Error Estimation Technique Research Articles

A uniformly convergent hybrid difference scheme for a system of singularly perturbed initial value problems

ABSTRACTA system of second-order singularly perturbed initial value problems with weak hypotheses on the coefficients is considered. The equations have diffusion parameters of different magnitudes, which gives rise to overlapping boundary layers. The structure of these layers is analysed, and this leads to the construction of a Shishkin-type mesh. On this mesh a hybrid difference scheme is proposed, which is a combination of the second-order difference schemes on the fine mesh and the midpoint upwind scheme on the coarse mesh. By applying the truncation error estimate techniques and a difference analogue of Gronwall's inequality, it is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameters. Numerical experiments are provided to validate the theoretical results.

Iterative Component Mode Synthesis Using a Priori and a Posteriori Criteria

An iterative algorithm of component mode synthesis for a rational reduced-order modeling is proposed. The present algorithm consists of three steps: initial mode selection with a priori criterion, model reduction using the selected modes, and a posteriori error evaluation. The algorithm is based on the Craig–Bampton method, which is one of the most widely used techniques. Specifically, the present algorithm uses, in a judicious manner, both an a priori mode selection criterion employing the moment-matching strategy and an a posteriori error estimation technique in evaluating the relative eigenvalue errors. If desired, an optional step of boundary interface reduction offers an additional possibility of obtaining a more compact model. The proposed iterative algorithm is recommended to researchers as well as commercialized software users in developing linear reduced-order structural dynamics models. Numerical examples are presented for its practical usage and performance.

Modeling mechanical behavior of an additively manufactured metal structure with local texture variations: a study on model form error

The repetitive rapid solidification that occurs in metal additive manufacturing (AM) processes creates microstructures distinctly different from wrought materials. Local variability in AM microstructures (either by design or unintentional) raises questions as to how AM structures should be modeled at the part-scale to minimize modeling error. The key goal of this work is to demonstrate a posteriori error estimation applied to an AM part. It is assumed that the actual microstructure is unknown and an approximate, spatially uniform material model is used. Error bounds are calculated for many reference models based on AM microstructures with elongated grain morphology and localized or global fiber textures during a post-processing step. The current findings promote confidence that a posteriori model form error estimation could be used effectively in mechanical performance simulations of AM parts to quickly obtain quantitative error metrics between an approximate model result and many microstructure-based reference models. The a posteriori error estimation introduces significant time savings compared to computing the full reference model solutions. Tight bounds on model form error are obtained when texture variations in the reference models occur on large length scales. For materials with property variation at small length scales, multi-scale error estimation techniques are needed to properly account for the many interfaces present between areas with different properties.

Analysis of Robustness of Variational Multiscale Error Estimators for the Forward Propagation Study

Error estimation and control play a crucial role in the accuracy and reliability of numerical simulations. Variational multiscale (VMS) methods have emerged as effective techniques for error estimation in forward propagation studies. The robustness and accuracy of VMS error estimators under different conditions and parameters need to be thoroughly investigated. This research paper focuses on the analysis of the robustness of variational multiscale error estimators for forward propagation studies. The study explores the performance and limitations of VMS error estimators under various scenarios, including different mesh resolutions, material properties, and boundary conditions. The research aims to provide insights into the effectiveness and reliability of VMS error estimators, contributing to the improvement of forward propagation simulations. The analysis of robustness of variational multiscale error estimators for the forward propagation study addresses the important challenge of accurately assessing and quantifying the errors in computational models used for forward propagation. Variational multiscale error estimators have gained significant attention as they provide a reliable framework for error estimation in complex physical simulations. However, the robustness of these estimators, particularly in the context of forward propagation, remains an open question. To analyze the robustness of variational multiscale error estimators for the forward propagation study. We begin by formulating the problem and defining the mathematical framework for error estimation. Next, we investigate the performance and reliability of variational multiscale error estimators under different scenarios, including variations in model parameters, mesh resolution, and boundary conditions. We evaluate the accuracy and consistency of the estimators by comparing the estimated errors with reference solutions or known analytical solutions.
 We examine the influence of different factors, such as spatial and temporal discretization schemes, on the robustness of variational multiscale error estimators. We assess the sensitivity of the estimators to modeling assumptions and potential sources of error, such as numerical approximation and solution regularization. The effect of different physical phenomena, such as material nonlinearity or complex boundary conditions, on the performance of the estimators. The findings of this provide valuable insights into the robustness and limitations of variational multiscale error estimators for the forward propagation study. Understanding the performance characteristics and potential weaknesses of these estimators is crucial for accurately assessing the reliability and uncertainty of computational models used in forward propagation. The analysis of robustness contributes to the development of improved error estimation techniques and enhances the confidence in the results obtained from computational simulations. This focuses on the analysis of robustness of variational multiscale error estimators for the forward propagation study. By investigating their performance under different scenarios and evaluating their sensitivity to various factors, we provide a comprehensive understanding of the reliability and limitations of these estimators. The insights gained from this analysis have implications for improving the accuracy and confidence in computational models used for forward propagation in diverse fields, including engineering, physics, and computational sciences.

An Asymptotics-Based Adaptive Finite Element Method for Kohn–Sham Equation

In Radovitzky and Ortiz (Comput Methods Appl Mech Eng 172(1–4):203–240, 1999), an error estimation technique for nonlinear PDEs is presented to adaptively generating mesh, based on the reduction of the order of the approximate polynomial. In this paper, following a similar analysis framework, we propose an a posteriori error estimation for Kohn–Sham equation by coarsening mesh. An upper bound for the difference of the total energies on two successively refined meshes is derived by the numerical solutions on two meshes through an asymptotic analysis, which finally generates an a posteriori error estimation. A variety of numerical tests show that such an a posteriori error estimation works very well in our h-adaptive finite element method framework. In addition, to further improve the efficiency, we solve a Poisson equation instead of the Kohn–Sham equation on the coarsened mesh. The effectiveness of this improvement is analyzed and numerically examined.

Error estimation of the physical and mechanical characteristics measurements by indentation

The active application in the practice of testing the indentation methods, in particular to measure the physical and mechanical properties of metals, polymers, biological technologies demands to development techniques for the measurement error estimation. At the same time existing traditional measurement error evaluation system, based on the using of the reference blocks, is not always suitable for use in testing and research laboratories. The aim of this work was development the technique for estimating the indirect measurements error of materials physical and mechanical characteristics that can be applied in practice and based on the existing standards. Checking of the proposed approach using the experimental values of the hardness and elastic modulus obtained during static indentation for various metals.It is shown that since the initial information about the material is an indentation curve representing the dependence of the load versus penetration depth of the indenter into the material tested, then it is better to confirm the metrological characteristics of the indentation measuring devices using the applied force and achieved displacement, but to estimate the accuracy of determining the properties through the error of indirect measurements. The equations for calculating the hardness and modulus of elasticity are derived. It allows to determine the component value most influencing the error magnitude. The calculation of error on the base of the value of boundary of a random and non-exclusive systematic error was carrying out.The advantage of the developed technique is the fact that the measurement of the physical and mechanical characteristics is based on the experimental data and does not require the creation of the additional metrological assurance. The proposed approach seems appropriate to extend for the determination of the measurement error of other characteristics: the yield point, the strain hardening exponent, creep, relaxation, determined by the indentation methods.

On the overestimation of random forest's out-of-bag error.

The ensemble method random forests has become a popular classification tool in bioinformatics and related fields. The out-of-bag error is an error estimation technique often used to evaluate the accuracy of a random forest and to select appropriate values for tuning parameters, such as the number of candidate predictors that are randomly drawn for a split, referred to as mtry. However, for binary classification problems with metric predictors it has been shown that the out-of-bag error can overestimate the true prediction error depending on the choices of random forests parameters. Based on simulated and real data this paper aims to identify settings for which this overestimation is likely. It is, moreover, questionable whether the out-of-bag error can be used in classification tasks for selecting tuning parameters like mtry, because the overestimation is seen to depend on the parameter mtry. The simulation-based and real-data based studies with metric predictor variables performed in this paper show that the overestimation is largest in balanced settings and in settings with few observations, a large number of predictor variables, small correlations between predictors and weak effects. There was hardly any impact of the overestimation on tuning parameter selection. However, although the prediction performance of random forests was not substantially affected when using the out-of-bag error for tuning parameter selection in the present studies, one cannot be sure that this applies to all future data. For settings with metric predictor variables it is therefore strongly recommended to use stratified subsampling with sampling fractions that are proportional to the class sizes for both tuning parameter selection and error estimation in random forests. This yielded less biased estimates of the true prediction error. In unbalanced settings, in which there is a strong interest in predicting observations from the smaller classes well, sampling the same number of observations from each class is a promising alternative.

Performance assessment of swarm-assisted mean error estimation-based fault detection technique for transmission line protection

In this paper, a fast and computationally efficient fault detection algorithm is proposed for power transmission network protection using mean error estimation technique. In order to avoid false detection, instead of adopting the traditional trial and error-based approach, particle swarm optimization-assisted optimal threshold setting method is introduced in this paper. Performance of the proposed fault detection technique is tested for numerous fault and non-fault events generated on a two-bus test power system through MATLAB/SIMULINK software. The observations on results clearly show that using the proposed method even critical faults can be detected securely within a quarter cycle of the 50 Hz power system. The comparative assessment with two existing methods confirms the effectiveness of the proposed scheme.

A collocation approach for solving two-dimensional second-order linear hyperbolic equations

In this study, a collocation approach is introduced to solve second-order two-dimensional hyperbolic telegraph equation under the initial and boundary conditions. The method is based on the Bessel functions of the first kind, matrix operations and collocation points. The method is constructed in four steps for the considered problem. In first step we construct the fundamental relations for the solution method. By using the collocation points and matrix operations, second step gives the constructing of the main matrix equation. In third step, matrix forms are created for the initial and boundary conditions. We compute the approximate solutions by combining second and third steps. Algorithm of the proposed method is given. Later, error estimation technique is presented and the approximate solutions are improved. Numerical applications are included to demonstrate the validity and applicability of the presented method.

Adaptive reduction of constitutive model-form error using a posteriori error estimation techniques

In engineering practice, models are typically kept as simple as possible for ease of setup and use, computational efficiency, maintenance, and overall reduced complexity to achieve robustness. In solid mechanics, a simple and efficient constitutive model may be favored over one that is more predictive, but is difficult to parameterize, is computationally expensive, or is simply not available within a simulation tool. In order to quantify the modeling error due to the choice of a relatively simple and less predictive constitutive model, we adopt the use of a posteriori model-form error-estimation techniques. Based on local error indicators in the energy norm, an algorithm is developed for reducing the modeling error by spatially adapting the material parameters in the simpler constitutive model. The resulting material parameters are not material properties per se, but depend on the given boundary-value problem. As a first step to the more general nonlinear case, we focus here on linear elasticity in which the “complex” constitutive model is general anisotropic elasticity and the chosen simpler model is isotropic elasticity. The algorithm for adaptive error reduction is demonstrated using two examples: (1) A transversely-isotropic plate with hole subjected to tension, and (2) a transversely-isotropic tube with two side holes subjected to torsion.

Improved statistical fluctuation analysis for measurement-device-independent quantum key distribution with four-intensity decoy-state method.

Recently Zhang et al [ Phys. Rev. A95, 012333 (2017)] developed a new approach to estimate the failure probability for the decoy-state BB84 QKD system when taking finite-size key effect into account, which offers security comparable to Chernoff bound, while results in an improved key rate and transmission distance. Based on Zhang et al's work, now we extend this approach to the case of the measurement-device-independent quantum key distribution (MDI-QKD), and for the first time implement it onto the four-intensity decoy-state MDI-QKD system. Moreover, through utilizing joint constraints and collective error-estimation techniques, we can obviously increase the performance of practical MDI-QKD systems compared with either three- or four-intensity decoy-state MDI-QKD using Chernoff bound analysis, and achieve much higher level security compared with those applying Gaussian approximation analysis.

An efficient numerical method for a two-point boundary value problem with a Caputo fractional derivative

In this paper a two-point boundary value problem with a Caputo fractional derivative is considered. By using a shooting method based on the secant iterative method, the boundary value problem is turned into an initial value problem. Then the initial value problem is transformed into an equivalent integral–differential equation with a weakly singular kernel. An integral discretization scheme on the uniform mesh is developed to approximate the integral–differential equation. By applying the truncation error estimate techniques and a discrete analogue of Gronwall’s inequality, it is proved that the numerical scheme is first-order convergent in the discrete maximum norm. Numerical experiments verify the theoretical results.

A numerical method for the solutions of the HIV infection model of CD4+T-cells

In this paper, the human immunodeficiency virus (HIV) infection model of CD[Formula: see text][Formula: see text]T-cells is considered. In order to numerically solve the model problem, an operational method is proposed. For that purpose, we construct the operational matrix of integration for bases of Taylor polynomials. Then, by using this matrix operation and approximation by polynomials, the HIV infection problem is transformed into a system of algebraic equations, whose roots are used to determine the approximate solutions. An important feature of the method is that it does not require collocation points. In addition, an error estimation technique is presented. We apply the present method to two numerical examples and compare our results with other methods.

A robust numerical method for a fractional differential equation

This paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order α ∈ (0, 1). First the fractional differential equation is transformed into an equivalent Volterra integral equation of the second kind with a weakly singular kernel. Based on the a priori information about the exact solution, an integral discretization scheme on an a priori chosen adapted mesh is proposed. By applying the truncation error estimate techniques and a discrete analogue of Gronwall’s inequality, it is proved that the numerical method is first-order convergent in the discrete maximum norm. Numerical results indicate that this method is more accurate and robust than finite difference methods when α is close to 0.

A residual-based h-adaptive reconstructed discontinuous Galerkin method for the compressible Euler equations on unstructured grids

In this paper, a residual-based h-adaptive high-order reconstructed DG (rDG) method based on the hybrid reconstruction strategy is developed for solving the compressible Euler equations on unstructured grids. The proposed method combines the advantages of high-order rDG discretization with appropriate residual-based error estimation techniques and h-adaptive refinement strategies, which exhibits its superior potential compared to the underlying DG methods. To be specific, a third-order hybrid rDG(p1p2) method has been carefully designed and evaluated on incompatible quadrilateral grids with hanging nodes in order to preserve 2-exactness property during the implementation of mesh refinement and coarsening. A residual-based error estimator is used as local error indicator during the h-adaptive procedures. A number of test cases are presented to assess the performance of the high-order h-adaptive rDG method. The hybrid reconstruction strategy combined with h-adaptive techniques presented in this work demonstrates promise for improving the level of accuracy and reducing the computational cost for numerical simulations of compressible inviscid flows.

Strict upper and lower bounds for quantities of interest in static response sensitivity analysis

• A CRE-based technique is proposed for goal-oriented error estimation in sensitivity analysis. • Strict bounds for quantities associated with sensitivity derivatives are obtained. • The bounding property is exemplified by model problems. In this paper, a goal-oriented error estimation technique for static response sensitivity analysis is proposed based on the constitutive relation error (CRE) estimation for finite element analysis (FEA). Strict upper and lower bounds of various quantities of interest (QoI) that are associated with the response sensitivity derivative fields are acquired. Numerical results are presented to assess the strict bounding properties of the proposed technique.

A Numerical Approach with Residual Error Estimation for Eolution of High-order Linear Differential-difference Equations by Using Gegenbauer Polynomials

The main aim of this study is to apply the Gegenbauer polynomials for the solution of high-order linear differential difference equations with functional arguments under initial-boundary conditions.The technique we have used is essentially based on the truncated Gegenbauer series and its matrix representations along with collocation points. Also, by using the Mean-Value Theorem and residual function, an efficient error estimation technique is proposed and some illustrative examples are presented to demonstrate the validity and applicability of the method.

A Numerical Method for Solving Second-Order Linear Partial Differential Equations Under Dirichlet, Neumann and Robin Boundary Conditions

The aim of this paper is to give a collocation method to solve second-order partial differential equations with variable coefficients under Dirichlet, Neumann and Robin boundary conditions. By using the Bessel functions of the first kind, the matrix operations and the collocation points, the method is constructed and it transforms the partial differential equation problem into a system of algebraic equations. The unknown coefficients of the assuming solution are determined by solving this system. The algorithm of the proposed method is presented. Also, error estimation technique is introduced and the approximate solutions are improved by means of it. To show the validity and applicability of the presented method, we solve numerical examples and give the comparison of solutions and comparisons of the errors (actual and estimation).

Parameter-uniform hybrid difference scheme for solutions and derivatives in singularly perturbed initial value problems

A second-order initial value problem with a small parameter multiplying the first- and second-order derivatives is considered. The precise knowledge about the behavior of the exact solution is analyzed. Based on the behavior of the exact solution, a hybrid finite difference scheme on a Shishkin mesh is proposed, which is a combination of the second-order difference scheme on the fine mesh and the modified midpoint upwind scheme on the coarse mesh. By applying the truncation error estimate techniques and a difference analogue of Gronwall’s inequality we prove that the scheme is almost second-order convergent for numerical solutions and scaled numerical derivatives. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

IntegratedMRF: random forest-based framework for integrating prediction from different data types.

IntegratedMRF is an open-source R implementation for integrating drug response predictions from various genomic characterizations using univariate or multivariate random forests that includes various options for error estimation techniques. The integrated framework was developed following superior performance of random forest based methods in NCI-DREAM drug sensitivity prediction challenge. The computational framework can be applied to estimate mean and confidence interval of drug response prediction errors based on ensemble approaches with various combinations of genetic and epigenetic characterizations as inputs. The multivariate random forest implementation included in the package incorporates the correlations between output responses in the modeling and has been shown to perform better than existing approaches when the drug responses are correlated. Detailed analysis of the provided features is included in the Supplementary Material . The framework has been implemented as a package IntegratedMRF , which can be downloaded from https://cran.r-project.org/web/packages/IntegratedMRF/index.html , where further explanation of the package is available. ranadip.pal@ttu.edu. Supplementary data are available at Bioinformatics online.