Error Estimates Research Articles

Optimal Point-Wise Error Estimate of Two Second-Order Accurate Finite Difference Schemes for the Heat Equation with Concentrated Capacity

Optimal Point-Wise Error Estimate of Two Second-Order Accurate Finite Difference Schemes for the Heat Equation with Concentrated Capacity

Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation

In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the H1-norm directly, which represents an advancement over the L∞-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.

Investigation of the approximate solution of one class of curvilinear integral equations by the projection method

UDC 517.9 We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined.

Bernstein Polynomials for Solving Fractional Differential Equations with Two Parameters

This work presents a general framework for solving generalized fractional differential equations based on operational matrices of the generalized Bernstein polynomials. This method effectively obtains approximate analytical solutions of many fractional differential equations. The generalized fractional derivative of the Caputo type with two parameters and its properties are studied. Using orthonormal Bernstein polynomials has led to the development of fractional polynomials, which offer an approximate solution for ordinary fractional differential equations. The approach employs the generalized orthogonal Bernstein polynomials (FOBPs) and constructs their operational matrices for fractional integration and derivative in the generalized Caputo sense to achieve this objective. Operational matrices convert ordinary differential equations into a system of algebraic equations, which can be solved using Newton’s method. The convergence analysis and error estimate associated with the proposed problem have been investigated using the approximation of generalized orthogonal Bernstein polynomials (FOBPs). The effect of the new parameters of the fractional derivative is presented in several examples. Finally, several examples are included to clarify the proposed technique’s validity, efficiency, and applicability via generalized orthogonal Bernstein polynomials (FOBPs) approximation.

Note on the radical inflation in the estimates of error variance in measurement models

This note discusses the radical technical inflation in error variance and the related standard error of test scores from both conceptual and empirical viewpoints. This technical inflation arises as a direct consequence of the technical underestimation of item-score correlation by the product–moment coefficient of correlation (PMC), which is embedded in the traditional estimators of reliability such as coefficients alpha, theta, omega, or rho (maximal reliability). Specifically, in educational settings where compilations usually include both easy and difficult items, the estimate by PMC may substantially deviate from the true association between an item and the score. Consequently, the use of traditional estimators of reliability leads to technically inflated estimates of standard errors, as the error variance related to these traditional measurement models is significantly inflated, resulting in deflated reliability estimates. In educational testing, employing deflation-corrected standard errors, calculated using deflation-corrected reliability estimators, would provide a more accurate measure of the test score’s true precision.

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.

A microwave temperature sounder (MWTS)–microwave humidity sounder fusion cloud detection method for land data from the FengYun‐3D MWTS and its impact on error estimation

AbstractThe presence of clouds greatly disrupting the estimation of observation errors in satellite data. Therefore, cloud detection is a necessary step in satellite radiance data assimilation, not only to identify clear sky data and cloudy data, but also, more importantly, to accurately specify observation errors in different scenarios during all‐sky data assimilation. However, the assimilation of the microwave temperature sounder‐2 (MWTS‐2) onboard the FengYun‐3D satellite has been restricted by the low accuracy of cloud detection due to the absence of low‐frequency channels, especially for land areas where complex underlying surface brings more difficulties to the identification of cloudy data. The microwave humidity sounder‐2 (MWHS‐2), which is designed for water vapor sounding, is usually equipped on the same satellite as MWTS‐2. The sensitivity of the MWHS‐2 to water vapor enables it to provide more accurate information about clouds over land. Based on the advantage of the two instruments measuring on the same platform, an effective cloud detection method for MWTS‐2 data over land is established in this study after collocating the data of the MWTS‐2 and MWHS‐2. The results show that the new method achieves a cloud detection rate exceeding 75%, with a significantly reduced false detection rate compared with the traditional scattering index method, which is decreased from 40% to approximately 26%. More importantly, in non‐precipitating cloud regions, the cloud index built by the new method has a significant linear correlation with the difference between observed and simulated brightness temperatures (OMB) of MWTS‐2 data. Statistical analysis reveals that the cloud index can be used to effectively rectify the systematic OMB bias caused by those clouds in non‐precipitation cloud areas, which demonstrates the promising application of this cloud index in all‐sky MWTS‐2 data assimilation over land.

Pointwise error estimate of the LDG method for 2D singularly perturbed reaction-diffusion problem

Pointwise error estimate of the LDG method for 2D singularly perturbed reaction-diffusion problem

An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel

This work studies an integro-fractional differential equation (I-FrDE) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (I-DE) into a mixed integral equation (MIE) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this MIE, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (TMM) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate.

A novel control strategy for PWM rectifier based on underactuated characteristics

In this paper, a double closed loop controller based on equivalent input disturbances (EID) error perturbation is proposed to solve the problem of abnormal operation of three-phase pulse width modulation (PWM) rectifier caused by load and external conditions. By analysing the buffeting problem and response time problem caused by traditional sliding mode in the current control process, a new current control method of terminal fuzzy sliding mode is designed. The system error is estimated by EID error estimation, and fuzzy control is introduced to improve the accuracy of error estimation, which solves the current buffeting and response problem. In order to solve the problem of response rate and harmonics in the voltage controller design, the extended state observer (ESO) in active disturbance rejection control (ADRC) control is improved, EID error estimation is introduced, the response rate and anti-disturbance ability of the system are further improved, and the harmonics of the system are suppressed. The experimental results show the effectiveness and superiority of the proposed method.

Implementing the procedure of measurement phase difference of the model signals using a virtual laboratory practicum

As this virtual laboratory practicum, which is a tool for mastering theoretical principles and practical skills in digital signal processing, was developed, users (mainly students) were given the possibility to parameterize harmonic signals. It includes the determination of the number of samples, sampling rate, amplitudes, initial phases, and other characteristics necessary for accurate metrological measurements. This publication presents the structure of the virtual workshop and the mathematical model underlying it, with an emphasis on aspects of analyzing the procedures for measuring the phase difference of harmonic signal models using the Discrete Fourier Transform. Particular attention is focused on the phase correction algorithm, which is aimed at improving the accuracy of measurements of the phase difference of signals. The presented methodology describes in detail the measurement process, estimation of errors arising from limited measurement time due to non-synchronization, the influence of the initial phase values of signals, and shows how the synchronization process contributes to a significant increase in the accuracy of measuring informative parameters.

A hybrid finite element method for moving-habitat models in two spatial dimensions

Moving-habitat models track the density of a population whose suitable habitat shifts as a consequence of climate change. Whereas most previous studies in this area consider 1-dimensional space, we derive and study a spatially 2-dimensional moving-habitat model via reaction-diffusion equations. The population inhabits the whole space. The suitable habitat is a bounded region where population growth is positive; the unbounded complement of its closure is unsuitable with negative growth. The interface between the two habitat types moves, depicting the movement of the suitable habitat poleward. Detailed modelling of individual movement behaviour induces a nonstandard discontinuity in the density across the interface. For the corresponding semi-discretised system we prove well-posedness for a constant shifting velocity before constructing an implicit-explicit hybrid finite element method. In this method, a Lagrange multiplier weakly imposes the jump discontinuity across the interface. For a stationary interface, we derive optimal a priori error estimates over a conformal mesh with nonconformal discretisation. We demonstrate with numerical convergence tests that these results hold for the moving interface. Finally, we demonstrate the strength of our hybrid finite element method with two biologically motivated cases, one for a domain with a curved boundary and the other for non-constant shifting velocity.

Exponential Quasi-Synchronization of Fractional-Order Fuzzy Cellular Neural Networks via Impulsive Control

This paper investigates the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks with parameters mismatch via impulsive control. Firstly, under the framework of the generalized Caputo fractional-order derivative, a new fractional-order impulsive differential inequality is established. Secondly, based on this fractional-order impulsive differential inequality, a general criterion for the quasi-synchronization of fractional-order systems is obtained. Then, specific to the fractional-order fuzzy cellular neural network model in this paper, the criteria and error estimation of the exponential quasi-synchronization of fractional-order fuzzy cellular neural networks can be obtained. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

The Uniqueness and Iterative Properties of Positive Solution for a Coupled Singular Tempered Fractional System with Different Characteristics

In this paper, we focus on the uniqueness and iterative properties of positive solution for a coupled p-Laplacian system of singular tempered fractional equations with differential order and characteristics. Firstly, the system is converted to an integral equation, and then, a coupled iterative technique and some suitable growth conditions are proposed; furthermore, some elaborate results about the uniqueness and iterative properties of positive solutions of the system are established, which include the uniqueness, the convergence analysis, the asymptotic behavior, and error estimation, as well as the convergence rate of the positive solution. The interesting points of this paper are that the order of the system of equations is different and the nonlinear terms of the system possess the opposite monotonicity and allow for stronger singularities at space variables.

Pilot spoofing attack detection and channel estimation for secure massive MIMO

AbstractPilot spoofing attack (PSA) is an active eavesdropping attack in massive multiple‐input multiple‐output systems, where the eavesdroppers transmit the same pilot sequence as the legitimate user does to the base station to confuse the normal channel estimation during the uplink channel training phase. With the contaminated channel estimations, more information will be leaked to eavesdroppers in the downlink transmission phase. However, it is a challenging issue to detect the PSA attack due to the similarity of the received signals and the variations of the wireless channels. Here, a new PSA detection scheme by using the difference of two different estimators is presented, that is, the least square estimator and the minimum mean square error estimator, without requiring any priori of eavesdroppers. Following that, a new data‐aided channel estimation scheme is proposed to eliminate the PSA effect. Simulation results demonstrate that the proposed PSA detection scheme outperforms the conventional energy detector, and more accurate legitimate channel estimation and higher sum secrecy rates can be obtained with the proposed scheme.

Filtered Lie-Trotter splitting for the “good” Boussinesq equation: Low regularity error estimates

We investigate a filtered Lie-Trotter splitting scheme for the “good” Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in H s H^{s} for 0 > s ≤ 2 0>s\leq 2 , overcoming the constraint of s > 1 / 2 s>1/2 imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order τ s / 2 \tau ^{s/2} in L 2 L^2 for such levels of regularity. Our analytical findings are supported by numerical experiments.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

GNSS interference detection: Methodology utilising ADS-B NACp indicator and GPS almanac data

Abstract The threat of GNSS interference poses a great danger to many critical infrastructure systems including air navigation. With a focus on mitigating this threat, this paper proposes a methodology for detecting GNSS interference. The methodology utilises the quality indicator NACp transmitted in ADS-B messages and GPS almanac data for interference detection. The NACp indicator enables estimation of the position error derived from GPS, which is compared with the HDOP value of the GPS satellite constellation. Based on this comparison, the developed detection algorithm determines whether the aircraft is affected by jamming. The detection methodology is evaluated on datasets obtained during deliberate experiments with GPS jamming. The proposed methodology provides a way to detect GNSS interference, facilitating mitigation of its impact on air traffic operation.

Study and Analysis of the Thunder Source Location Error Based on Acoustic Ray-Tracing

Error analysis and estimation of thunder source location results is a prerequisite for obtaining accurate location results of thunder sources, which is of great significance for a deeper understanding of the physical process of lightning channel discharges. Most of the thunder source location algorithms are based on the simplified model of the straight-line propagation of acoustic waves to determine the location of the thunder source; however, the acoustic wave is affected by the inhomogeneity of the atmosphere medium in the propagation process and its acoustic ray will be bent. Temperature and humidity are the main factors affecting the vertical distribution of the velocity of sound in the atmosphere, therefore, it is necessary to study the changes in location errors under the models of uniform vertical distribution of temperature only and uniform vertical distribution of humidity only. This paper focuses on the theory of acoustic ray-tracing in neglecting the presence of the wind and the acoustic attenuation and the theoretical derivation of the location error of thunder source inversion for the three models is carried out by using MATLAB R2019b programming. Then, simulation analysis and comparative study on the variation law of thunder source location error with the height of the source, ground temperature, ground humidity, and array position under the three models are carried out. The results of the study show that the maximum location error can be obtained from the straight-line propagation model, the location error obtained from the model of uniform vertical distribution of temperature only is the second, and the location error obtained from the model of uniform vertical distribution of humidity only is the least and can be negligible compared to the first two models. In the trend of error variation, the variation of location error with temperature and humidity is relatively flat in the first two models; however, the variation of location error with the height of the thunder source is more drastic, which can be more than 80%. The location error obtained from the array inversion closer to the thunder source increases linearly with the height of the thunder source, the location error obtained from the more distant array inversion shows a fast-decreasing trend at the height of the thunder source from 500 to 3500 m, and a flat trend above 3500 m. The location error varies relatively smoothly with the height of the thunder source, the ground temperature, and the ground humidity in the model of uniform vertical distribution of humidity only. In addition, the position of the array also has an important effect on lightning location. The further the horizontal distance from the source, the greater the location error will be obtained in the first two models, and when the thunder source is at a low height and detected at a long distance, the location error will be very large, so relevant data should be modified in actual observation.

A posteriori error estimation and adaptivity for temporal multiscale problems

AbstractIn science and engineering, problems over multiple scales in time often arise. Two examples are material damage in oscillating structures or plaque growth in pulsating blood vessels. Here the long term effects are of interest but they depend on the coupled fast‐changing physical processes which must be taken into account. One method to efficiently simulate these problems is to split them into an averaged slow scale process and a localized fast scale process. We apply the dual weighted residual method on a generalized formulation for this type of problem to obtain an error estimator which can be used for adaptive refinement. Special care must be taken to separate the slow scale and fast scale influences on the estimator.