Iterative Convergence Research Articles

Traction–separation law for bridged cracks at immiscible polymers interface

A traction–separation law for bonds at cracks bridged zones at immiscible polymers interface is developed. It is assumed that bonds at the crack bridged zone between different polymers are formed by bundles of the polymer molecules–promoter of adhesion. It is taken into account that molecules of the polymer-promoter can form one-stitch and many-stitch bonds between polymers. Nonlinear deformation curves of bonds at the crack bridged zone were obtained on this basis. The model of different materials joint with bridged interface crack is used to simulate immiscible polymers adhesive bonding. The system of singular integral-differential equations is applied for evaluation of bonds traction for the polymers interface bridged crack. The analysis of the bond traction–separation law parameters influence on the numerical iterative solution convergence of the integral-differential equations system has been performed. Evaluations of the polymer joint fracture toughness were performed using the non-local two-parameter criterion for quasi-static growth of bridged cracks.

Agriculture crop yield prediction using inertia based cat swarm optimization

Crop yield prediction is among the most important and main sources of income in the Indian economy. In this paper, the improved cat swarm optimization (ICSO) based recurrent neural network (RNN) model is proposed for crop yield prediction using time series data. The inertia weight parameter is added to position equation that is selected randomly, and a new velocity equation is produced which enhances the searching ability in the best cat area. By using inertia weight, the ICSO enhances performance of feature selection and obtains better convergence in minimum iteration. The RNN is applied to produce direct graph using sequence of data and decides current layer output by involving all other existing calculations. The performance of the model is estimated using coefficient of determination (R2), root mean square error (RMSE), mean squared error (MSE), and mean absolute error (MAE) on the yield from the years 2011 to 2021 with an annual prediction for 120 records of approximately 8 million nuts. The evaluated result shows that the proposed ICSO-RNN model delivers metrics such as R2, MAE, MSE, and RMSE values of 0.99, 0.77, 0.68, and 0.82 correspondingly, which ensures accurate yield prediction when compared with the existing methods which are hybrid reinforcement learning-random forest (RL-RF) and machine learning (ML) methods.

Inertial Krasnoselskii-Mann Iterations

We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with estimates for the non-asymptotic rate at which the residuals vanish. Strong and linear convergence are obtained in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and show that passing from one regime to the other is a continuous process in terms of the hypotheses on the parameters. Numerical illustrations are provided for an inertial primal-dual method and an inertial three-operator splitting algorithm, whose performance is superior to that of their non-inertial counterparts.

A new inexact gradient descent method with applications to nonsmooth convex optimization

The paper proposes and develops a novel inexact gradient method (IGD) for minimizing C 1 -smooth functions with Lipschitzian gradients, i.e. for problems of C 1 , 1 optimization. We show that the sequence of gradients generated by IGD converges to zero. The convergence of iterates to stationary points is guaranteed under the Kurdyka-Łojasiewicz (KL) property of the objective function with convergence rates depending on the KL exponent. The newly developed IGD is applied to designing two novel gradient-based methods of nonsmooth convex optimization such as the inexact proximal point methods (GIPPM) and the inexact augmented Lagrangian method (GIALM) for convex programs with linear equality constraints. These two methods inherit global convergence properties from IGD and are confirmed by numerical experiments to have practical advantages over some well-known algorithms of nonsmooth convex optimization.

Can neural networks benefit from objectives that encourage iterative convergent computations? A case study of ResNets and object classification.

Recent work has suggested that feedforward residual neural networks (ResNets) approximate iterative recurrent computations. Iterative computations are useful in many domains, so they might provide good solutions for neural networks to learn. However, principled methods for measuring and manipulating iterative convergence in neural networks remain lacking. Here we address this gap by 1) quantifying the degree to which ResNets learn iterative solutions and 2) introducing a regularization approach that encourages the learning of iterative solutions. Iterative methods are characterized by two properties: iteration and convergence. To quantify these properties, we define three indices of iterative convergence. Consistent with previous work, we show that, even though ResNets can express iterative solutions, they do not learn them when trained conventionally on computer-vision tasks. We then introduce regularizations to encourage iterative convergent computation and test whether this provides a useful inductive bias. To make the networks more iterative, we manipulate the degree of weight sharing across layers using soft gradient coupling. This new method provides a form of recurrence regularization and can interpolate smoothly between an ordinary ResNet and a "recurrent" ResNet (i.e., one that uses identical weights across layers and thus could be physically implemented with a recurrent network computing the successive stages iteratively across time). To make the networks more convergent we impose a Lipschitz constraint on the residual functions using spectral normalization. The three indices of iterative convergence reveal that the gradient coupling and the Lipschitz constraint succeed at making the networks iterative and convergent, respectively. To showcase the practicality of our approach, we study how iterative convergence impacts generalization on standard visual recognition tasks (MNIST, CIFAR-10, CIFAR-100) or challenging recognition tasks with partial occlusions (Digitclutter). We find that iterative convergent computation, in these tasks, does not provide a useful inductive bias for ResNets. Importantly, our approach may be useful for investigating other network architectures and tasks as well and we hope that our study provides a useful starting point for investigating the broader question of whether iterative convergence can help neural networks in their generalization.

Robustness improvement of optimal control in terms of RBFNN with empirical model reduction and transfer learning

This paper proposes a method to compute solutions of optimal controls for dynamic systems in terms of radial basis function neural networks (RBFNN) with Gaussian neurons. The RBFNN is used to compute the value function from Hamilton–Jacobi–Bellman equation with the policy iteration. The concept of dominant system is introduced to create initial coefficients of the neural networks to stabilise unstable systems and guarantee the convergence of policy iteration. Model reduction and transfer learning techniques are introduced to improve robustness of RBFNN optimal control, and reduce computational time. Numerical and experimental results show that the resulting optimal control has excellent control performance in stabilisation and trajectory tracking, and much-improved robustness to disturbances and model uncertainties even when the system responses move to a domain of the state space that is larger than the domain where the neural networks are trained.

On iterative methods based on Sherman-Morrison-Woodbury splitting

We consider a new splitting based on the Sherman-Morrison-Woodbury formula, which is particularly effective with iterative methods for the numerical solution of large linear systems and especially for systems involving matrices that are perturbations of circulant or block circulant matrices. Such matrices typically arise in the discretization of differential equations using finite element or finite difference methods. We prove the convergence of the new iteration without making any assumptions regarding the symmetry or diagonal-dominance of the matrix, which are limiting factors for most classical iterative methods.To illustrate the efficacy of the new iteration we present various applications. These include extensions of the new iteration to block matrices that arise in certain saddle point problems as well as two-dimensional finite difference discretizations. The new method exhibits fast convergence in all of the test cases we used. It has minimal storage requirements, straightforward implementation and compatibility with nearly circulant matrices via the Fast Fourier Transform. Remarkably, the new method was tested against very large matrices demonstrating extremely fast convergence. For these reasons it can be a valuable tool for the solution of various finite elements and finite differences discretizations of differential equations.

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

Solving Surface-volume Integral Equations for PEC and Inhomogeneous/Anisotropic Materials with Multibranch Basis Functions

Multibranch basis functions have been confirmed to be effective for local refinement of domain decomposition methods in the application of solving surface and volume integral equations. Surface-volume integral equations (SVIEs) are applied for solving the hybrid electromagnetic scattering problems involving perfect electric conductors (PEC) and dielectrics, especially inhomogeneous and anisotropic media. In this paper, multibranch Rao-Wilton-Glisson basis functions (MB-RWGs) are applied in conjunction with multibranch Schaubert-Wilton-Glisson basis functions (MB-SWGs) for solving the SVIEs. Block diagonal preconditioners (BDPs) are used to accelerate the iteration convergence based on generalized minimum residual (GMRES) algorithms. The numerical results demonstrate the accuracy of the multibranch basis functions in solving SVIEs, and also show that proper BDPs can accelerate the iteration convergency.

A Variant of the Second-Moment Method for k-Eigenvalue Calculations

The second-moment (SM) method is a linear variant of the quasi-diffusion (QD) method for accelerating the iterative convergence of Sn source calculations. It has several significant advantages relative to the QD method, diffusion synthetic acceleration, and nonlinear diffusion acceleration. Here, we define a variant of this method for k-eigenvalue calculations that retains the advantages of the original method, and we computationally demonstrate the efficacy of the method for simple example calculations. In particular, this method has two important properties. First, it is a linear acceleration scheme requiring only the solution of a pure k-eigenvalue diffusion equation with a corrective source term as opposed to a k-eigenvalue drift-diffusion equation. Second, unconditional stability is achieved even when the diffusion equation is not discretized in a manner consistent with the Sn spatial discretization. We are unaware of any other scheme that has these properties. We also show a connection between our method and the k-eigenvalue acceleration technique of Barbu and Adams, which motivated us to develop our SM method.

Polynomial convergence of iterations of certain random operators in Hilbert space

Abstract We study the convergence of a random iterative sequence of a family of operators on infinite-dimensional Hilbert spaces, inspired by the stochastic gradient descent (SGD) algorithm in the case of the noiseless regression. We identify conditions that are strictly broader than previously known for polynomial convergence rate in various norms, and characterize the roles the randomness plays in determining the best multiplicative constants. Additionally, we prove almost sure convergence of the sequence.

Strong convergence of Halpern's type iteration for α-nonexpansive semigroup in Banach spaces and CAT(0) spaces

Strong convergence of Halpern's type iteration for α-nonexpansive semigroup in Banach spaces and CAT(0) spaces

Optimizing Topology in Satellite–UAV Collaborative IoT: A Graph Partitioning Simulated Annealing Approach

Currently, space-based information networks, represented by satellite Internet, are rapidly developing. UAVs can serve as airborne mobile terminals, representing a novel node in satellite IoT, offering more accurate and robust data streaming for connecting global satellite–UAV collaborative IoT systems. It is characterized by high-speed dynamics, with node distances and visibility constantly changing over time. Therefore, there is a need for faster and higher-quality topology optimization research. A reliable, secure, and adaptable network topology optimization algorithm has been proposed to handle various complex scenarios. Additionally, considering the dynamic and time-varying nature of these types of networks, the concept of time slices has been introduced to accelerate the iterative efficiency of problem-solving. Experimental results demonstrate that the proposed algorithm is expected to exhibit better convergence and performance in subsequent iterations compared with traditional solutions. Besides being a solution for topology optimization, the proposed algorithm offers a new way of thinking, enabling the handling of larger satellite–UAV collaborative IoT systems.

Short-term gas load prediction based on GRA-IPSO-LSTM

With the improvement of natural gas production, supply and marketing mechanism, the effective prediction of natural gas short-term load is of great significance for rational allocation of resources and solving urban energy security. Grey relation analysis (GRA) was used to determine the main controlling factors affecting the short-term load of natural gas. The inertia weight and acceleration factor of the particle swarm optimization (PSO) algorithm were improved by applying a nonlinear change strategy. The hyperparameters suitable for the LSTM model were found by using an optimization algorithm, and a combination model of GRA-IPSO-LSTM was formed. Other models are compared to verify the accuracy and reliability of the model. The results show that, according to the size of the grey correlation degree, the factors that have little influence on the daily load can be deleted step by step to reduce the complexity of the subsequent prediction model. The iteration speed, convergence accuracy, and optimization quality of the IPSO algorithm are improved, which reduces the limitation of the artificial selection of LSTM model hyperparameters. The MAPE, RMSE, and R of the GRA-IPSO-LSTM combined model are 5461.02 m3, 0.49%, and 0.99932, which significantly improve the prediction accuracy, proving that the combined model in this paper can be used for accurate prediction of short-term natural gas load.

Design and Development of Intelligent Learning System for University Innovation and Entrepreneurship Based on Knowledge Visualisation

To build a comprehensive learning resource system covering learning process, learning users, learning resources and other aspects, an intelligent learning system for innovation and entrepreneurship in universities was constructed and designed combined with knowledge visualisation theory. The system includes student information record module, system recommendation calculation module, learning material module, and knowledge visualisation module. The research focuses on the system recommendation calculation module, including the User-based Collaborative Filtering Algorithm (UCF) and the Joint Recommendation (JR) algorithm of the application content recommendation algorithm. The best neighbour number of JR is 20, and the best transmission path length is 2. The recommendation effect is optimal under this setting. The convergence iterations of the training and test set of the joint recommendation algorithm are 80 times and 100 times, respectively. For the training set and test set, the accuracy, precision, sensitivity, recall, running time, error, and other performance evaluation indicators of the joint recommendation algorithm are better than the corresponding values of other recommendation algorithms. The system has a good design effect and experience effect in the application of the online learning platform.

Improve refinement approach iterative method for solution linear equition of sparse matrices

In this paper, systems of linear equations on sparse matrices investigated through modified improve method using Gauss-Seidel and successive overrelaxation (SOR) approach. Taking into adapted convergence rate on the Improve refinement Gauss-seidel outperformed the prior two Gauss-Seidel methods in terms of rate of convergence and number of iterations required to solve the problem by applying a modified version of the Gauss-Seidel approach. to observe the effectiveness of this method, the numerical example is given. The main findings in this study, that Gauss seidel improvement refinement gives optimum spectral radius and convergence rate. Similarly, the SOR improved refinement method gives. Considering their performance, using parameters such as time to converge, number of iterations required to converge and spectral radius level of accuracy. However, SOR works with relaxation values so that it greatly affects the convergence rate and spectral radius results if given greater than 1.

Fixed Point and Convergence Results for Contractive-Type Self-Mappings of Metric Spaces with Graphs

It is known that a strict contraction on a complete metric space with a graph possesses a fixed point. In the present paper, we show that this property holds for single valued and set-valued self-mappings of metric spaces with graphs that are of the contractive type. We also show the convergence of iterates of these mappings to fixed points. In particular, our results are true for metric spaces with symmetric graphs.

Finite-size effects in periodic coupled cluster calculations

We provide the first rigorous study of the finite-size error in the simplest and representative coupled cluster theory, namely the coupled cluster doubles (CCD) theory, for gapped periodic systems. Given exact Hartree-Fock orbitals and their corresponding orbital energies, we demonstrate that the correlation energy obtained from the approximate CCD method, after a finite number of fixed-point iterations over the amplitude equation, exhibits a finite-size error scaling as O(Nk−13). Here Nk is the number of discretization points in the Brillouin zone and characterizes the system size. Under additional assumptions ensuring the convergence of the fixed-point iterations, we demonstrate that the CCD correlation energy also exhibits a finite-size error scaling as O(Nk−13). Our analysis shows that the dominant error lies in the coupled cluster amplitude calculation, and the convergence of the finite-size error in energy calculation can be boosted to O(Nk−1) with accurate amplitudes. This also provides the first proof of the scaling of the finite-size error in the third order Møller-Plesset perturbation theory (MP3) for periodic systems.

BDBM: A distributed network simplification method for solving task allocation problems

Existing task allocation algorithms often fail to consider communication load and do not meet the scalability requirements of large-scale systems in practical applications. Furthermore, many general network simplification algorithms are centralized and not designed for task allocation scenarios, thus losing their superiority when applied to solve task allocation problems in distributed systems. Inspired by recent network simplification algorithms, this paper introduces a novel network simplification algorithm called bid-based distributed broken-motifs (BDBM), which simplifies the communication network by reducing the number of closed-loop triangles. The BDBM algorithm keeps those communication edges that facilitate the deconfliction based on the initial bids, thus avoiding the surge in the number of convergence iterations caused by the simplification. In addition, CBBA is distributed and requires only two iterations to complete the communication network simplification. Theoretical analysis confirms that the simplified network of BDBM remains connected and can be used in combination with improved task allocation algorithms based on consensus-based bundle algorithm (CBBA). In terms of experiments, we conducted comprehensive experiments in different settings and found that the proposed algorithm does not lead to worse allocation solutions. The statistical results also show that BDBM outperforms state-of-the-art network simplification algorithms in terms of solving efficiency and scalability in assisting CBBA solve large-scale complex task allocation problems.

Distributed Voltage Security Enhancement Using Measurement-Based Voltage Sensitivities

The existing susceptance-based methods adopt the constant susceptance matrix as the voltage sensitivities and have been widely applied in voltage security enhancement of power systems. However, the existing methods may be inefficient under the heavy loading condition in which the fundamental of decoupled power flow fails to hold. In this case, the inaccurate voltage sensitivities in the existing method lead to the inefficient voltage security enhancement with more required iterations of the numerical convergence. In order to improve such deficiency, the distributed voltage security enhancement method using the measurement-based voltage sensitivities is proposed in this paper. By using real-time measurements of phasor measurement units (PMUs), the multi-port equivalent models of the power grid are built and used to derive the measurement-based voltage sensitivities and the system transmission capability sensitivities. By using the measurement-based sensitivities, the proposed method can be formulated as the measurement-based optimization problem and further implemented in the distributed framework through using alternating direction method of multipliers (ADMM). Simulations on IEEE 118-bus and IEEE 300-bus test systems are presented to validate the effectiveness of the proposed method.