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Статті в журналах з теми "Error approximation"

Автор: Grafiati

Опубліковано: 23 вересня 2022

Оновлено: 27 січня 2023

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1

Heltai, Luca, and Wenyu Lei. "A priori error estimates of regularized elliptic problems." Numerische Mathematik 146, no. 3 (September 29, 2020): 571–96. http://dx.doi.org/10.1007/s00211-020-01152-w.

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Abstract Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $$H^1$$ H 1 and $$L^2$$ L 2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

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2

Kato, Seiji, Fred G. Rose, and Thomas P. Charlock. "Computation of Domain-Averaged Irradiance Using Satellite-Derived Cloud Properties." Journal of Atmospheric and Oceanic Technology 22, no. 2 (February 1, 2005): 146–64. http://dx.doi.org/10.1175/jtech-1694.1.

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Abstract The respective errors caused by the gamma-weighted two-stream approximation and the effective thickness approximation for computing the domain-averaged broadband shortwave irradiance are evaluated using cloud optical thicknesses derived from 1 h of radiance measurements by the Moderate Resolution Imaging Spectrometer (MODIS) over footprints of Clouds and the Earth’s Radiant Energy System (CERES) instruments. Domains are CERES footprints of which dimension varies approximately from 20 to 70 km, depending on the viewing zenith angle of the instruments. The average error in the top-of-atmosphere irradiance at a 30° solar zenith angle caused by the gamma-weighted two-stream approximation is 6.1 W m−2 (0.005 albedo bias) with a one-layer overcast cloud where a positive value indicates an overestimate by the approximation compared with the irradiance computed using the independent column approximation. Approximately one-half of the error is due to deviations of optical thickness distributions from a gamma distribution and the other half of the error is due to other approximations in the model. The error increases to 14.7 W m−2 (0.012 albedo bias) when the computational layer dividing the cloud layer is increased to four. The increase is because of difficulties in treating the correlation of cloud properties in the vertical direction. Because the optical thickness under partly cloudy conditions, which contribute two-thirds of cloudy footprints, is smaller, the error is smaller than under overcast conditions; the average error for partly cloudy condition is −2.4 W m−2 (−0.002 albedo bias) at a 30° solar zenith angle. The corresponding average error caused by the effective thickness approximation is 0.5 W m−2 for overcast conditions and −21.5 W m−2 (−0.018 albedo bias) for partly cloudy conditions. Although the error caused by the effective thickness approximation depends strongly on the optical thickness, its average error under overcast conditions is smaller than the error caused by the gamma-weighted two-stream approximation because the errors at small and large optical thicknesses cancel each other. Based on these error analyses, the daily average error caused by the gamma-weighted two-stream and effective thickness approximations is less than 2 W m−2.

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3

Peköz, Erol A. "Stein's method for geometric approximation." Journal of Applied Probability 33, no. 3 (September 1996): 707–13. http://dx.doi.org/10.2307/3215352.

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The Stein–Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.

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4

Peköz, Erol A. "Stein's method for geometric approximation." Journal of Applied Probability 33, no. 03 (September 1996): 707–13. http://dx.doi.org/10.1017/s0021900200100142.

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Анотація:

The Stein–Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.

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5

Howard, Roy M. "Arbitrarily Accurate Analytical Approximations for the Error Function." Mathematical and Computational Applications 27, no. 1 (February 9, 2022): 14. http://dx.doi.org/10.3390/mca27010014.

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A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation erf(x)≈1 for |x|>xo and with xo optimally chosen. Two generalizations are possible; the first is based on demarcating the integration interval into m equally spaced subintervals. The second, is based on utilizing a larger fixed subinterval, with a known integral, and a smaller subinterval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Furthermore, the initial approximations, and those arising from the first generalization, can be utilized as inputs to a custom dynamic system to establish approximations with better convergence properties. Indicative results include those of a fourth-order approximation, based on four subintervals, which leads to a relative error bound of 1.43 × 10−7 over the interval [0, ∞]. The corresponding sixteenth-order approximation achieves a relative error bound of 2.01 × 10−19. Various approximations that achieve the set relative error bounds of 10−4, 10−6, 10−10, and 10−16, over [0, ∞], are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for exp(−x2) that have significantly higher convergence properties than a Taylor series approximation. Fourth, the definition of a complementary demarcation function eC(x) that satisfies the constraint eC2(x)+erf2(x)=1. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modeled by the error function.

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6

Şimşek, Burçin, and Satish Iyengar. "Approximating the Conway-Maxwell-Poisson normalizing constant." Filomat 30, no. 4 (2016): 953–60. http://dx.doi.org/10.2298/fil1604953s.

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The Conway-Maxwell-Poisson is a two-parameter family of distributions on the nonnegative integers. Its parameters ? and ? model the intensity and the dispersion, respectively. Its normalizing constant is not always easy to compute, so good approximations are needed along with an assessment of their error. Shmueli, et al. [11] derived an approximation assuming that ? is an integer, and gave an estimate of the relative error. Their numerical work showed that their approximation performs well in some parameter ranges but not in others. Our aims are to show that this approximation applies to all real ? > 0; to provide correction terms to this approximation; and to give different approximations for ? very small and very large. We then investigate the error terms numerically to assess our approximations. In parameter ranges for which Shmueli?s approximation does poorly we show that our correction terms or alternative approximations give considerable improvement.

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7

GREPL, MARTIN A. "CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 22, no. 03 (March 2012): 1150015. http://dx.doi.org/10.1142/s0218202511500151.

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We present reduced basis approximations and associated a posteriori error bounds for parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii ) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct "affine" coefficient-function approximations of the "nonaffine" (or nonlinear) parametrized functions. We consider linear time-invariant as well as linear time-varying nonaffine functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. Numerical results are presented to confirm and test our approach.

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8

GERNER, ANNA-LENA, and KAREN VEROY. "REDUCED BASISA POSTERIORIERROR BOUNDS FOR THE STOKES EQUATIONS IN PARAMETRIZED DOMAINS: A PENALTY APPROACH." Mathematical Models and Methods in Applied Sciences 21, no. 10 (October 2011): 2103–34. http://dx.doi.org/10.1142/s0218202511005672.

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We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method, built upon the penalty formulation for saddle point problems, provides error bounds not only for the velocity but also for the pressure approximation, while simultaneously admitting affine geometric variations with relative ease. The essential ingredients are: (i) dimension reduction through Galerkin projection onto a low-dimensional reduced basis space; (ii) stable, good approximation of the pressure through supremizer-enrichment of the velocity reduced basis space; (iii) optimal and numerically stable approximations identified through an efficient greedy sampling method; (iv) certainty, through rigorous a posteriori bounds for the errors in the reduced basis approximation; and (v) efficiency, through an offline-online computational strategy. The method is applied to a flow problem in a two-dimensional channel with a (parametrized) rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter, and that the effects of the penalty parameter are relatively benign.

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9

Yan, Xiaoyu, Jie Chen, Holger Nies, and Otmar Loffeld. "Analytical Approximation Model for Quadratic Phase Error Introduced by Orbit Determination Errors in Real-Time Spaceborne SAR Imaging." Remote Sensing 11, no. 14 (July 12, 2019): 1663. http://dx.doi.org/10.3390/rs11141663.

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Research on real-time spaceborne synthetic aperture radar (SAR) imaging has emerged as satellite computation capability has increased and applications of SAR imaging products have expanded. The orbit determination data of a spaceborne SAR platform are essential for the SAR imaging procedure. In real-time SAR imaging, onboard orbit determination data cannot achieve a level of accuracy that is equivalent to the orbit ephemeris in ground-based SAR processing, which requires a long processing time using common ground-based SAR imaging procedures. It is important to study the influence of errors in onboard real-time orbit determination data on SAR image quality. Instead of the widely used numerical simulation method, an analytical approximation model of the quadratic phase error (QPE) introduced by orbit determination errors is proposed. The proposed model can provide approximation results at two granularities: approximations with a satellite’s true anomaly as the independent variable and approximations for all positions in the satellite’s entire orbit. The proposed analytical approximation model reduces simulation complexity, extent of calculations, and the processing time. In addition, the model reveals the core of the process by which errors are transferred to QPE calculations. A detailed comparison between the proposed method and a numerical simulation method proves the correctness and reliability of the analytical approximation model. With the help of this analytical approximation model, the technical parameter iteration procedure during the early-stage development of an onboard real-time SAR imaging mission will likely be accelerated.

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10

Moon, Seonghyeon, and Kwanghee Ko. "A point projection approach for improving the accuracy of the multilevel B-spline approximation." Journal of Computational Design and Engineering 5, no. 2 (October 31, 2017): 173–79. http://dx.doi.org/10.1016/j.jcde.2017.10.004.

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Abstract In this study, we present a method for improving the accuracy of the multilevel B-spline approximation (MBA) method. We combine a point projection method with the MBA method for reducing the approximation error by directly adjusting the control points in the local area. An initial surface is generated by the MBA method, and grid points are produced on the surface. These grid points are projected onto the scattered point set, and the distances between the grid points and the projected points are computed. The control points are then modified based on the distances. The proposed method shows better approximations even with the same number of control points and ensures C2-continuity. The experimental results with examples verify the validity of the proposed method. Highlights We propose a method for improving the multilevel B-spline approximation method. We use a point projection method for computing the amount of errors. The computed errors are directly applied to the control points for reducing the approximation error.

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11

Veeser, Andreas. "Positivity Preserving Gradient Approximation with Linear Finite Elements." Computational Methods in Applied Mathematics 19, no. 2 (April 1, 2019): 295–310. http://dx.doi.org/10.1515/cmam-2018-0017.

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AbstractPreserving positivity precludes that linear operators onto continuous piecewise affine functions provide near best approximations of gradients. Linear interpolation thus does not capture the approximation properties of positive continuous piecewise affine functions. To remedy, we assign nodal values in a nonlinear fashion such that their global best error is equivalent to a suitable sum of local best errors with positive affine functions. As one of the applications of this equivalence, we consider the linear finite element solution to the elliptic obstacle problem and derive that its error is bounded in terms of these local best errors.

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12

Hande, P., L. Tong, and A. Swami. "Flat fading approximation error." IEEE Communications Letters 4, no. 10 (October 2000): 310–11. http://dx.doi.org/10.1109/4234.880818.

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13

Nenashev, Vadim A., Igor G. Khanykov, and Mikhail V. Kharinov. "A Model of Pixel and Superpixel Clustering for Object Detection." Journal of Imaging 8, no. 10 (October 6, 2022): 274. http://dx.doi.org/10.3390/jimaging8100274.

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The paper presents a model of structured objects in a grayscale or color image, described by means of optimal piecewise constant image approximations, which are characterized by the minimum possible approximation errors for a given number of pixel clusters, where the approximation error means the total squared error. An ambiguous image is described as a non-hierarchical structure but is represented as an ordered superposition of object hierarchies, each containing at least one optimal approximation in g0 = 1,2,..., etc., colors. For the selected hierarchy of pixel clusters, the objects-of-interest are detected as the pixel clusters of optimal approximations, or as their parts, or unions. The paper develops the known idea in cluster analysis of the joint application of Ward’s and K-means methods. At the same time, it is proposed to modernize each of these methods and supplement them with a third method of splitting/merging pixel clusters. This is useful for cluster analysis of big data described by a convex dependence of the optimal approximation error on the cluster number and also for adjustable object detection in digital image processing, using the optimal hierarchical pixel clustering, which is treated as an alternative to the modern informally defined “semantic” segmentation.

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14

ODEN, J. TINSLEY, SERGE PRUDHOMME, TIM WESTERMANN, JON BASS, and MARK E. BOTKIN. "ERROR ESTIMATION OF EIGENFREQUENCIES FOR ELASTICITY AND SHELL PROBLEMS." Mathematical Models and Methods in Applied Sciences 13, no. 03 (March 2003): 323–44. http://dx.doi.org/10.1142/s0218202503002520.

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In this paper, a method for deriving computable estimates of the approximation error in eigenvalues or eigenfrequencies of three-dimensional linear elasticity or shell problems is presented. The analysis for the error estimator follows the general approach of goal-oriented error estimation for which the error is estimated in so-called quantities of interest, here the eigenfrequencies, rather than global norms. A general theory is developed and is then applied to the linear elasticity equations. For the shell analysis, it is assumed that the shell model is not completely known and additional errors are introduced due to modeling approximations. The approach is then based on recovering three-dimensional approximations from the shell eigensolution and employing the error estimator developed for linear elasticity. The performance of the error estimator is demonstrated on several test problems.

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15

Mahadevan, Sankaran, and Ramesh Rebba. "Inclusion of Model Errors in Reliability-Based Optimization." Journal of Mechanical Design 128, no. 4 (January 8, 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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16

Song, Hanjie, Yingjie Gao, Jinhai Zhang, and Zhenxing Yao. "Long-offset moveout for VTI using Padé approximation." GEOPHYSICS 81, no. 5 (September 2016): C219—C227. http://dx.doi.org/10.1190/geo2015-0094.1.

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The approximation of normal moveout is essential for estimating the anisotropy parameters of the transversally isotropic media with vertical symmetry axis (VTI). We have approximated the long-offset moveout using the Padé approximation based on the higher order Taylor series coefficients for VTI media. For a given anellipticity parameter, we have the best accuracy when the numerator is one order higher than the denominator (i.e., [[Formula: see text]]); thus, we suggest using [4/3] and [7/6] orders for practical applications. A [7/6] Padé approximation can handle a much larger offset and stronger anellipticity parameter. We have further compared the relative traveltime errors between the Padé approximation and several approximations. Our method shows great superiority to most existing methods over a wide range of offset (normalized offset up to 2 or offset-to-depth ratio up to 4) and anellipticity parameter (0–0.5). The Padé approximation provides us with an attractive high-accuracy scheme with an error that is negligible within its convergence domain. This is important for reducing the error accumulation especially for deeper substructures.

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17

Bercu, Gabriel. "New Refinements for the Error Function with Applications in Diffusion Theory." Symmetry 12, no. 12 (December 6, 2020): 2017. http://dx.doi.org/10.3390/sym12122017.

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In this paper we provide approximations for the error function using the Padé approximation method and the Fourier series method. These approximations have simple forms and acceptable bounds for the absolute error. Then we use them in diffusion theory.

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18

KNEZEVIC, DAVID J., NGOC-CUONG NGUYEN, and ANTHONY T. PATERA. "REDUCED BASIS APPROXIMATION ANDA POSTERIORIERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS." Mathematical Models and Methods in Applied Sciences 21, no. 07 (July 2011): 1415–42. http://dx.doi.org/10.1142/s0218202511005441.

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In this paper we present reduced basis (RB) approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold — to provide dimension reduction; an efficient proper orthogonal decomposition–Greedy sampling method for identification of optimal and numerically stable approximations — to yield rapid convergence; accurate (online) calculation of the solution-dependent stability factor by the successive constraint method — to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the RB approximation and associated outputs — to provide certainty in our predictions; and an offline–online computational decomposition strategy for our RB approximation and associated error bound — to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional "complex" enclosure — a square with a small rectangle cutout — parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the RB approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.

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19

Portman, V. T., R. D. Weill, and V. G. Shuster. "Higher-Order Approximation in Accuracy Computations of Machine Tools and Robots." Journal of Manufacturing Science and Engineering 119, no. 4B (November 1, 1997): 732–42. http://dx.doi.org/10.1115/1.2836817.

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Second and higher-order approximations for machine accuracy calculations are necessary when the precision of the error estimation obtained on the basis of the first order approximations is poor or when first order approximations are not influencing the output accuracy of the machine. This latter case is widely present in the calculations of the functional accuracy of machines, in particular when estimating the machine tool set-up errors or when calculating the influence of machine setting displacements. In the case of machining operations, the second order approximation for the normal position error of the real surface relative to the nominal one is shown to depend on the second fundamental form of the nominal surface. As a real world application, the setting of a grinding machine for a crowned conic surface grinding operation is calculated.

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20

Pecci, Filippo, Edo Abraham, and Ivan Stoianov. "Quadratic head loss approximations for optimisation problems in water supply networks." Journal of Hydroinformatics 19, no. 4 (April 17, 2017): 493–506. http://dx.doi.org/10.2166/hydro.2017.080.

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This paper presents a novel analysis of the accuracy of quadratic approximations for the Hazen–Williams (HW) head loss formula, which enables the control of constraint violations in optimisation problems for water supply networks. The two smooth polynomial approximations considered here minimise the absolute and relative errors, respectively, from the original non-smooth HW head loss function over a range of flows. Since quadratic approximations are used to formulate head loss constraints for different optimisation problems, we are interested in quantifying and controlling their absolute errors, which affect the degree of constraint violations of feasible candidate solutions. We derive new exact analytical formulae for the absolute errors as a function of the approximation domain, pipe roughness and relative error tolerance. We investigate the efficacy of the proposed quadratic approximations in mathematical optimisation problems for advanced pressure control in an operational water supply network. We propose a strategy on how to choose the approximation domain for each pipe such that the optimisation results are sufficiently close to the exact hydraulically feasible solution space. By using simulations with multiple parameters, the approximation errors are shown to be consistent with our analytical predictions.

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21

Barbour, A. D., and Simon Tavaré. "A Rate for the Erdős-Turán Law." Combinatorics, Probability and Computing 3, no. 2 (June 1994): 167–76. http://dx.doi.org/10.1017/s0963548300001097.

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The Erdős-Turán law gives a normal approximation for the order of a randomly chosen permutation of n objects. In this paper, we provide a sharp error estimate for the approximation, showing that, if the mean of the approximating normal distribution is slightly adjusted, the error is of order log−1/2n.

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22

Ali, Farhad. "Difference Approximations at Internal Physical Boundaries." International Journal of Modern Physics C 08, no. 06 (December 1997): 1267–85. http://dx.doi.org/10.1142/s0129183197001132.

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Two sample difference approximations at internal physical boundaries, in magnetic field calculations, are analyzed in the light of truncation and discretization error expansions. Existence of asymptotic expansions for discretization error and their use in error estimation is studied in the light of approximations of the resulting variational equations. Two test problems are presented to illustrate validity of the error analysis. Better difference approximations are shown to lead to better approximation of the associated variational equations.

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23

Ernst, W. R. "Approximation of the error function." Computers & Chemical Engineering 16, no. 3 (March 1992): 225–26. http://dx.doi.org/10.1016/0098-1354(92)85008-v.

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24

Kemp, Gordon C. R. "The Joint Distribution of Forecast Errors in the AR(1) Model." Econometric Theory 7, no. 4 (December 1991): 497–518. http://dx.doi.org/10.1017/s0266466600004734.

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Анотація:

Second-order asymptotic expansion approximations to the joint distributions of dynamic forecast errors and of static forecast errors in the stationary Gaussian pure AR(1) model are derived. The approximation to the dynamic forecast errors distribution can be expressed as a multivariate normal distribution with modified mean vector and covariance matrix, thus generalizing the results of Phillips [12]. However, the approximation to the static forecast errors distribution includes skewness and kurtosis terms. Thus the class of multivariate normal distributions does not provide as good approximations (in terms of error convergence rates) to the distributions of the static forecast errors as to the distributions of the dynamic forecast errors. These results cast some doubt on the appropriateness of model validation procedures, such as Chow tests, which use the static forecast errors and implicitly assume that these have a distribution which is well approximated by a multivariate normal.

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25

Eck, Bradley J., and Martin Mevissen. "Quadratic approximations for pipe friction." Journal of Hydroinformatics 17, no. 3 (December 29, 2014): 462–72. http://dx.doi.org/10.2166/hydro.2014.170.

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Анотація:

This paper develops quadratic approximations to the classical models of Darcy–Weisbach and Hazen–Williams for frictional head loss in water pipes. Key elements of the technique are selecting the approximation domain and minimizing the relative error. A comparison for individual pipes over a range of diameter, roughness, and Reynolds number showed that the approximation provides accuracy consistent with the experimental error in the classical equations. Two benchmark water distribution networks are also considered. In these systems, flows and pressures computed using the approximation were consistent with simulations using the classical models. Applications of the approximation include mathematical optimization problems where polynomial expressions are desirable.

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26

Langer, Ulrich, Sergey Repin, and Monika Wolfmayr. "Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems." Computational Methods in Applied Mathematics 15, no. 3 (July 1, 2015): 353–72. http://dx.doi.org/10.1515/cmam-2015-0012.

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AbstractThe paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests confirm the efficiency of the a posteriori error bounds derived.

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27

Hu, Qiya, and Rongrong Song. "A variant of the plane wave least squares method for the time-harmonic Maxwell’s equations." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 1 (January 2019): 85–103. http://dx.doi.org/10.1051/m2an/2018043.

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In this paper we are concerned with the plane wave method for the discretization of time-harmonic Maxwell’s equations in three dimensions. As pointed out in Hiptmair et al. (Math. Comput. 82 (2013) 247–268), it is difficult to derive a satisfactory L2 error estimate of the standard plane wave approximation of the time-harmonic Maxwell’s equations. We propose a variant of the plane wave least squares (PWLS) method and show that the new plane wave approximations yield the desired L2 error estimate. Moreover, the numerical results indicate that the new approximations have sightly smaller L2 errors than the standard plane wave approximations. More importantly, the results are derived for more general models in inhomogeneous media.

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28

Marinov, Marin B., Nikolay Nikolov, Slav Dimitrov, Todor Todorov, Yana Stoyanova, and Georgi T. Nikolov. "Linear Interval Approximation for Smart Sensors and IoT Devices." Sensors 22, no. 3 (January 26, 2022): 949. http://dx.doi.org/10.3390/s22030949.

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In this work, we introduce and use an innovative approach for adaptive piecewise linear interval approximation of sensor characteristics, which are differentiable functions. The aim is to obtain a discreet type of inverse sensor characteristic, with a predefined maximum approximation error, with minimization of the number of points defining the characteristic, which in turn is related to the possibilities for using microcontrollers with limited energy and memory resources. In this context, the results from the study indicate that to overcome the problems arising from the resource constraints of smart devices, appropriate “lightweight” algorithms are needed that allow efficient connectivity and intelligent management of the measurement processes. The method has two benefits: first, low-cost microcontrollers could be used for hardware implementation of the industrial sensor devices; second, the optimal subdivision of the measurement range reduces the space in the memory of the microcontroller necessary for storage of the parameters of the linearized characteristic. Although the discussed computational examples are aimed at building adaptive approximations for temperature sensors, the algorithm can easily be extended to many other sensor types and can improve the performance of resource-constrained devices. For prescribed maximum approximation error, the inverse sensor characteristic is found directly in the linearized form. Further advantages of the proposed approach are: (i) the maximum error under linearization of the inverse sensor characteristic at all intervals, except in the general case of the last one, is the same; (ii) the approach allows non-uniform distribution of maximum approximation error, i.e., different maximum approximation errors could be assigned to particular intervals; (iii) the approach allows the application to the general type of differentiable sensor characteristics with piecewise concave/convex properties.

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29

Gibson, M., and T. Hain. "Error Approximation and Minimum Phone Error Acoustic Model Estimation." IEEE Transactions on Audio, Speech, and Language Processing 18, no. 6 (August 2010): 1269–79. http://dx.doi.org/10.1109/tasl.2009.2032607.

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30

DUNCA, ADRIAN, and VOLKER JOHN. "FINITE ELEMENT ERROR ANALYSIS OF SPACE AVERAGED FLOW FIELDS DEFINED BY A DIFFERENTIAL FILTER." Mathematical Models and Methods in Applied Sciences 14, no. 04 (April 2004): 603–18. http://dx.doi.org/10.1142/s0218202504003374.

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This paper analyzes finite element approximations of space averaged flow fields which are given by filtering, i.e. averaging in space, the solution of the steady state Stokes and Navier–Stokes equations with a differential filter. It is shown that [Formula: see text], the error of the filtered velocity [Formula: see text] and the filtered finite element approximation of the velocity [Formula: see text], converges under certain conditions of higher order than [Formula: see text], the error of the velocity and its finite element approximation. It is also proved that this statement stays true if the L2-error of finite element approximations of [Formula: see text] and [Formula: see text] is considered. Numerical tests in two and three space dimensions support the analytical results.

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31

Kraus, Johannes, Svetoslav Nakov, and Sergey I. Repin. "Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson–Boltzmann Equation." Computational Methods in Applied Mathematics 20, no. 2 (April 1, 2020): 293–319. http://dx.doi.org/10.1515/cmam-2018-0252.

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AbstractWe consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson–Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [19] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.

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32

Lipponen, A., V. Kolehmainen, S. Romakkaniemi, and H. Kokkola. "Correction of approximation errors with Random Forests applied to modelling of aerosol first indirect effect." Geoscientific Model Development Discussions 6, no. 2 (April 19, 2013): 2551–83. http://dx.doi.org/10.5194/gmdd-6-2551-2013.

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Abstract. In atmospheric models, due to their computational time or resource limitations, physical processes have to be simulated using reduced models. The use of a reduced model, however, induces errors to the simulation results. These errors are referred to as approximation errors. In this paper, we propose a novel approach to correct these approximation errors. We model the approximation error as an additive noise process in the simulation model and employ the Random Forest (RF) regression algorithm for constructing a computationally low cost predictor for the approximation error. In this way, the overall simulation problem is decomposed into two separate and computationally efficient simulation problems: solution of the reduced model and prediction of the approximation error realization. The approach is tested for handling approximation errors due to a reduced coarse sectional representation of aerosol size distribution in a cloud droplet activation calculation. The results show a significant improvement in the accuracy of the simulation compared to the conventional simulation with a reduced model. The proposed approach is rather general and extension of it to different parameterizations or reduced process models that are coupled to geoscientific models is a straightforward task. Another major benefit of this method is that it can be applied to physical processes that are dependent on a large number of variables making them difficult to be parameterized by traditional methods.

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33

Ganji, S. S., M. G. Sfahani, S. M. Modares Tonekaboni, A. K. Moosavi, and D. D. Ganji. "Higher-Order Solutions of Coupled Systems Using the Parameter Expansion Method." Mathematical Problems in Engineering 2009 (2009): 1–20. http://dx.doi.org/10.1155/2009/327462.

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We consider periodic solution for coupled systems of mass-spring. Three practical cases of these systems are explained and introduced. An analytical technique called Parameter Expansion Method (PEM) was applied to calculate approximations to the achieved nonlinear differential oscillation equations. Comparing with exact solutions, the first approximation to the frequency of oscillation produces tolerable error 3.14% as the maximum. By the second iteration the respective error became 1/5th, as it is 0.064%. So we conclude that the first approximation of PEM is so benefit when a quick answer is required, but the higher order approximation gives a convergent precise solution when an exact solution is required.

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34

Arora, Sandeep, Dharamvir Singh Ahlawat, and Dharambir Singh. "Estimation of Lattice Constants and Band Gaps of Group-III Nitrides using Local and Semi Local Functionals." Oriental Journal of Chemistry 34, no. 4 (July 16, 2018): 2137–43. http://dx.doi.org/10.13005/ojc/3404055.

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We performed the optimization of lattice constants of Group- III nitrides (InN, AlN, GaN) in wurtzite and Zinc blende structures using various semilocal exchange correlation functional in generalized gradient approximations (GGA) namely PBE, WC, PBEsol in addition to local density approximation (LDA) functional. We used these optimized lattice parameters to predict the band gap values using modified Becke Johnson exchange potential with original and improved parameterization as suggested by David Koller for semiconductors having band gap values below 7eV. Among the different functionals considered, PBEsol optimize the lattice parameters with smallest mean error (0.00639 Å) relative to experimental values, while WC approximation with a slightly greater value of mean error (0.00513 Å). It is shown that mBJLDA approximation improves the band gap for the materials studied when compared with LDA and GGA results. It is also shown that LDA optimized parameters with mBJLDA approximation, which leads to mean error of 0.162 eV reproduces the experimental band gap in most efficient way.

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35

Fehér, Áron, Lorinc Márton, and Mihály Pituk. "Approximation of a Linear Autonomous Differential Equation with Small Delay." Symmetry 11, no. 10 (October 15, 2019): 1299. http://dx.doi.org/10.3390/sym11101299.

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A linear autonomous differential equation with small delay is considered in this paper. It is shown that under a smallness condition the delay differential equation is asymptotically equivalent to a linear ordinary differential equation with constant coefficients. The coefficient matrix of the ordinary differential equation is a solution of an associated matrix equation and it can be written as a limit of a sequence of matrices obtained by successive approximations. The eigenvalues of the approximating matrices converge exponentially to the dominant characteristic roots of the delay differential equation and an explicit estimate for the approximation error is given.

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36

Bakhshaii, Atoossa, and Roland Stull. "Saturated Pseudoadiabats—A Noniterative Approximation." Journal of Applied Meteorology and Climatology 52, no. 1 (January 2013): 5–15. http://dx.doi.org/10.1175/jamc-d-12-062.1.

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AbstractTwo noniterative approximations are presented for saturated pseudoadiabats (also known as moist adiabats). One approximation determines which moist adiabat passes through a point of known pressure and temperature, such as through the lifting condensation level on a skew T or tephigram. The other approximation determines the air temperature at any pressure along a known moist adiabat, such as the final temperature of a rising cloudy air parcel. The method used to create these statistical regressions is a relatively new variant of genetic programming called gene-expression programming. The correlation coefficient between the resulting noniterative approximations and the iterated data such as plotted on thermodynamic diagrams is over 99.97%. The mean absolute error is 0.28°C, and the root mean square error is 0.44 within a thermodynamic domain bounded by −30° < θw ≤ 40°C, P > 20 kPa, and −60° ≤ T ≤ 40°C, where θw, P, and T are wet-bulb potential temperature, pressure, and air temperature.

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37

Alhejaili, Weaam, Alvaro H. Salas, and Samir A. El-Tantawy. "Novel Approximations to the (Un)forced Pendulum–Cart System: Ansatz and KBM Methods." Mathematics 10, no. 16 (August 12, 2022): 2908. http://dx.doi.org/10.3390/math10162908.

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In the present investigation, some novel analytical approximations to both unforced and forced pendulum–cart system oscillators are obtained. In our investigation, two accurate and effective approaches, namely, the ansatz method with equilibrium point and the Krylov–Bogoliubov–Mitropolsky (KBM) method, are implemented for analyzing pendulum–cart problems.The obtained results are compared with the Runge–Kutta (RK4) numerical approximation. The obtained approximations using both ansatz and KBM methods show good coincidence with RK4 numerical approximation. In addition, the global maximum error is estimated as compared to RK4 numerical approximation.

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38

ELBER, GERSHON, and ELAINE COHEN. "ERROR BOUNDED VARIABLE DISTANCE OFFSET OPERATOR FOR FREE FORM CURVES AND SURFACES." International Journal of Computational Geometry & Applications 01, no. 01 (March 1991): 67–78. http://dx.doi.org/10.1142/s0218195991000062.

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Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the control polygon/mesh and then attempts to estimate the error of the approximated offset over a region. Most of the current offset algorithms estimate the error using a finite set of samples taken from the region and therefore can not guarantee the offset approximation is within a given tolerance over the whole curve or surface. This paper presents new methods to globally bound the error of the approximated offset of freeform curves and surfaces and then automatically derive new approximations with improved accuracy. These tools can also be used to develop a global error bound for a variable distance offset operation and to detect and trim out loops in the offset.

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39

Gourary, Mark, Sergey Rusakov, Mikhail Zharov, and Sergey Ulyanov. "Computational Algorithms for Reducing Rational Transfer Functions’ Order." SPIIRAS Proceedings 19, no. 2 (April 23, 2020): 330–56. http://dx.doi.org/10.15622/sp.2020.19.2.4.

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A problem of reducing a linear time-invariant dynamic system is considered as a problem of approximating its initial rational transfer function with a similar function of a lower order. The initial transfer function is also assumed to be rational. The approximation error is defined as the standard integral deviation of the transient characteristics of the initial and reduced transfer function in the time domain. The formulations of two main types of approximation problems are considered: a) the traditional problem of minimizing the approximation error at a given order of the reduced model; b) the proposed problem of minimizing the order of the model at a given tolerance on the approximation error. Algorithms for solving approximation problems based on the Gauss-Newton iterative process are developed. At the iteration step, the current deviation of the transient characteristics is linearized with respect to the coefficients of the denominator of the reduced transfer function. Linearized deviations are used to obtain new values of the transfer function coefficients using the least-squares method in a functional space based on Gram-Schmidt orthogonalization. The general form of expressions representing linearized deviations of transient characteristics is obtained. To solve the problem of minimizing the order of the transfer function in the framework of the least squares algorithm, the Gram-Schmidt process is also used. The completion criterion of the process is to achieve a given error tolerance. It is shown that the sequence of process steps corresponding to the alternation of coefficients of polynomials of the numerator and denominator of the transfer function provides the minimum order of transfer function. The paper presents an extension of the developed algorithms to the case of a vector transfer function with a common denominator. An algorithm is presented with the approximation error defined in the form of a geometric sum of scalar errors. The use of the minimax form for error estimation and the possibility of extending the proposed approach to the problem of reducing the irrational initial transfer function are discussed. Experimental code implementing the proposed algorithms is developed, and the results of numerical evaluations of test examples of various types are obtained.

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40

Vukanic, Jovan, Dusan Arsenovic, and Dragomir Davidovic. "A new way of obtaining analytic approximations of Chandrasekhar’s H function." Nuclear Technology and Radiation Protection 22, no. 2 (2007): 38–43. http://dx.doi.org/10.2298/ntrp0702038v.

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Applying the mean value theorem for definite integrals in the non-linear integral equation for Chandrasekhar?s H function describing conservative isotropic scattering, we have derived a new, simple analytic approximation for it, with a maximal relative error below 2.5%. With this new function as a starting-point, after a single iteration in the corresponding integral equation, we have obtained a new, highly accurate analytic approximation for the H function. As its maximal relative error is below 0.07%, it significantly surpasses the accuracy of other analytic approximations.

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41

Jawecki, Tobias, Winfried Auzinger, and Othmar Koch. "Computable upper error bounds for Krylov approximations to matrix exponentials and associated $${\varvec{\varphi }}$$-functions." BIT Numerical Mathematics 60, no. 1 (September 11, 2019): 157–97. http://dx.doi.org/10.1007/s10543-019-00771-6.

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Abstract An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of $$\varphi $$φ-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

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42

Perić, Zoran, Aleksandar Marković, Nataša Kontrec, Jelena Nikolić, Marko D. Petković, and Aleksandra Jovanović. "Two Interval Upper-Bound Q-Function Approximations with Applications." Mathematics 10, no. 19 (October 1, 2022): 3590. http://dx.doi.org/10.3390/math10193590.

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The Gaussian Q-function has considerable applications in numerous areas of science and engineering. However, the fact that a closed-form expression for this function does not exist encourages finding approximations or bounds of the Q-function. In this paper, we determine analytically two novel interval upper bound Q-function approximations and show that they could be used efficiently not only for the symbol error probability (SEP) estimation of transmission over Nakagami-m fading channels, but also for the average symbol error probability (ASEP) evaluation for two modulation formats. Specifically, we determine analytically the composition of the upper bound Q-function approximations specified at disjoint intervals of the input argument values so as to provide the highest accuracy within the intervals, by utilizing the selected one of two upper bound Q-function approximations. We show that a further increase of the accuracy, achieved in the case with two upper-bound approximations composing the interval approximation, can be obtained by forming a composite interval approximation of the Q-function that assumes another extra interval and by specifying the third form for the upper-bound Q-function approximation. The proposed analytical approach can be considered universal and widely applicable. The results presented in the paper indicate that the proposed Q-function approximations outperform in terms of accuracy other well-known approximations carefully chosen for comparison purposes. This approximation can be used in numerous theoretical communication problems based on the Q-function calculation. In this paper, we apply it to estimate the average bit error rate (ABER), when the transmission in a Nakagami-m fading channel is observed for the assumed binary phase-shift keying (BPSK) and differentially encoded quadrature phase-shift keying (DE-QPSK) modulation formats, as well as to design scalar quantization with equiprobable cells for variables from a Gaussian source.

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43

YIN, PENG-YENG. "GENETIC ALGORITHMS FOR POLYGONAL APPROXIMATION OF DIGITAL CURVES." International Journal of Pattern Recognition and Artificial Intelligence 13, no. 07 (November 1999): 1061–82. http://dx.doi.org/10.1142/s0218001499000598.

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In this paper, three polygonal approximation approaches using genetic algorithms are proposed. The first approach approximates the digital curve by minimizing the number of sides of the polygon and the approximation error should be less than a prespecified tolerance value. The second approach minimizes the approximation error by searching for a polygon with a given number of sides. The third approach, which is more practical, determines the approximating polygon automatically without any given condition. Moreover, a learning strategy for each of the proposed genetic algorithm is presented to improve the results. The experimental results show that the proposed approaches have better performances than those of existing methods.

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44

Zhumekenov, Abylay, Rustem Takhanov, Alejandro J. Castro, and Zhenisbek Assylbekov. "Approximation error of Fourier neural networks." Statistical Analysis and Data Mining: The ASA Data Science Journal 14, no. 3 (March 23, 2021): 258–70. http://dx.doi.org/10.1002/sam.11506.

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45

Murray, Rua. "Approximation error for invariant density calculations." Discrete & Continuous Dynamical Systems - A 4, no. 3 (1998): 535–57. http://dx.doi.org/10.3934/dcds.1998.4.535.

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46

Del Corso, G. M. "Randomized error estimation for eigenvalue approximation." Calcolo 37, no. 1 (March 1, 2000): 21–46. http://dx.doi.org/10.1007/s100920050002.

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47

Levis, F. E. "The error in discrete Φ-approximation". Mathematical and Computer Modelling 34, № 5-6 (вересень 2001): 469–77. http://dx.doi.org/10.1016/s0895-7177(01)00076-0.

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48

Ceretani, Andrea N., Natalia N. Salva, and Domingo A. Tarzia. "Approximation of the modified error function." Applied Mathematics and Computation 337 (November 2018): 607–17. http://dx.doi.org/10.1016/j.amc.2018.05.054.

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49

Namboothiri, V. N. Narayanan, and M. S. Shunmugam. "Form error evaluation using L1-approximation." Computer Methods in Applied Mechanics and Engineering 162, no. 1-4 (August 1998): 133–49. http://dx.doi.org/10.1016/s0045-7825(97)00338-1.

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50

Allen, Edward J. "On the Error in Padé Approximation." SIAM Review 29, no. 2 (June 1987): 299. http://dx.doi.org/10.1137/1029049.

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