, and K. H. Finken. "Numerical modelling of error field penetration." Nuclear Fusion 48, no. 2 (2008): 024007. http://dx.doi.org/10.1088/0029-5515/48/2/024007.

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27

Kuperberg, Greg. "Numerical Cubature Using Error-Correcting Codes." SIAM Journal on Numerical Analysis 44, no. 3 (2006): 897–907. http://dx.doi.org/10.1137/040615572.

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28

Estep, D. J., S. M. Verduyn Lunel, and R. D. Williams. "Error estimation for numerical differential equations." IEEE Antennas and Propagation Magazine 38, no. 2 (1996): 71–76. http://dx.doi.org/10.1109/74.500237.

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Mercer, Peter R. "Error Estimates For Numerical Integration Rules." College Mathematics Journal 36, no. 1 (2005): 27–34. http://dx.doi.org/10.1080/07468342.2005.11922107.

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Marinov, Tchavdar, Joe Omojola, Quintel Washington, and LaQunia Banks. "Behavior of the Numerical Integration Error." Applied Mathematics 05, no. 10 (2014): 1412–26. http://dx.doi.org/10.4236/am.2014.510133.

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31

Laurie, Dirk P. "Practical error estimation in numerical integration." Journal of Computational and Applied Mathematics 12-13 (May 1985): 425–31. http://dx.doi.org/10.1016/0377-0427(85)90036-6.

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32

Connors, Jeffrey M., Jeffrey W. Banks, Jeffrey A. Hittinger, and Carol S. Woodward. "A Method to Calculate Numerical Errors Using Adjoint Error Estimation for Linear Advection." SIAM Journal on Numerical Analysis 51, no. 2 (2013): 894–926. http://dx.doi.org/10.1137/110845100.

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33

Mahadevan, Sankaran, and Ramesh Rebba. "Inclusion of Model Errors in Reliability-Based Optimization." Journal of Mechanical Design 128, no. 4 (2006): 936–44. http://dx.doi.org/10.1115/1.2204973.

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This paper proposes a methodology to estimate errors in computational models and to include them in reliability-based design optimization (RBDO). Various sources of uncertainties, errors, and approximations in model form selection and numerical solution are considered. The solution approximation error is quantified based on the model itself, using the Richardson extrapolation method. The model form error is quantified based on the comparison of model prediction with physical observations using an interpolated resampling approach. The error in reliability analysis is also quantified and included in the RBDO formulation. The proposed methods are illustrated through numerical examples.

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34

Bychkov, Yuri, Elena Solovyeva, and Sergei Scherbakov. "Analytical-Numerical Calculation Algorithm of Algebraic Equations Roots with Specified Limits of Errors." SPIIRAS Proceedings 18, no. 6 (2019): 1491–514. http://dx.doi.org/10.15622/sp.2019.18.6.1491-1514.

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This paper proposes an algorithm for calculating approximate values of roots of algebraic equations with a specified limit of absolute errors. A mathematical basis of the algorithm is an analytical-numerical method of solving nonlinear integral-differential equations with non-stationary coefficients. The analytical-numerical method belongs to the class of one-step continuous methods of variable order with an adaptive procedure for choosing a calculation step, a formalized estimate of the error of the performed calculations at each step and the error accumulated during the calculation. The proposed algorithm for calculating the approximate values of the roots of an algebraic equation with specified limit absolute errors consists of two stages. The results of the first stage are numerical intervals containing the unknown exact values of the roots of the algebraic equation. At the second stage, the approximate values of these roots with the specified limit absolute errors are calculated. As an example of the use of the proposed algorithm, defining the roots of the fifth-order algebraic equation with three different values of the limiting absolute error is presented. The obtained results allow drawing the following conclusions. The proposed algorithm enables to select numeric intervals that contain unknown exact values of the roots. Knowledge of these intervals facilitates the calculation of the approximate root values under any specified limiting absolute error. The algorithm efficiency, i.e., the guarantee of achieving the goal, does not depend on the choice of initial conditions. The algorithm is not iterative, so the number of calculation steps required for extracting a numerical interval containing an unknown exact value of any root of an algebraic equation is always restricted. The algorithm of determining a certain root of the algebraic equation is computationally completely autonomous.

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35

Harťanský, R., V. Smieško, and L. Maršálka. "Numerical Analysis of Isotropy Electromagnetic Sensor Measurement Error." Measurement Science Review 13, no. 6 (2013): 311–14. http://dx.doi.org/10.2478/msr-2013-0046.

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Abstract The article deals with classification and quantification of electromagnetic field measurement errors in case an isotropic sensor as a field probe is used. The focus is mainly on the error of measurement method, resulting from mutual interaction of the field probe sensors associated with the origin of the so-called mutual impedance.

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Marburg, Steffen. "A Pollution Effect in the Boundary Element Method for Acoustic Problems." Journal of Theoretical and Computational Acoustics 26, no. 02 (2018): 1850018. http://dx.doi.org/10.1142/s2591728518500184.

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The pollution effect is a well-known and well-investigated phenomenon of the finite element method for wave problems in general and for acoustic problems in particular. It is understood as the problem that a local mesh refinement cannot compensate the numerical error which is generated and accumulated in other regions of the model. This is the case for the phase error of the finite element method which leads to dispersion resulting in very large numerical errors for domains with many waves in them and is of particular importance for low order elements. Former investigations have shown that a pollution effect resulting from dispersion is unlikely for the boundary element method. However, numerical damping in the boundary element method can account for a pollution effect. A further investigation of numerical damping reveals that it has similar consequences as the phase error of the finite element method. One of these consequences is that the number of waves within the domain may be controlling the discretization error in addition to the size and the order of the boundary elements. This will be demonstrated in computational examples discussing traveling waves in rectangular ducts. Different lengths, element types and mesh sizes are tested for the boundary element collocation method. In addition to the amplitude error which is due to numerical damping, a rather small phase error is observed. This may indicate numerical dispersion.

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Li, Guangning, Dinesh Bhatia, Min Xu, and Jian Wang. "Grid Convergence Analysis for MUSCL-based Numerical Scheme in Shockwave-containing Flows." MATEC Web of Conferences 257 (2019): 02001. http://dx.doi.org/10.1051/matecconf/201925702001.

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This paper investigated the influence of limiter functions widely utilized in MUSCL-type (Monotone Upstream-centred Schemes for Conservation Laws) upwind numerical schemes on the solution accuracy of shockwave-containing flows. An incident shock interacting with laminar boundary layer developed on a flat plate was numerically simulated with the in-house developed code. A mixed-order grid convergence study was performed to assess the spatial errors of different limiters in simulating the selected shockwave-containing flow on flat plate. The conclusions are that, limiter functions implemented in the current in-house code play the critical roles in accurately predicting shockwave-containing flows. The mixed-order error estimator based on grid convergence study was proved to be applicable to evaluate the spatial errors of shockwave-containing flows, where the shock could reduce the nominal second- or third-order accuracy to first-order. The mixed-order estimator is conservative in the sense that the actual error is less than the error estimated, in the examined case.

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Banks, J. W., J. A. F. Hittinger, J. M. Connors, and C. S. Woodward. "Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport." Computer Methods in Applied Mechanics and Engineering 213-216 (March 2012): 1–15. http://dx.doi.org/10.1016/j.cma.2011.11.021.

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Suciu, Nicolae, Florin A. Radu, and Emil Cătinaş. "Iterative schemes for coupled flow and transport in porous media -- Convergence and truncation errors." Journal of Numerical Analysis and Approximation Theory 53, no. 1 (2024): 264–89. http://dx.doi.org/10.33993/jnaat531-1429.

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Nonlinearities of coupled flow and transport problems for partially saturated porous media are solved with explicit iterative L-schemes. Their behavior is analyzed with the aid of the computational orders of convergence. This approach allows highlighting the influence of the truncation errors in the numerical schemes on the convergence of the iterations. Further, by using manufactured exact solutions, error-based orders of convergence of the iterative schemes are assessed and the convergence of the numerical solutions is demonstrated numerically through grid-convergence tests.

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ÜNAL, Osman, and Nuri AKKAŞ. "An Innovative Approach for Numerical Solution of the Unsteady Convection-Dominated Flow Problems." Karadeniz Fen Bilimleri Dergisi 12, no. 2 (2022): 1069–80. http://dx.doi.org/10.31466/kfbd.1165640.

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In this study, convection-diffusion equation is solved numerically using four different space discretization methods namely first-order upwinding, second-order central difference, cubic (partially upwinded) and cubic-TVD (Total Variation Diminishing) techniques. All methods are compared with the analytical solution. The first-order method is not close to the analytical solution due to the numerical dispersion. The higher-order techniques reduce numerical dispersion. However, they cause another numerical error, unphysical oscillation. This study proposes an innovative approach on cubic-TVD method to eliminate undesired oscillations. Proposed model decreases numerical errors significantly compared to previously developed techniques. Moreover, numerical results of presented model quite close to the analytical solution. Finally, all Matlab codes of numerical and analytical solutions for convection-diffusion equation are added to Appendix in order to facilitate other researchers’ work.

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Aves, Mark A., David F. Griffiths, and Desmond J. Higham. "Does Error Control Suppress Spuriosity?" SIAM Journal on Numerical Analysis 34, no. 2 (1997): 756–78. http://dx.doi.org/10.1137/s0036142994276980.

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Axelsson, O., and L. Kolotilina. "Monotonicity and Discretization Error Estimates." SIAM Journal on Numerical Analysis 27, no. 6 (1990): 1591–611. http://dx.doi.org/10.1137/0727093.

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Liu, Wenbin, and Ningning Yan. "On Quasi-Norm Interpolation Error Estimation And A Posteriori Error Estimates for p-Laplacian." SIAM Journal on Numerical Analysis 40, no. 5 (2002): 1870–95. http://dx.doi.org/10.1137/s0036142901393589.

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Estep, Donald. "A Posteriori Error Bounds and Global Error Control for Approximation of Ordinary Differential Equations." SIAM Journal on Numerical Analysis 32, no. 1 (1995): 1–48. http://dx.doi.org/10.1137/0732001.

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Zhu, Chun Jiu, Feng Qiang Nan, and Yi Liang. "Numerical Simulation of Extrusion Molding of Single - Hole Propellant." Materials Science Forum 917 (March 2018): 269–75. http://dx.doi.org/10.4028/www.scientific.net/msf.917.269.

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In order to improve the simulation accuracy of single hole nitroguanidine propellant extrusion molding process, the influence of wall slip on the extrusion process was studied. The finite element method was used to simulate the extrusion forming of single pore nitroguanidine. The power law constitutive equation considering the effect of wall slip was constructed to calculate the slip factor and non-Newtonian index , the influence of the slip coefficient on the pressure field, velocity field and shear velocity field in the flow channel were analyzed. compared with the actual results, the error of the middle aperture is 1.37%, the arc thickness error is 0.96% and the outer diameter error is 1.12%.Those errors are less then 1.5%. So, the simulation results basically meet the production needs.

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Oliveira, S. P., and G. Seriani. "Effect of Element Distortion on the Numerical Dispersion of Spectral Element Methods." Communications in Computational Physics 9, no. 4 (2011): 937–58. http://dx.doi.org/10.4208/cicp.071109.080710a.

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AbstractSpectral element methods are well established in the field of wave propagation, in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error. The latter is experimentally acknowledged, but has been theoretically shown only in limited cases, such as Cartesian meshes. It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions. In the present work, we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance. We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries.

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Basnet, Ganesh Bahadur, Madhav Prasad Poudel, and Resham Prasad Paudel. "Error Estimates in the Maximum Norm for the Solution of Poisson’s Equation Approximated by the Five-Point Laplacian Using the Discrete Maximum Principle." Journal of Nepal Mathematical Society 6, no. 2 (2024): 28–37. http://dx.doi.org/10.3126/jnms.v6i2.63020.

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In this paper, we study error estimates in the maximum norm in the context of solving Poisson’s equation numerically when approximated the Five-Point Laplacian method using the discrete maximum principle. The primary objective is to assess the accuracy of this numerical approach in solving Poisson’s equation and to provide insights into the behavior of error estimates. We focus on the estimates of maximum norm of the discrete functions defined on a grid in a unit square as well as in a square of side s, and estimate errors measured in the maximum norm.

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Sjöberg, L., and M. Bagherbandi. "A Numerical Study of the Analytical Downward Continuation Error in Geoid Computation by EGM08." Journal of Geodetic Science 1, no. 1 (2011): 2–8. http://dx.doi.org/10.2478/v10156-010-0001-8.

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A Numerical Study of the Analytical Downward Continuation Error in Geoid Computation by EGM08Today the geoid can be conveniently determined by a set of high-degree spherical harmonics, such as EGM08 with a resolution of about 5'. However, such a series will be biased when applied to the continental geoid inside the topographic masses. This error we call the analytical downward continuation (DWC) error, which is closely related with the so-called topographic potential bias. However, while the former error is the result of both analytical continuation of the potential inside the topographic masses and truncation of a series, the latter is only the effect of analytical continuation.This study compares the two errors for EGM08, complete to degree 2160. The result shows that the topographic bias ranges from 0 at sea level to 5.15 m in the Himalayas region, while the DWC error ranges from -0.08 m in the Pacific to 5.30 m in the Himalayas. The zero-degree effects of the two are the same (5.3 cm), while the rms of the first degree errors are both 0.3 cm. For higher degrees the power of the topographic bias is slightly larger than that for the DWC error, and the corresponding global rms values reaches 25.6 and 25.3 cm, respectively, at nmax=2160. The largest difference (20.5 cm) was found in the Himalayas. In most cases the DWC error agrees fairly well with the topographic bias, but there is a significant difference in high mountains. The global rms difference of the two errors clearly indicates that the two series diverge, a problem most likely related with the DWC error.

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Jeong, Darae, Seungsuk Seo, Hyeongseok Hwang, Dongsun Lee, Yongho Choi, and Junseok Kim. "Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/359028.

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We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. We use a finite difference method to numerically solve the equation. To show the efficiency of the given boundary condition, several numerical examples are presented. In numerical test, we investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error. Numerical results show that linear boundary condition is accurate and efficient among the other boundary conditions.

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Appadu, A. R., and H. H. Gidey. "Time-Splitting Procedures for the Numerical Solution of the 2D Advection-Diffusion Equation." Mathematical Problems in Engineering 2013 (2013): 1–20. http://dx.doi.org/10.1155/2013/634657.

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We perform a spectral analysis of the dispersive and dissipative properties of two time-splitting procedures, namely, locally one-dimensional (LOD) Lax-Wendroff and LOD (1, 5) [9] for the numerical solution of the 2D advection-diffusion equation. We solve a 2D numerical experiment described by an advection-diffusion partial differential equation with specified initial and boundary conditions for which the exact solution is known. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion and dissipation errors. Lastly, an optimization technique is implemented to find the optimal value of temporal step size that minimizes the dispersion error for both schemes when the spatial step is chosen as 0.025, and this is validated by numerical experiments.

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