Error Norm Research Articles

On convergence rates of Kaczmarz-type methods with different selection rules of working rows

The Kaczmarz method is a classical while effective iteration method for solving very large-scale consistent systems of linear equations, and the randomized Kaczmarz method is an important and valuable variant of the Kaczmarz method. By theoretically analyzing and numerically experimenting several criteria typically adopted in the non-randomized and the randomized Kaczmarz method for selecting the working row, we derive sharper upper bounds for the convergence rates of some of the correspondingly induced Kaczmarz-type methods including those with respect to the maximal residual, maximal distance, and distance selection rules of the working row, and, for this whole suite of iteration methods consisting of the Kaczmarz methods with respect to the uniform, non-uniform, residual, distance, maximal residual, and maximal distance selection rules of the working row, we reveal their comparable relationships in terms of both mean-squared distance and mean-squared error, and show their computational effectiveness and numerical robustness based upon implementing a large number of test examples. Here the mean-squared distance is defined as the mean-value of the squared Euclidean norm of the current update increment of the iteration, and the mean-squared error is defined as the mean-value of the squared Euclidean norm of the current error that is the difference between the current iterate and the true solution of the target linear system.

Prediction Distribution for Simultaneous Auto-Regressive Model with Multivariate Student-T Error under the Bayesian Approach

Prediction distribution is a basis for predictive inferences applied in many real world situations. The Bayesian approach under uniform prior is employed in this paper to derive the prediction distribution for Simultaneous Auto-regressive model with multivariate Student-t error distribution. Conditional on a set of realized responses, a single and a set of future responses have a univariate and multivariate Student-t distributions respectively, whose degrees of freedom depend on the size of the realized sample and the dimension of the auto-regression parameters' vector but do not depend on the degrees of freedom of the error distribution. Results are identical to those obtained under normal error distribution by a range of statistical approaches such as the normal distribution, autocorrelations and classical methods. This indicates not only the inference robustness with respect to departures from normal error to multivariate Student-t error distributions, but also indicates that the Bayesian approach with uniform prior is competitive with other statistical methods in the derivation of prediction distribution.

A kernel‐based method for solving the time‐fractional diffusion equation

AbstractIn this paper, we focus on the development and study of a numerical method based on the idea of kernel‐based approximation and finite difference discretization to obtain the solution for the time‐fractional diffusion equation. Using the theory of reproducing kernel, reproducing kernel functions with a polynomial form will be established in polynomial reproducing kernel spaces spanned by the Chebychev basis polynomials. In the numerical method, first the time‐fractional derivative term in the aforementioned equation is approximated by using the finite difference scheme. Then, by the help of collocation method based on reproducing kernel approximation, we will illustrate how to derive the numerical solution in polynomial reproducing kernel space. Finally, to support the accuracy and efficiency of the numerical method, we provide several numerical examples. In numerical experiments, the quality of approximation is calculated by absolute error and discrete error norms.

Fast adaptive sampling with operation time control

The rational choice of sampling points is crucial for supervised machine learning algorithms. Adaptive sampling ensures optimal distribution of points in the multidimensional space and provides higher approximation accuracy with respect to the random sampling. Most approaches to the adaptive sampling maximize the function evaluating the compromise between local exploitation and global exploration. However, existing accurate methods suffer from long sampling time, while fast methods are not accurate. Only a few reported methods are available as open source implementations. We introduce a fast adaptive sampling algorithm with accuracy comparable to that of the best-known adaptive sampling methods and demonstrate a comparative study of different approaches to minimization of approximation errors. The method scales linearly with number of sampling points and supports batch generation of new points. The optimization function is the integral norm of an approximation error, the Lp norm, where the norm parameter p regulates sampling, inclining it either towards local exploitation or, conversely, global exploration. The small Lp norm is achieved by reducing the function approximation error and the size of the region with large variation after adding a new sampling point. Our solution is similar to kriging in terms of the choice between local exploitation and global exploration. The difference is in error estimation which depends on the values of the function in sampling points in contrast to homoscedastic variance estimate of kriging. Fast and accurate adaptive sampling is of interest to supervised machine learning tasks such as training a model interatomic potential, quantitative analysis of spectra, self-driving chemical laboratories, and many others. The source code for the new adaptive sampling algorithm MinLpE is available on Github and is distributed under a LGPL license.

Finite-time pinning impulsive synchronization of stochastic complex networks with mixed delays

This paper mainly considers the finite-time synchronization (FTS) of stochastic complex networks with mixed delays via pinning impulsive control. Firstly, a novel pinning impulsive controller is introduced by selecting the nodes of the stochastic complex network with large error norms, which not only saves control resources, but also ensures that stochastic complex network with mixed delays can realize FTS. It is worth noting that the finite-time control strategy constructed is not affected by the sign function. Secondly, a uniform FTS criterion under synchronizing impulses and desynchronizing impulses is derived based on the idea of average impulsive interval. Moreover, the obtained sufficient condition are further extended to general stochastic complex networks and stochastic complex networks with mixed delays under delayed impulses. Finally, the feasibility of the theory results is verified by a numerical example.

Design and research of power system Beidou timing and positioning module based on K‐means clustering and gross error processing

AbstractThe clock synchronisation of the power system is to realise the clock synchronisation of the whole network. The clock synchronisation network is composed of the clock synchronisation systems of power grids at all levels. Dispatching agencies, power plants, and substations have their clock synchronisation systems. By using computer technology, communication technology, and network technology, combined with the topological structure of the power grid and geographic information of the geographic information system, automatically process the key index data to ensure that the distribution network project can manage the power consumption and distribution 24 h without interruption. Considering the particularity of power, the accuracy of the receiver is strictly required, and so are the positioning, speed measurement, and time accuracy. Test indicators study a set of accurate and reliable test and evaluation methods. A gross error processing method based on the k‐means algorithm is proposed. Experiments show that gross errors can be well identified and eliminated in one‐dimensional and multidimensional data. Considering that invalid data may be hidden in the test data, to improve the identification accuracy without affecting the detection of normal gross errors, based on the proposed k‐means algorithm, the number of visible satellites is added. Because the magnitude difference between the number of visible satellites and the original three‐dimensional positioning error data is relatively large, it is normalised first. The processing data is extended from three‐dimensional to four‐dimensional. The experimental simulation shows that it can not only identify invalid data but also ensure a good effect of gross error elimination, reduce possible economic losses, bring significant direct and indirect economic benefits, and verify the feasibility and promotional value of the online monitoring platform through practice.

Petrov–Galerkin approximation of time-fractional coupled Korteweg–de Vries equation for propagation of long wave in shallow water

The time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) describe various interesting real-world phenomena including wave propagation and the description of shallow water waves on a viscous fluid. This paper presents an accurate and robust numerical technique to solve the TFCKdVE. The cubic B-spline is introduced as a basis function and a quadratic B-spline is used as a test function in a finite element method (FEM) is known as Petrov–Galerkin method. The temporal fractional part is simplified via L1 formula, while the B-spline is employed for the space approximation. The Lax–Richtmyer stability criterion is applied to analyze the stability of the proposed scheme. Four test problems are solved to check performance and validation of the scheme. The accuracy and efficiency of the proposed method are checked via various error norms. The obtained results show good agreement with the exact solutions and earlier work available in the literature.

Fibonacci Wavelet Method for the Numerical Solution of Nonlinear Reaction-Diffusion Equations of Fisher-Type

This article aims to propose an efficient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-diffusion equation of Fisher-type. The underlying numerical scheme starts by formulating operational matrices of integration corresponding to the Fibonacci wavelets. Besides, we study the error analysis and convergence theorem of the proposed technique. Subsequently, a set of algebraic equations are formed corresponding to the given problem, which could be handled via any conventional method, for instance, the Newton iteration technique. To demonstrate the efficiency of the proposed wavelet-based numerical method, we compare the obtained absolute,L∞,L2, and root mean square (RMS) error norms with the existing Lie symmetry method and cubic trigonometricB-spline (CTB) differential quadrature method in tabular form. From the numerical outcomes, it is ascertained that the proposed numerical technique is computationally more effective and yields precise outcomes in comparison to the existing ones.

Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation

<abstract> <p>This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using the $L_2$ and $ {L_\infty } $ error norms. The obtained results are compared with a previous existing method to test the accuracy of the proposed method.</p> </abstract>

Numerical solution for Benjamin-Bona-Mahony-Burgers equation with Strang time-splitting technique

In the present manuscript, the Benjamin-Bona-Mahony-Burgers (BBMB) equation will be handled numerically by Strang time-splitting technique. While applying this technique, collocation method based on quintic B-spline basis functions is applied. In line with our purpose, after splitting the BBM-Burgers equation given with appropriate initial boundary conditions into two subequations containing the derivative in terms of time, the quintic B-spline based collocation finite element method (FEM) for spatial discretization and the suitable finite difference approaches for time discretization is applied to each subequation and hereby two different systems of algebraic equations are obtained. Four test problems are utilized to test the efficiency and reliability of the presented method. The error norms $L_2$ and $L_\infty$ with mass, energy, and momentum conservation constants $I_1$,$I_2$ and $I_3$, respectively, are computed. To do a comparison with the other studies in the literature, the newly found approximate solutions are exhibited in both tabular and graphical formats. Also, stability analysis of numerical approach by the von Neumann method is researched.

A novel B-spline collocation method for Hyperbolic Telegraph equation

<abstract><p>The present study is concerned with the construction of a new high-order technique to establish approximate solutions of the Telegraph equation (TE). In this technique, a novel optimal B-spline collocation method based on quintic B-spline (QBS) basis functions is constructed to discretize the spatial domain and fourth-order implicit method is derived for time integration. Test problems are considered to verify the theoretical results and to demonstrate the applicability of the suggested technique. The error norm $ L_{\infty } $ and the rate of spatial and temporal convergence are computed and compared with those of techniques available in the literature. The obtained results show the improvement and efficiency of the proposed scheme over the existing ones. Also, it is obviously observed that the experimental rate of convergence is almost compatible with the theoretical rate of convergence.</p></abstract>

FOUR NUMERICAL SCHEMES FOR SOLUTION OF BURGERS' EQUATION VIA OPERATOR SPLITTING TRIGONOMETRIC CUBIC B-SPLINE COLLOCATION METHOD

In this study, we have used operator splitting methods for numerical solutions of the Burgers' equation by given four different numerical schemes. To set these schemes, we divide the Burgers equation into two sub-problems according to the time term, as linear $ U_{t}=\mathcal{L}(U) $ and nonlinear $ U_{t}=\mathcal{N}(U) $. Then, numerical schemes have been obtained by the finite element method using trigonometric cubic B-spline basis for each sub-problem. Splitting $ \mathcal{L}\circ\mathcal{N} $, $ \mathcal{N}\circ\mathcal{L} $ Lie-Trotter and $ \mathcal{L}\circ\mathcal{N} \circ \mathcal{L} $, $ \mathcal{N\circ L\circ N} $ Strang splitting solution schemes have been used to obtain the solution of the main equation. Numerical results calculated with these schemes have been compared among themselves in terms of $ L_{2} $, $ L_{\infty} $ error norms and CPU time. Furthermore, the numerical results have been compared with some studies that solved the equation directly with the same method. It has been observed that the numerical results obtained with the proposed schemes are in agreement with the exact solution and other studies in the literature. All calculations are obtained using Matlab Version R2015a.

A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models

<abstract><p>The numerical solution of the time-fractional Black-Scholes model for European and American options is presented using a local meshless collocation approach based on hybrid Gaussian-cubic radial basis functions with polynomials is presented. The approach is then expanded to a nonlinear time-fractional model for an option with transaction costs in a market with low liquidity. The spatial derivatives of the models are discretized using the proposed meshless technique. Numerical experiments are carried out for the American option, European option, and nonlinear transaction cost option models. In order to evaluate the effectiveness and precision of the suggested meshless approach, $ L_{\infty} $ and $ L_{rel} $ error norms are utilized. Both call and put option volatility is explored. A non-uniform grid customized around the strike price region is also used to determine the prices of European call and American put options. The methods described in literature are compared with the numerical results.</p></abstract>

A high-order convergence analysis for semi-Lagrangian scheme of the Burgers' equation

<abstract><p>In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete $ \ell_2 $-norm error $ \mathcal{O}(h^2+\triangle x^4+ \triangle x^{p+1}/h) $ by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme.</p></abstract>

A novel numerical manner for non-linear coupled variable order reaction-diffusion equation

In this work, an efficient variable order Bernstein collocation technique, which is based on Bernstein polynomials, is applied to a non-linear coupled system of variable order reaction-diffusion equations with given initial and boundary conditions. The operational matrix of Bernstein polynomials is derived for variable order derivatives w.r.t. time and space. The Bernstein operational matrix and collocation technique are applied to the concerned non-linear physical model to achieve a system of non-linear algebraic equations, which are further solved by using Newton method. A few examples are presented to demonstrate the accuracy and stability of the scheme by comparing L2 and L? norm errors between the obtained numerical solutions and existing solutions. The important feature of this article is the graphical exhibitions of the effects of variable order derivatives on the solutions of the considered non-linear coupled reaction-diffusion equation for different particular cases.

Stock Price Prediction of Tesla & Apple using LSTM

The stock market has always piqued academics' interest in predicting. Since they were originally created, machine learning algorithms have been increasingly popular for predicting changes in the stock market. The intricacy and unstable nature of the stock market make it challenging to predict stock values. In recent years, stock value forecasting has used both sentiment analysis and machine learning techniques. In this work, we present a study to anticipate the stock prices of Tesla and Apple using sentiment analysis and long short-term memory (LSTM). Using historical stock prices and sentiment data from Twitter, our LSTM model will be trained and put to the test. To evaluate the proposed model, metrics like normal error, mean square error (MSE), and mean absolute error (MAE) are taken into account. The obtained results showed very promising results and can be incorporated in stock price prediction applications.

Tensor Robust Principal Component Analysis From Multilevel Quantized Observations

We consider Quantized Tensor Robust Principal Component Analysis (Q-TRPCA), which aims to recover a low-rank tensor and a sparse tensor from noisy, quantized, and sparsely corrupted measurements. A nonconvex constrained maximum likelihood (ML) estimation method is proposed for Q-TRPCA. We provide an upper bound on the Frobenius norm of tensor estimation error under this method. Making use of tools in information theory, we derive a theoretical lower bound on the best achievable estimation error from unquantized measurements. Compared with the lower bound, the upper bound on the estimation error is nearly order-optimal. We further develop an efficient convex ML estimation scheme for Q-TRPCA based on the tensor nuclear norm (TNN) constraint. This method is more robust to sparse noises than the latter nonconvex ML estimation approach. Conducting experiments on both synthetic data and real-world data, we show the effectiveness of the proposed methods.

Adaptive Implicit Inverse Control for a Class of Butterfly-Like Hysteretic Nonlinear Systems and Its Application to Dielectric Elastomer Actuators

In this article, a butterfly-like Prandtl–Ishlinskii (PI) hysteresis model and a novel neural network based adaptive implicit inverse control scheme to describe and control the butterfly-like hysteresis are proposed. The main contributions are: 1) a butterfly-like PI model is developed for the purpose of predicting the hysteresis effects and the model is feasible for controller design; 2) an implicit inverse control scheme especially for mitigating the butterfly-like hysteresis is implemented, which avoids the construction of the direct inverse of the butterfly-like PI model; 3) an adaptive implicit inverse control approach, which integrates the neural network and the implicit inverse technique into the output-feedback control is developed for eliminating the butterfly-like hysteresis and an arbitrarily small <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{\infty }$</tex-math></inline-formula> norm of tracking error is achieved. The proposed modeling and control methods are validated experimentally via the dielectric elastomer actuator based motion control platform.

Распознавание выборочных распределений среди системы эталонов: метод ближайшего соседа

The value of the sample distribution identification error of a multidimensional discrete random variable among a library of reference patterns is studied, depending on the dimension of the random vector, the sample length and the distance between two reference distributions in the norms C and L1. It is shown that the recognition error in the L1 norm is significantly lower than in C. Reference distributions of n-grams for texts are considered as a practical application. It turned out that the accuracy of identification is mainly determined by the individual characteristics of the standards, and not by the distances between them. An algorithm has been developed to test the system of standards for recognition accuracy.

On the Robust Estimation of Spatial Autoregressive Models

The spatial autoregressive (SAR) model is commonly used for cross-sectional data with spatial dependence in the response variable. A robust M-estimator for the SAR model with independent normal random errors is proposed, which is resistant to extreme contamination or outliers in the data. The proposed estimator is defined by a set of robust estimating functions derived by modifying the estimating functions for the standard maximum likelihood (ML) estimator using the Huber function. An iterative algorithm based on iteratively reweighted least squares and a fixed point method is provided to compute the robust M-estimator. The proposed estimator is shown to be Fisher-consistent under the core model, has a bounded influence function, and is asymptotically normally distributed. A closed-form expression for the asymptotic covariance matrix estimator under the core model is derived, and used to study the asymptotic relative efficiency of the proposed estimator. Simulation studies demonstrate that the proposed robust M-estimator greatly outperforms the ML estimator as well as several other non-robust estimators of the SAR model, in terms of both estimation and inference under different contamination scenarios. Additionally, the proposed estimator offers comparable performance to some robust estimators of the SAR model under contamination. Applying the robust M-estimator for the SAR model to the U.S. federal grants and U.S. Nielsen Local Television View datasets reveals strong positive spatial dependence in the grant value among counties, insignificant positive effect of news watching on grant distribution, and evidence of outliers in the data. This results in different conclusions from those obtained using the ML estimator.