Error Estimates Research Articles

Probabilistic Error Estimate for Numerical Discretization of High-Index Saddle Dynamics with Inaccurate Models

Probabilistic Error Estimate for Numerical Discretization of High-Index Saddle Dynamics with Inaccurate Models

Finite element method for fractional order parabolic obstacle problem with nonlinear source term

This paper considers the finite element method for a fractional order parabolic obstacle problem in time with a nonlinear source term. An error estimation of the spatially semi-discrete problem is discussed. A fully discrete problem is proposed by introducing the finite difference approximation for the time variable. The L2-unconditional stability and convergence of the approximate solution have been proven.

Advanced Ensemble Framework for Diabetes Outcome Forecasting

Abstract: Diabetes is one of the chronic diseases, which is increasing from year to year. Developing an automated system that can detect diabetes patients plays an important role in medical science. A stack classifier model is designed for the detection of diabetes by combining three base estimators such as Random Forest Classifier, LightGBM Classifier, and K-Nearest Neighbors Classifier, and Logistic Regression as meta-classifiers. The data preprocessing includes transforming categorical variables into numerical format. Each base learner is trained on the preprocessed data and predictions are made. In the meta learner stage, Logistic Regression is trained to make predictions based on the predictions of the base learners. The goal of the meta learner is to learn how to combine the predictions of the base learners to make a more accurate final prediction on whether the patient is diabetic or not. The use of a stacking classifier improves prediction accuracy compared to using a single classifier. The developed model gives an accuracy of 98%. The 5-fold cross validation is used to get a more robust estimation of generalization error. Thus, the developed model offers a means to enhance early detection and elevate the quality of care for diabetic patients.

Exact Integral Formulas for False Discovery Rate and the Variance of False Discovery Proportion.

Multiple hypothesis testing is an integral component of data analysis for large-scale technologies such as proteomics, transcriptomics, or metabolomics, for which the false discovery rate (FDR) and positive FDR (pFDR) have been accepted as error estimation and control measures. The pFDR is the expectation of false discovery proportion (FDP), which refers to the ratio of the number of null hypotheses to that of all rejected hypotheses. In practice, the expectation of ratio is approximated by the ratio of expectation; however, the conditions for transforming the former into the latter have not been investigated. This work derives exact integral expressions for the expectation (pFDR) and variance of FDP. The widely used approximation (ratio of expectations) is shown to be a particular case (in the limit of a large sample size) of the integral formula for pFDR. A recurrence formula is provided to compute the pFDR for a predefined number of null hypotheses. The variance of FDP was approximated for a practical application in peptide identification using forward and reversed protein sequences. The simulations demonstrate that the integral expression exhibits better accuracy than the approximate formula in the case of a small number of hypotheses. For large sample sizes, the pFDRs obtained by the integral expression and approximation do not differ substantially. Applications to proteomics data sets are included.

An Adaptive Discrete Integral Terminal Sliding Mode Control Method for a Two-Joint Manipulator

In response to the trajectory tracking control problem of manipulators under measurement disturbances, a novel multi-input multi-output discrete integral terminal sliding mode control scheme is proposed. Initially, this scheme establishes a dynamic model of a two-joint manipulator based on the Lagrangian dynamics analysis method. Subsequently, a discrete integral terminal sliding mode control law based on the dynamic model of the two joints is designed, incorporating delayed estimation of unknown disturbances and discretization errors in the manipulator system. To enhance the trajectory tracking accuracy of the control scheme and suppress the impact of sliding mode chattering on the manipulator system, an adaptive switching term is introduced into the discrete integral terminal sliding mode control law. The paper derives an adaptive discrete integral terminal sliding mode control scheme and provides stability proof for the proposed approach. Simulation experiments are conducted to compare the proposed adaptive discrete integral terminal sliding mode control scheme with classical discrete sliding mode control schemes and discrete integral terminal sliding mode control schemes. The simulation results demonstrate that the designed adaptive discrete integral terminal sliding mode control scheme maintains trajectory tracking errors within 0.004 radians for each joint of the manipulator, with minimal changes in control torque for each joint. The absolute integral of the control torque variations is calculated at 5.85×103, which is lower than other control schemes, thereby validating the effectiveness and superiority of the proposed approach.

A neural network inverse problem solution for a MIMO magnetic levitation actuator

A multiple-input and multiple-output planar actuator is proposed, which utilizes mechanically driven stator magnet arrays to levitate a permanent magnet mover. A state of levitation and actuation is obtained by mechanically altering the orientation of the stator magnets to control the forces and torques on the mover. A challenge for the design and control of the actuator is inverting the relationship between the force and stator magnet rotation angles, as there is no closed-form analytical solution. In this study, a feed-forward neural network is applied to model the forward relation between stator magnet angle input and a force and torque output to reduce the forward computation time for the design process and for error estimation in real-time applications. Additionally, the neural network is considered for inverting the solution for a motion profile sampled at 1000 Hz. The developed forward model is able to calculate the forces and torques on the mover a factor 10 faster than the equivalent charge or Fourier model with an absolute error of 3 mN and 0.1 mNm for the forces and torques, respectively, and a feed-forward neural network is able to accurately learn an inverse solution for small motion profiles.

Zero‐sum game‐based H∞ optimal finite‐time prescribed performance control for nonlinear multiagent systems with actuator faults

AbstractThis article proposes an optimal leader‐follower consensus control protocol with finite‐time prescribed performance for nonlinear multiagent systems suffering from the actuator fault based on the zero‐sum game framework. First, to guarantee the specified performance of the system within a finite time frame, a novel error transformation function (ETF) is constructed. The ETF proposed in this paper can adjust the mapping range of error transformation to achieve better error convergence performance, thereby enhancing the flexibility of the controller design. Then, to compensate for the negative impact of actuator fault, an error‐related item is separated from the optimal performance index function to design an online fault estimator, which can obtain a better error estimation accuracy. Furthermore, to lessen the need for difficult‐to‐verify persistent excitation signal in adaptive dynamic programming algorithm, both historical and real‐time data are applied to adjust the updating law of neural network weight. Moreover, it demonstrates that the signals within the closed‐loop system are bounded and the expected finite‐time prescribed performance can be ensured in the presence of faults. Finally, a numerical example of multiple single‐link robots manipulator system is executed to validate the efficacy of the developed scheme.

Convergence and error estimates of a penalization finite volume method for the compressible Navier–Stokes system

Abstract In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we apply the penalty approach to control domain-related discretization errors. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

Space-time CutFEM on overlapping meshes I: simple continuous mesh motion

AbstractWe present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/“on top” of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

Complexity analysis of quasi continuous level Monte Carlo

Continuous level Monte Carlo is an unbiased, continuous version of the celebrated multilevel Monte Carlo method. The approximation level is assumed to be continuous resulting in a stochastic process describing the quantity of interest. Continuous level Monte Carlo methods allow naturally for samplewise adaptive mesh refinements, which are indicated by (goal-oriented) error estimators. The samplewise refinement levels are drawn in the estimator from an exponentially-distributed random variable. Unfortunately in practical examples this results in higher costs due to high variance in the samples. In this paper we propose a variant of continuous level Monte Carlo, where a quasi Monte Carlo sequence is utilized to ''sample'' the exponential random variable. We provide a complexity theorem for this novel estimator and show that this theoretically and practically results in a variance reduction of the whole estimator.

Simultaneous estimation of genotype error and uncalled deletion rates in whole genome sequence data.

Genotype data include errors that may influence conclusions reached by downstream statistical analyses. Previous studies have estimated genotype error rates from discrepancies in human pedigree data, such as Mendelian inconsistent genotypes or apparent phase violations. However, uncalled deletions, which generally have not been accounted for in these studies, can lead to biased error rate estimates. In this study, we propose a genotype error model that considers both genotype errors and uncalled deletions when calculating the likelihood of the observed genotypes in parent-offspring trios. Using simulations, we show that when there are uncalled deletions, our model produces genotype error rate estimates that are less biased than estimates from a model that does not account for these deletions. We applied our model to SNVs in 77 sequenced White British parent-offspring trios in the UK Biobank. We use the Akaike information criterion to show that our model fits the data better than a model that does not account for uncalled deletions. We estimate the genotype error rate at SNVs with minor allele frequency > 0.001 in these data to be [Formula: see text]. We estimate that 77% of the genotype errors at these markers are attributable to uncalled deletions [Formula: see text].

The Recovery and Re-Calibration of a 13-Month Aerosol Extinction Profiles Dataset from Searchlight Observations from New Mexico, after the 1963 Agung Eruption

The recovery and re-calibration of a dataset of vertical aerosol extinction profiles of the 1963/64 stratospheric aerosol layer measured by a searchlight at 32°N in New Mexico, US, is reported. The recovered dataset consists of 105 aerosol extinction profiles at 550 nm that cover the period from December 1963 to December 1964. It is a unique record of the portion of the aerosol cloud from the March 1963 Agung volcanic eruption that was transported into the Northern Hemisphere subtropics. The data-recovery methodology involved re-digitizing the 105 original aerosol extinction profiles from individual Figures within a research report, followed by the re-calibration. It involves inverting the original equation used to compute the aerosol extinction profile to retrieve the corresponding normalized detector response profile. The re-calibration of the original aerosol extinction profiles used Rayleigh extinction profiles calculated from local soundings. Rayleigh and aerosol slant transmission corrections are applied using the MODTRAN code in transmission mode. Also, a best-estimate aerosol phase function was calculated from observations and applied to the entire column. The tropospheric aerosol phase function from an AERONET station in the vicinity of the searchlight location was applied between 2.76 to 11.7 km. The stratospheric phase function, applied for a 12.2 to 35.2 km altitude range, is calculated from particle-size distributions measured by a high-altitude aircraft in the vicinity of the searchlight in early 1964. The original error estimate was updated considering unaccounted errors. Both the re-calibrated aerosol extinction profiles and the re-calibrated stratospheric aerosol optical depth magnitudes showed higher magnitudes than the original aerosol extinction profiles and the original stratospheric aerosol optical depth, respectively. However, the magnitudes of the re-calibrated variables show a reasonable agreement with other contemporary observations. The re-calibrated stratospheric aerosol optical depth demonstrated its consistency with the tropics-to-pole decreasing trend, associated with the major volcanic eruption stratospheric aerosol pattern when compared to the time-coincident stratospheric aerosol optical depth lidar observations at Lexington at 42° N.

Splines finite element solver for one-dimensional time-dependent Maxwell's equations via Fourier Transform Discretization

In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational method based on higher-order spline interpolations is used to approximate the solution relatively to the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are studied and proved, and the study is also provided by an error estimate and the order of convergence of the proposed method. Also, we use the critical Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and for all numerical computations, we consider several quadrature methods. Finally, we give some experiments to illustrate the success of such an approach.

Taekwondo win-loss determining factors using data mining-based decision tree analysis: focusing on game analysis for evidence-based coaching

BackgroundPurpose In this study, the purpose of this study is to identify the determinants of winning and losing in taekwondo by applying decision tree analysis, one of the data mining techniques, based on the 2022 World taekwondo championships women’s competition.Methods272 women’s games in the taekwondo championships in Guadalajara held by the WT in 2022 were used. For data processing, an independent sample t-test was performed for differences in game content variables according to the win/lose group, and a decision tree analysis was performed to confirm game content variables affecting the win/lose group. To check the predictive power of the model, classification accuracy, standard error, and misclassification estimates were calculated. All statistical significance levels were set at 0.05.ResultsFirst, it was found that there was no statistically significant difference only in body attack (attempt) and number of kicking variables according to the winning and losing groups(p > .05), and there were differences in all other game content variables(p < .05). Second, as a result of conducting a decision tree analysis to confirm the determinants of winning and losing in taekwondo sparring, winning situation, tie situation, and number of kicks were identified as important variables.ConclusionThe World taekwondo championships are analyzed in the currently changed taekwondo competition rules to identify important factors, and at the same time, based on this, data-based coaching is expected to improve performance.

Optimal Approximation of Unique Continuation

AbstractWe consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.

An exactly divergence-free hybridized discontinuous Galerkin method for the generalized Boussinesq equations with singular heat source

The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k−1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer’s fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator. 1991 Mathematics Subject Classification 65N12, 65N30, 65N50, 76N05.

Polytropic Dynamical Systems with Time Singularity

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

An Augmented Mixed DG Scheme for the Electric Field

In this paper, a new augmented mixed DG formulation for the numerical approximation of the electrostatic field was introduced and studied. Its error analysis was carried out and an optimal error estimates as a function of the mesh size was obtained. Some numerical tests confirming the theoretical convergence were given.

Алгоритм совместной оценки и компенсации ошибок временной и частотной синхронизации сигналов с возможностью регулирования точности оценки

An analysis of the influence of synchronization errors on signal processing is presented, a review of existing approaches for estimating and compensating for time and frequency synchronization errors is given, a new approach is proposed that allows joint evaluation of time and frequency offsets, providing a more accurate estimate compared to existing analogues and, moreover, regulating accuracy of fractional error estimation of both time and frequency synchronization.

Prevalence of visual impairment and estimation of refractive errors among school children in Kakamega, Kenya.

To investigate the prevalence of visual impairment (VI) and provide an estimation of uncorrected refractive errors in school-aged children, conducted by optometry students as a community service. The study was cross-sectional. Totally 3343 participants were included in the study. The initial examination involved assessing the uncorrected distance visual acuity (UDVA) and visual acuity (VA) while using a +2.00 D lens. The inclusion criteria for a subsequent comprehensive cycloplegic eye examination, performed by an optometrist, were as follows: a UDVA<0.6 decimal (0.20 logMAR) and/or a VA with +2.00 D ≥0.8 decimal (0.96 logMAR). The sample had a mean age of 10.92±2.13y (range 4 to 17y), and 51.3% of the children were female (n=1715). The majority of the children (89.7%) fell within the age range of 8 to 14y. Among the ethnic groups, the highest representation was from the Luhya group (60.6%) followed by Luo (20.4%). Mean logMAR UDVA choosing the best eye for each student was 0.29±0.17 (range 1.70 to 0.22). Out of the total, 246 participants (7.4%) had a full eye examination. The estimated prevalence of myopia (defined as spherical equivalent ≤-0.5 D) was found to be 1.45% of the total sample. While around 0.18% of the total sample had hyperopia value exceeding +1.75 D. Refractive astigmatism (cil<-0.75 D) was found in 0.21% (7/3343) of the children. The VI prevalence was 1.26% of the total sample. Among our cases of VI, 76.2% could be attributed to uncorrected refractive error. Amblyopia was detected in 0.66% (22/3343) of the screened children. There was no statistically significant correlation observed between age or gender and refractive values. The primary cause of VI is determined to be uncorrected refractive errors, with myopia being the most prevalent refractive error observed. These findings underscore the significance of early identification and correction of refractive errors in school-aged children as a means to alleviate the impact of VI.