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In this paper, we introduce an iteration process to approximate a fixed point of a contractive self-mapping. The comparison theorem indicates that our iteration process is faster than the other existing iteration processes in the literature. We also obtain convergence and stability theorems of this iterative process for a contractive self-mapping. Numerical examples show that our iteration process for approximating a fixed point of a contractive self-mapping is faster than the existing methods. Based on this process, we finally present a new modified Newton-Raphson method for finding the roots of a function and generate some nice polynomiographs.

Machine Learning (ML) solutions are nowadays distributed, according to the so-called server/worker architecture. One server holds the model parameters while several workers train the model. Clearly, such architecture is prone to various types of component failures, which can be all encompassed within the spectrum of a Byzantine behavior. Several approaches have been proposed recently to tolerate Byzantine workers. Yet all require trusting a central parameter server. We initiate in this paper the study of the ``general'' Byzantine-resilient distributed machine learning problem where no individual component is trusted. We show that this problem can be solved in an asynchronous system, despite the presence of $\frac{1}{3}$ Byzantine parameter servers and $\frac{1}{3}$ Byzantine workers (which is optimal). We present a new algorithm, ByzSGD, which solves the general Byzantine-resilient distributed machine learning problem by relying on three major schemes. The first, Scatter/Gather, is a communication scheme whose goal is to bound the maximum drift among models on correct servers. The second, Distributed Median Contraction (DMC), leverages the geometric properties of the median in high dimensional spaces to bring parameters within the correct servers back close to each other, ensuring learning convergence. The third, Minimum-Diameter Averaging (MDA), is a statistically-robust gradient aggregation rule whose goal is to tolerate Byzantine workers. MDA requires loose bound on the variance of non-Byzantine gradient estimates, compared to existing alternatives (e.g., Krum). Interestingly, ByzSGD ensures Byzantine resilience without adding communication rounds (on a normal path), compared to vanilla non-Byzantine alternatives. ByzSGD requires, however, a larger number of messages which, we show, can be reduced if we assume synchrony., This is a merge of arXiv:1905.03853 and arXiv:1911.07537; arXiv:1911.07537 will be retracted

With China's economy entering the stage of high-quality development, manufacturing energy carbon emission efficiency has become the focus of academic attention. It is of great significance to study the convergence of manufacturing energy carbon emission efficiency for realizing high-quality development of manufacturing in China. Based on the panel data of China's manufacturing sub-sectors, this paper measures and analyzes the evolution trend of manufacturing energy carbon emission and its efficiency. On this basis, this paper uses the coefficient of variation and convergence model to test the convergence of manufacturing energy carbon emission efficiency. The results show that China's manufacturing energy carbon emissions and its efficiency demonstrate an increasing trend. Coal was the main source of manufacturing energy carbon emissions. The manufacturing energy carbon emission efficiency does not have σ convergence, but has [Formula: see text] convergence, and its convergence has industry heterogeneity. The manufacturing energy carbon emission efficiency exits scale effect and technology effect, but not the effect of opening to the outside world and institutional effect, and its effect exists industry heterogeneity. By reducing carbon emissions, adopting differentiated policies, adjusting the industry scale, and enhancing the industry technology intensity, China's manufacturing can improve the energy carbon emission efficiency and promote high-quality economic development.

Economic growth in the information age is no longer a stage driven by unipolarity. It has entered a multi-polar driving stage characterized by integration, fusion, and integrated development on a larger scale between regions, and the trend of group competition with urban agglomerations as carriers has become increasingly obvious. This paper improves the neural network algorithm based on the needs of industrial economic integration in the digital age, and proposes an industry convergence analysis model based on the improved neural network algorithm. Moreover, this article combines industry models to analyze actual needs and constructs an industry convergence analysis model based on improved neural networks, and analyzes the integration of different industries. In addition, this article conducts experiments through multiple sets of data, and combines the neural network model of this article to conduct research. Through experimental research, we know that the model constructed in this paper can play an important role in the analysis of industry convergence.

This paper focuses on time of arrival (TOA) measurement for wireless personal area network (WPAN) under the circumstances that dense multipath and non-line of sight (NLOS), which exploits a non-coherent receiver for millimeter wave (mm-Wave) pulses. For the localization enhancement, an improved statistics fingerprint analysis (SFA) method is presented for energy-assisted TOA estimate and the ranging error mitigation using artificial neural network (ANN) by fourth order cumulants (FOCs) technique. The corresponding mean absolute error (MAE) is utilized as a measurement to access the performance of the developed method. Simulation results indicate SFA can outperform TOA estimation scheme significantly proposed in WPAN previously, show that SFA can effectively enhance the accuracy of TOA estimates evidently. The advantages of SFA such as low complexity implementation, fast convergence, high accuracy, etc. make it an attractive positioning method in WPAN compared with a coherent receiver.

The traditional least squares (LS) and gradient descent (GD) algorithms can estimate the parameters of the regression models. They can be inefficient when the models have complex structures: (1) the unknown parameters in the information vector make the algorithm be impossible to update the parameters; (2) the zigzagging nature of the gradient descent algorithm and the complex structures lead to slow convergence rates; and (3) the step-size and derivative function calculations may be unsolvable for complex nonlinear models. This paper proposes two kinds of algorithms for exponential nonlinear models. The first is the two-stage algorithm, which decomposes the complex model into a linear part and a nonlinear part, where the linear part is estimated using the LS algorithm and the nonlinear part is identified based on the GD algorithm. The second is the particle swarm optimization algorithm which can simultaneously obtain all the parameters. To increase the convergence rates, the Aitken method is also introduced. The simulation results demonstrate the effectiveness of the proposed algorithms.

This article deals with an efficient numerical technique to solve a class of multi-term time fractional Volterra–Fredholm partial integro-differential equations of first kind. The fractional derivatives are defined in Caputo sense. The Adomian decomposition method is used to construct the scheme. For simplicity of the analysis, the model problem is converted into a multi-term time fractional Volterra–Fredholm partial integro-differential equation of second kind. In addition, the convergence analysis and the condition for existence and uniqueness of the solution are provided. Several numerical examples are illustrated in support of the theoretical analysis.

In this work, we construct two numerical approximations methods to deal with approximations of a linear Fredholm integro-differential equation. In order to show the existence and uniqueness of the solution we start by the reformulation of the integro-differential equation to a system of integral equations. We explain the general framework of the projection method which helps to clarify the basic ideas of the Collocation and Kantorovich methods. We introduce a new iterative method to avoid inversing the block operator matrix. Next, we apply the iterative projection methods and we present theorems to show the convergence of the constructed solutions to the exact solution. Finally, to observe the error behavior of the two methods, we give two numerical examples.

This paper aims to establish higher order convergence of the (inexact) Newton’s method for solving generalized equations composed of the sum of a single-valued mapping and a set-valued mapping between arbitrary Banach spaces without Lipschitz conditions. Imposing Holder calmness property on the gradient of the single-valued mapping instead of Lipschitz continuity, by virtue of the contraction mapping principle, we establish exact relationship between the order of calmness for the gradient and the order of local convergence for the (inexact) Newton’s method. Furthermore, we extend the obtained results to a restricted version of the Newton’s method, which ensures that every sequence generated by this method converges to a solution of the generalized equation. Numerical examples are provided to illustrate the theoretical results.

Surrogate-assisted many-objective optimization is to locate Pareto optimal solutions using a limited number of function evaluations. Most existing surrogate-assisted evolutionary algorithms are designed to embed in a specific many-objective evolutionary algorithm. The Pareto-based bi-indicator infill sampling criterion has been proven to be effective in saving expensive evaluations in surrogate-assisted multi-objective evolutionary algorithms. It introduces two indicators measuring the convergence and diversity as two optimization objectives. In this paper, we extend our previous work to propose a generic framework with a Pareto-based bi-indicator infill sampling criterion for expensive many-objective optimization. The proposed framework gives a general method for traditional MOEAs to solve the expensive optimization task. We incorporate the proposed framework into the reference vectors guided evolutionary algorithm and compare it with another surrogate-assisted reference vectors guided evolutionary algorithm. Empirical studies on DTLZ problems with more than three objectives demonstrate the effectiveness and superiority of the proposed framework and the incorporated algorithm.

The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense. First, an inexact projection on the feasible set is computed, allowing for an appropriate relative error tolerance. Second, an inexact non-monotone line search scheme is employed to compute a step size which defines the next iteration. It is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual scaled gradient projection method employing monotone line searches., Comment: 30 pages, 14 figures

The aim of this work is to formulate and analyze nonstandard finite difference (NSFD) schemes for a recognized virus-patch dynamic model. We introduce a simple and efficient approach to study the global asymptotic stability (GAS) of the constructed NSFD schemes. This approach is based on the Lyapunov stability theorem for discrete-time dynamical systems in combination with a well-known result on the GAS of discrete-time nonlinear cascade systems. As an important consequence, we obtain dynamically consistent NSFD schemes that provide reliable numerical approximations for the virus-patch dynamic model regardless of chosen step sizes. In addition, the convergence and error bounds for the constructed NSFD schemes are also investigated. Finally, a set of numerical examples is performed to support and illustrate the theoretical results. The numerical results show that some typical standard Runge–Kutta schemes fail to preserve the positivity and GAS of the continuous model for some given step sizes, meanwhile, the NSFD schemes preserve these properties for the same step sizes.

In order to improve the security of image information in the information age, we propose an image encryption scheme in this paper, which is achieved by using finite-time cluster synchronization of two-layer complex dynamic networks. Considering the cluster characteristics of nodes in complex networks and the time convergence of synchronization, the method of finite-time cluster synchronization, which plays an important role in information encryption/decryption, is investigated and the influence of stochastic disturbance is taken into consideration as well. A two-layer encryption system is proposed, in which the drive layer is used for encryption, and the response layer is used for decryption. The encryption process can be mainly divided into three steps: scrambling preprocessing, encryption function operation, and diffusion. The encryption scheme uses a symmetric encryption method, and both the encryption keys and the decryption keys are disposable and dynamic, which can increase the complexity and security of the keys. A detailed theoretical analysis for the capability of encryption scheme, which has the advantages of great cipher set, fast encryption and decryption speed, good encryption effect, is provided and the time for information encryption/decryption is controllable and predictable. The availability and practicability of the encryption scheme are verified by simulation results.

It is shown that the solution to the boundary - initial value problem for a Kelvin–Voigt fluid of order one depends continuously upon the Kelvin–Voigt parameters, the viscosity, and the viscoelastic coefficients. Convergence of a solution is also shown.

Melanoma is one of the main causes of cancer-related deaths. The development of new computational methods as an important tool for assisting doctors can lead to early diagnosis and effectively reduce mortality. In this work, we propose a convolutional neural network architecture for melanoma diagnosis inspired by ensemble learning and genetic algorithms. The architecture is designed by a genetic algorithm that finds optimal members of the ensemble. Additionally, the abstract features of all models are merged and, as a result, additional prediction capabilities are obtained. The diagnosis is achieved by combining all individual predictions. In this manner, the training process is implicitly regularized, showing better convergence, mitigating the overfitting of the model, and improving the generalization performance. The aim is to find the models that best contribute to the ensemble. The proposed approach also leverages data augmentation, transfer learning, and a segmentation algorithm. The segmentation can be performed without training and with a central processing unit, thus avoiding a significant amount of computational power, while maintaining its competitive performance. To evaluate the proposal, an extensive experimental study was conducted on sixteen skin image datasets, where state-of-the-art models were significantly outperformed. This study corroborated that genetic algorithms can be employed to effectively find suitable architectures for the diagnosis of melanoma, achieving in overall 11% and 13% better prediction performances compared to the closest model in dermoscopic and non-dermoscopic images, respectively. Finally, the proposal was implemented in a web application in order to assist dermatologists and it can be consulted at http://skinensemble.com.

We consider a system in which the charged particle orbits under the influence of the electromagnetic field of three dipoles located on a system of three celestial bodies. Using well-known bivariate iterative scheme, known as Newton–Raphson (NR) iterative scheme, we numerically evaluated the positions of the stationary points (SPs) or equilibrium points (EPs) or libration points (LPs) and the linked basins of convergence (BoCs), and we also evaluated their linear stability. Moreover, we unveiled how the parameters, entering the effective potential function, affect the convergence dynamics of the system. Moreover, we also unveiled how the involved parameters affect the geometry of the zero velocity curves (ZVCs). Further, the correlation with the required number of iterations and the regions of convergence as well as the probability distributions associated to the BoCs is illustrated. In order to quantify the degree of final-state uncertainty of the BoCs, the basin entropy (BE) and for the fractality of boundaries of BoCs, the boundary basin entropy (BBE) are computed.

In this paper, we summarize results on convergence rates of various kernel based non- and semiparametric estimators, focusing on the impact of insufficient distributional smoothness, possibly unknown smoothness and even non-existence of density. In the presence of a possible lack of smoothness and the uncertainty about smoothness, methods of safeguarding against this uncertainty are surveyed with emphasis on nonconvex model averaging. This approach can be implemented via a combined estimator that selects weights based on minimizing the asymptotic mean squared error. In order to evaluate the finite sample performance of these and similar estimators we argue that it is important to account for possible lack of smoothness.

The moving asymptotes function is widely applied as sequential explicit convex approximation in topology optimization, due to the controllable conservatism and convergence. In virtue of these advantages, it is supposed that the efficiency of optimality criteria method will become higher if constraint functions are approximated by moving asymptotes function instead of linear function. This work presents an optimality criteria method for topology optimization under multiple constraints where the constraint functions are approximated by moving asymptotes function. The dual feasibility condition is customarily adopted to establish explicit update scheme of topological variable, where hypothesis of active topological variable set is avoided, in the case that gradient of objective function is positive. The complementary slackness condition and primal feasibility condition are combined into a simplified dual programming to solve for the Lagrange multipliers, where hypothesis of active constraint set is avoided. Three benchmark examples under multiple displacement, stress, compliance or eigenfrequency constraints are solved by the presented optimality criteria method, the results are compared to the method of moving asymptotes and the optimality criteria method with linear constraint approximation.

In this paper, we discuss the modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems under a weakened condition, and present the general convergence conditions for the method in terms of spectral radius and matrix norm, respectively. Moreover, for some special cases of the method, we propose the concrete convergence conditions and optimal parameters. These convergence theories improve the existing results to some extent. The numerical experiments verify the validity and practicality of the presented results.

In this paper, an adaptive modified weak Galerkin (AMWG) method is considered to solve second-order elliptic problem. Under the assumption of a penalty parameter, by showing reliability of error estimator, comparison of solutions and reduction of error estimator, the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops, is proved to be a contraction, namely, the adaptive algorithm is convergent. Numerical experiments are implemented to support the theoretical results.

In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known "fast iterative shrinkage-thresholding algorithm" (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.

This paper provides a review of the consensus problem as one of the most challenging issues in the distributed control of the multi-agent systems (MASs). In this survey, firstly, the consensus algorithms for the agents with the single-integrator, double-integrator and high-order dynamic models were collected from various research works, and the convergence condition for each of these algorithms was explained. Secondly, all the consensus-related problems such as those in the sampled-data consensus, quantized consensus, random-network consensus, leader–follower consensus, finite-time consensus, bipartite consensus, group consensus/cluster consensus, and the scaled consensus were analyzed and compared with each other. Thirdly, we focused on the common control techniques used for the consensus problems in the presence of disturbance and divided all these control methods into two categories: robust control and adaptive control. Finally, we reviewed the most prevalent consensus applications in the MASs, including the subjects of rendezvous, formation control, axial alignment and the wireless sensor networks.

The differential evolution (DE) algorithm is an efficient random search algorithm based on swarm intelligence for solving optimization problems. It has the advantages of easy implementation, fast convergence, strong optimization ability and good robustness. However, the performance of DE is very sensitive to the design of different operators and the setting of control parameters. To solve these key problems, this paper proposes an improved self-adaptive differential evolution algorithm with a shuffled frog-leaping strategy (SFSADE). It innovatively incorporates the idea of the shuffled frog-leaping algorithm into DE, and at the same time, it cleverly introduces a new strategy of classification mutation, and also designs a new adaptive adjustment mechanism for control parameters. In addition, we have carried out a large number of simulation experiments on the 25 benchmark functions of CEC 2005 and two nonparametric statistical tests to comprehensively evaluate the performance of SFSADE. Finally, the results of simulation experiments and nonparametric statistical tests show that SFSADE is very effective in improving DE, and significantly improves the overall diversity of the population in the process of dynamic evolution. Compared with other advanced DE variants, its global search speed and optimization performance also has strong competitiveness.

We consider the finite element (FE) approximation of the two dimensional shallow water equations (SWE) by considering discretizations in which both space and time are established using a stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to establish stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We establish a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptive mesh refinements on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to some commonly applied benchmarks for the SWE.

Redundancy manipulators need favorable redundancy resolution to obtain suitable control actions to guarantee accurate kinematic control. Among numerous kinematic control applications, some specific tasks such as minimally invasive manipulation/surgery require the distal link of a manipulator to translate along such fixed point. Such a point is known as remote center of motion (RCM) to constrain motion planning and kinematic control of manipulators. Recurrent neural network (RNN) which possesses parallel processing ability, is a powerful alternative and has achieved success in conventional redundancy resolution and kinematic control with physical constraints of joint limits. However, up to now, there still is few related works on the RNNs for redundancy resolution and kinematic control of manipulators with RCM constraints considered yet. In this paper, for the first time, an RNN-based approach with a simplified neural network architecture is proposed to solve the redundancy resolution issue with RCM constraints, with a new and general dynamic optimization formulation containing the RCM constraints investigated. Theoretical results analyze and convergence properties of the proposed simplified RNN for redundancy resolution of manipulators with RCM constraints. Simulation results further demonstrate the efficiency of the proposed method in end-effector path tracking control under RCM constraints based on a redundant manipulator.

The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart–type finite element is used to approximate the velocity and the conforming piecewise linear element P1 is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments.

The objective of this paper is to find the numerical solution of two-dimensional multi-term time fractional mobile-immobile equation. The proposed technique is based on the arbitrary-order orthogonal spline collocation method for the spatial discretization, the L1 approximation for the Caputo fractional derivative, and a second-order backward differentiation formula in time. The stability and convergence are proved in detail. Then, the convergence analysis is validated by a number of numerical experiments.

In this paper we suggest a theoretical framework to study the generalized split null point problem (GSNPP) via a multi-step inertial based parallel hybrid projection algorithm in Hilbert spaces. The architecture and the suitable set of control conditions ensure the strong convergence of the algorithm. The possible applications of the algorithm are illustrated via various theoretical and numerical results. The key advantages of the algorithm are highlighted in comparison to the classical and non-inertial algorithm as well as (one-step and two-step) inertial variants of the algorithm for the GSNPP.

We propose a proximal point method for quasiconvex pseudomonotone equilibrium problems. The subproblems of the method are optimization problems whose objective is the sum of a strongly quasiconvex function plus the standard quadratic regularization term for optimization problems. We prove, under suitable additional assumptions, convergence of the generated sequence to a solution of the equilibrium problem, whenever the bifunction is strongly quasiconvex in its second argument, thus extending the validity of the convergence analysis of proximal point methods for equilibrium problems beyond the standard assumption of convexity of the bifunction in the second argument.

This paper puts forward an improved multi-objective bacterial colony chemotaxis (MOBCC) algorithm based on Pareto dominance. A time-varying step size tactic is adopted to increase the global and local searching abilities of the improved MOBCC algorithm. An external archive is created to keep previously found Pareto optimal solutions. A non-dominated sorting method integrating crowding distance assignment is applied to enhance the time efficiency of the improved MOBCC algorithm. A hybrid method combining bacterial individual mutation, oriented mutation of bacterial colony and local search of external archive is applied to enhance the convergence of the algorithm and maintain the diversity of solution set. The framework of MOEAs based on Pareto dominance is integrated into the improved MOBCC algorithm properly through replacements of the bacterial individuals in the bacterial colony, archive operation, and updating of the bacterial colony. The improved MOBCC algorithm is compared with three common multi-objective optimization algorithms SPEA2, NSGA-II and MOEA/D on fifteen test problems and evolution of optimization, and the experimental results confirm the validity of the improved MOBCC algorithm. Furthermore, the effects of the improved MOBCC algorithm’s parameters on the performance of the improved MOBCC algorithm are analyzed.

Nowadays, in different fields, tremendous attention is received by the Multi-agent systems for complex problem solutions with smaller task subdivision. Multiple inputs are utilized, e.g., history of actions, interactions with its neighboring agents by an agent. By the existing techniques for the task planning of the control structure the low efficiency is exhibited. By utilizing the sole numerical analysis method for a complicated distributed resource planning problem, the satisfactory optimal solution is impossible to obtain. In this paper, the control structure model is presented based on the multi-agents, in which the multi-agents superiority is exploited for complex task achievement. The collaboration of multi-agent framework is redefined, and the local conflict coordination mechanism is developed. Moreover, the high adaptability and superior cooperation are exhibited by the presented technique. The function value and its time–space complexity are analyzed, and it is obtained that the lower objective function value is achieved by the algorithm and the better convergence and adaptability are exhibited. The presented technique is 37–43% better than the Hierarchical Task Network Planning (HTN) technique for different time slots. The performance of the presented technique is 29–34% better compared to the Time Preference HTN technique in terms of function value. The performance of the proposed technique is better compared to the existing techniques in terms of obtained function values.

This paper aims to define and study unbounded fuzzy norm convergence in fuzzy Banach lattice, closely related to unbounded fuzzy order convergence. Some other theoretical concepts like fuzzy quasi-interior point and disjoint sequences are studied in relation to unbounded fuzzy norm convergence. Moreover, we defined a topology that is compatible with unbounded fuzzy norm convergence.

Long short-term memory (LSTM) neural network has been widely studied and applied in the real world. To obtain the LSTM neural network with better accuracy and more appropriate structure, the hybrid coding particle swarm optimization (HCPSO) algorithm is proposed. Firstly, the hybrid coding scheme is developed to represent the weights and structure of LSTM neural network, simultaneously. Then, the novel update mechanism is proposed to adjust the position of particles. Meanwhile, the discrete update strategy (DUS) and adaptive nonlinear moderate random search strategy (ANMRS) are proposed to enhance the convergence and global search capability of HCPSO, respectively. Finally, the effectiveness of HCPSO is demonstrated by multiple numerical examples. The experiment results show that the proposed HCPSO algorithm is more competitive in optimizing LSTM neural networks than other algorithms.

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

The application of Levy processes including major movements or jumps over a small period of time has proved to be an effective technique in financial research to catch certain unusual or extreme cases in stock price dynamics. Models that follow the Levy process are The FMLS, Kobol, and CGMY models. These models gradually grow the interest for study among researchers because of some of the best choices them. Therefore the topic of approaching these three different models has drawn yet more attention. In the current paper, we present these models’ numerical method. At first, The Riemann-Liouville tempered fractional derivative (RLTFD) with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the fourth kind (SCPFK). In the second step, we gain the semi-discrete design to solve the tempered fractional B-S model (TFBSM) by applying finite difference approximation. We’re going to show that this system is stable and $$\mathcal {O}(\delta \tau )$$ is the convergence order. In fact, a fast stabilized method will obtain to reduce the time from processing and the computation time per repetition. Then to get the full scheme, we use SCPFK to approximate the spatial fractional derivative. Finally, two numerical examples are presented to demonstrate the accuracy and usefulness of the developed system.

In this article, affine recurrent fractal interpolation function is constructed and its convergence analysis is established to understand the approximation properties. Besides, the existence of optimal recurrent fractal interpolation function for given continuous function is discussed. Further, shape preserving aspects of the recurrent fractal interpolation function are investigated by imposing the necessary conditions on the vertical scaling factors. Numerical examples are explored which support the theoretical results.

This article develops a method based on the generalized Gegenbauer–Humbert wavelets in concert with their operational matrices of fractional integration to deal with the fractional partial differential equations and find the approximate solutions of it. The goal is to show that the proposed method is appropriate for boundary and initial-boundary problems even though it is generalized form. The convergence of the method under study is investigated. The numerical results gained by the proposed method are considered and compared with other methods, to establish the effectiveness and accuracy.

This paper proposes a new flux control strategy for consequent-pole brushless hybrid excitation (CPBHE) machines to achieve a high efficiency and a fast convergence. Based on an analysis of the mathematical model of a CPBHE machine, a loss model of the CPBHE machine is derived, which demonstrates the feasibility of adopting d-axis flux to search for the optimal efficiency point. Furthermore, to improve the convergence speed of the online search algorithm and to make it suitable for the frequently varying operating conditions of electric vehicles (EVs), an adaptive step search algorithm based on the margin of the voltage limit combined with the offline table lookup method is proposed. Experimental results show that the proposed algorithm can optimize the efficiency of a motor system in the flux-weakening region and that it features a fast optimization process that makes it suitable for EV drives.

The shuffled frog leaping algorithm is easily trapped into local optimum and has the low optimization accuracy when it is used to optimize the complex functions problems. To overcome the above shortcomings, a shuffled frog leaping algorithm with contraction factor was proposed. By introducing acceleration factors c1 and c2, the ability of worst individual to learn from best individual within the submemeplexes or global best individual of the entire population was improved and the convergence rate of algorithm was accelerated. Under inserting the contraction factor χ, the convergence of algorithm was ensured. After performing local optimization of the self-learning operator on the worst individual, and taking full advantage of the useful information in the worst individuals, the self-learning ability of the individual and the optimization accuracy of the algorithm were improved. Simulation results illustrated that the enhanced algorithm performed better optimization performance than basic SFLA and other improved SFLAs. Finally, the proposed algorithm was used to optimize five problems of the mechanical design, and its validity and practicability were verified.

By combining multilayer perceptrons (MLPs) and radial basis function neural networks (RBF-NNs), an efficient multilayer RBF network is proposed in this work for regression problems. As an extension to the existing multilayer RBF network (RBF-MLP-I), the new multilayer RBF network (RBF-MLP-II) first nonlinearly transforms the multi-dimensional input data by adopting a set of multivariate basis functions. Then, linear weighted sums of these basis functions, i.e., the RBF approximations, are computed in the first hidden layer and used as the features of this layer. Subsequently, in the following hidden layers, each feature of the preceding hidden layer is fed into a univariate RBF characterized by the trainable scalar center and width, and then, RBF approximations are also applied to these basis functions. Finally, the features of the last hidden layer are linearly transformed to approximate the target output data. RBF-MLP-II reduces the number of parameters in basis functions and thus the network complexity of RBF-MLP-I. Verified by four regression problems, it is demonstrated that the proposed RBF-MLP-II exhibits the best approximation accuracy and fastest training convergence compared to conventional MLPs, RBF-NNs, and RBF-MLP-I.

Multi-modal semantic trajectory prediction is of great importance for location-based applications. However, predicting trajectory is not trivial facing three challenges: (1) It is difficult to integrate useful information from multi-modal and heterogeneous data in different granularity for effective feature fusion; (2) All kinds of dependencies existing in multi-modal semantic trajectories are closely coupled and dynamically evolved, forming complex dependencies for which are difficult to quantify; (3) During the model training, the distribution of each modal feature shifts in different directions, resulting in the distortion of dependencies, which is accompanied by slow convergence and inaccurate predictions. In this paper, the Complex Dependencies Auto-learning Prediction Model (CDAPM) is proposed to solve these problems. First, the effective and robust representation of each points is obtained by jointly embedding multi-modal information. Then, the dependencies attention module is proposed to calculate the dependencies weight matrix and auto-learn the contribution of each point. Also, it solves the problem of long-term dependency effectively. Position Encoding and LSTM are used to enhance the time relationship of trajectory. Finally, Mode Normalization is designed to maintain prediction accuracy by preventing the distortion of dependencies and significantly accelerate the convergence speed. Experiments on two real data sets show that CDAPM outperforms the state-of-the-art methods.

This work formulates an alternating direction implicit (ADI) Galerkin scheme for the nonlocal diffusion-wave equation in three-dimensional space. The L1 discretization formula is used to approximate the fractional Caputo derivative. A fully discrete Galerkin scheme is obtained via spatial discretization based on the finite element method. Then, an ADI algorithm is designed to reduce the computational cost. The stability and convergence analyses are derived via the energy method. Numerical experiments demonstrate the theoretical results.

In this study, a closed-loop $${D^\alpha }$$ -type iterative learning control (ILC) with a proportional D-type iterative learning updating law for the initial shift is applied to nonlinear conformable fractional order system. First, the system with the initial shift is introduced. Then, fractional order ILC (FOILC) frameworks that experience the initial shift problem for the path tracking of nonlinear conformable fractional order systems are addressed. Moreover, the sufficient condition for the convergence of tracking errors is obtained in the time domain by introducing $$\lambda $$ -norm and Holder’s inequality. Lastly, numerical examples are provided to illustrate the effectiveness of the proposed methods.

The authors analyze the two-stage Popov method with Bregman divergence and a new adaptive rule for choosing the step size, which does not require the Lipschitz constants to be known and operator values at additional points to be calculated. For variational inequalities with pseudo-monotone and Lipschitz continuous operators acting in a finite-dimensional normed linear space, the convergence theorem for the method is proved.

How to maintain a good balance between convergence and diversity is particularly important for the performance of the many-objective evolutionary algorithms. Especially, the many-objective optimization problem is a complicated Pareto front, the many-objective evolutionary algorithm can easily converge to a narrow of the Pareto front. An efficient environmental selection and normalization method are proposed to address this issue. The maximum angle selection method based on vector angle is used to enhance the diversity of the population. The maximum angle rule selects the solution as reference vector can work well on complicated Pareto front. A penalty-based adaptive vector distribution selection criterion is adopted to balance convergence and diversity of the solutions. As the evolution process progresses, the new normalization method dynamically adjusts the implementation of the normalization. The experimental results show that new algorithm obtains 30 best results out of 80 test problems compared with other five many-objective evolutionary algorithms. A large number of experiments show that the proposed algorithm has better performance, when dealing with numerous many-objective optimization problems with regular and irregular Pareto Fronts.

We provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper, we propose a somewhat different approach, in the sense that the algorithm itself is used in order to formulate a new optimality condition satisfied by its feasible limit points. With this tool at hand, we present several new properties and insights on limit points of augmented Lagrangian schemes, in particular, characterizing the strongest possible global convergence result for the safeguarded augmented Lagrangian method.

We have developed a machine-learning (ML) accelerated optimization method for challenging real-life well-location optimization (WLO) applications. The optimization protocol encompasses a stage at which the objective function is computed by a proxy ML model trained to accurately approximate the objective function that would be otherwise calculated using a computationally more intensive reservoir simulation run. As ML algorithms for proxy development, we have used two variants of the support-vector regression (SVR) technique, namely, e-SVR and ν-SVR. In the proposed proxy-accelerated optimization workflow, WLO starts with the conventional approach of using the reservoir simulator as the objective-function evaluator for a relatively small number of iterations of a global optimizer that conducts explorative search. This initial simulation investment is utilized not only to deliver optimization improvements but also to train an SVR proxy for the objective function of the investigated WLO problem. Subsequent to initial iterations, the SVR proxy is activated to serve as the objective-function evaluator for the global optimizer. Upon convergence, optimal model parameters estimated by the proxy-based optimization protocol are input to the flow simulator to access the full set of simulation results and also to re-evaluate the proxy accuracy. Optionally, the optimal model configuration can be supplied as initial guess to an efficient local optimizer to evaluate the further improvement potential for a few iterations using the reservoir simulator as the objective-function evaluator. The proposed SVR-accelerated optimization method is tested on a real field WLO problem. We have observed a 40 to 60% reduction in the overall computational cost with the SVR-accelerated protocol. It is important to note that the training time of the SVR proxy is negligibly short for small datasets (with 100s to 1000s data points) such as the ones generated over the course of WLO iterations. However, the main challenge associated with the proposed method is tuning the hyper-parameters of the SVR proxy for a given problem of interest and striking a good balance between accuracy and generalizability of proxy predictions. While some experimentation is still required with SVR hyper-parameters, especially in the first application, we accelerate this step through an auditable optimization process in our implementation. Results of our study demonstrate that the SVR-accelerated WLO method is applicable for real-life problems.