Iterative Convergence Research Articles

Adaptive weighted progressive iterative approximation based on coordinate decomposition.

During the iterative process of the progressive iterative approximation, it is necessary to calculate the difference between the current interpolation curve and the corresponding data points, known as the adjustment vector. To achieve more precise adjustments of control points, this paper decomposes the adjustment vector into its coordinate components and introduces a weight for each component. By dynamically adjusting these weights, we can accelerate the convergence of iterations and enhance approximation accuracy. During the iteration, the weight coefficients are flexibly adjusted based on the error of the current iteration step, demonstrating the flexibility and precision of the geometric iterative method in addressing geometric approximation problems. Numerical experiment results indicate that this vector decomposition technique is a critical mathematical operation for improving algorithm efficiency and precisely adjusting the shapes of curves or surfaces to approximate the data set.

Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation

Abstract The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus.

Convergence analysis of positive solution for Caputo–Hadamard fractional differential equation

By deriving the expression of Green function and some of its special properties and establishing appropriate substitution and appropriate cone, the existence of unique iterative positive, error estimation, and convergence rate of approximate solution are obtained for singular p-Laplacian Caputo–Hadamard fractional differential equation with infinite-point boundary conditions. Nonlinearities involve derivative terms that make our analysis difficult in the course of this research, and we deal with the difficulty of derivative terms by making appropriate substitutions. An example is given to demonstrate the validity of our main results.

Almost-Sure Convergence of Iterates and Multipliers in Stochastic Sequential Quadratic Optimization

Stochastic sequential quadratic optimization (SQP) methods for solving continuous optimization problems with nonlinear equality constraints have attracted attention recently, such as for solving large-scale data-fitting problems subject to nonconvex constraints. However, for a recently proposed subclass of such methods that is built on the popular stochastic-gradient methodology from the unconstrained setting, convergence guarantees have been limited to the asymptotic convergence of the expected value of a stationarity measure to zero. This is in contrast to the unconstrained setting in which almost-sure convergence guarantees (of the gradient of the objective to zero) can be proved for stochastic-gradient-based methods. In this paper, new almost-sure convergence guarantees for the primal iterates, Lagrange multipliers, and stationarity measures generated by a stochastic SQP algorithm in this subclass of methods are proved. It is shown that the error in the Lagrange multipliers can be bounded by the distance of the primal iterate to a primal stationary point plus the error in the latest stochastic gradient estimate. It is further shown that, subject to certain assumptions, this latter error can be made to vanish by employing a running average of the Lagrange multipliers that are computed during the run of the algorithm. The results of numerical experiments are provided to demonstrate the proved theoretical guarantees.

Prediction of Compressive Strength of Fly Ash-Recycled Mortar Based on Grey Wolf Optimizer-Backpropagation Neural Network.

The evaluation of the mechanical performance of fly ash-recycled mortar (FARM) is a necessary condition to ensure the efficient utilization of recycled fine aggregates. This article describes the design of nine mix proportions of FARMs with a low water/cement ratio and screens six mix proportions with reasonable flowability. The compressive strengths of FARMs were tested, and the influence of the water/cement ratio (w/c) and age on the compressive strength was analyzed. Meanwhile, a backpropagation neural network (BPNN) model optimized by the grey wolf optimizer (GWO), namely the GWO-BPNN model, was established to predict the compressive strength of FARM. The input layer of the model consisted of w/c, a cement/sand ratio, water reducer, age, and fly ash content, while the output layer was the compressive strength. The data set consisted of 150 sets from this article and existing research in the literature, of which 70% is used for model training and 30% for model validation. The results show that compared with the traditional BPNN, the coefficient of determination (R2) of GWO-BPNN increases from 0.85 to 0.93, and the mean squared error (MSE) of model training decreases from 0.018 to 0.015. Meanwhile, the convergence iterations of model validation decrease from 108 to 65. This indicates that GWO improved the prediction accuracy and computational efficiency of BPNN. The model results of characteristic heat, kernel density estimation, scatter matrix, and the SHAP value all indicated that the w/c was strongly negatively correlated with compressive strength, while the sand/cement ratio and age were strongly positively correlated with compressive strength. However, the relationship between the contents of fly ash, the water reducer, and the compressive strength was not obvious.

Analyzing market plans for enhanced energy hub efficiency: strategies for integrating multiple energy sources and collaboration

ABSTRACT In this study, we introduce the Archimedes optimization algorithm as a key component of our methodology. The Archimedes algorithm, renowned for its robustness and efficiency, is specifically tailored to handle complex optimization problems encountered in energy management systems. By leveraging its unique capabilities, we ensure precise solutions and rapid convergence even under challenging solution conditions characterized by uncertain energy demand forecasts. The proposed algorithm operates by iteratively refining solutions based on probabilistic load predictions and system constraints. It employs sophisticated optimization techniques to navigate the multi-dimensional solution space efficiently, thereby identifying optimal strategies for energy resource allocation within EHs. By incorporating the Archimedes algorithm into our framework, we can effectively capture the dynamic interactions between various energy carriers and adapt to changing market conditions in real-time. Furthermore, the Archimedes algorithm’s ability to provide near-optimal solutions without the need for exhaustive computational resources makes it particularly well-suited for practical implementation in energy management systems. Its efficiency ensures that decision-makers can obtain actionable insights and make informed decisions quickly, thereby enhancing the operational efficiency and economic viability of EHs. The proposed optimization algorithm outperforms alternatives like Archimedes optimization algorithm (APO) and genetic algorithm (GA). On average, it requires 25% fewer iterations for convergence. Moreover, it achieves a 90% convergence rate within just 50 iterations, compared to 70 iterations for APO and 60 iterations for GA, based on 30 distinct runs.

Optimized Path Planning and Scheduling in Robotic Mobile Fulfillment Systems Using Ant Colony Optimization and Streamlit Visualization

Context: In the age of rapid e-commerce growth; the Robotic Mobile Fulfillment Systems (RMFS) have become the major trend in warehouse automation. These systems involve the use of self- governed mobile chares to collect shelves as well as orders for deliveries with regard to optimization of task allocation and with reduced expenses. However, in a manner to implement such systems, one needs to find enhanced algorithms pertaining to resource mapping and the planning of movement of robots in sensitive environments. Problem Statement: Despite RMFS have certain challenges especially when it comes to the distribution of tasks and the overall distances that employees have to cover. Objective: The main goal of this paper is to propose a new compound optimization model based on RL-ACO to optimize the RMFS’s task assignment and navigation. Also, the direction of the study is to investigate how such methods can be applied to real-life warehouse automation and how effective such methods can be on a large scale. Methodology: This research introduces a new optimization model for RMFS selection which integrates reinforcement learning with Ant Colony Optimization (ACO). Specifically, a real gym environment was created to perform the order assignment and training in the way of robotic movement. Reinforcement Learning (RL) models were trained with Proximal Policy Optimization (PPO) for improving the dynamic control of robots and ACO was used for computing optimal shelf trajectories. The performance was also measured by policy gradient loss, travelled distance and time taken to complete the tasks. Results: The proposed framework showed potential in enhancing the efficiency of tasks required and the travel distances involved. In each of the RL models used the shortest paths were identified and the best route was determined to have a total distance of 102.91 units. Also, other values such as, value function loss and policy gradient loss showed learning and convergence in iterations. To build a global solution, ACO integration went a step forward in enabling route optimization through effective combinatorial problems solving. Implications: This research offers a practical, generalizable and flexible approach for the improvement of the operations of RMFS and thinking for warehouse automation.

A convergent multi-step efficient iteration method to solve nonlinear equation systems

A convergent multi-step efficient iteration method to solve nonlinear equation systems

An iterative constraint energy minimizing generalized multiscale finite element method for contact problem

This work presents an Iterative Constraint Energy Minimizing Generalized Multiscale Finite Element Method (ICEM-GMsFEM) for solving the contact problem with high contrast coefficients. The model problem can be characterized by a variational inequality, where we add a penalty term to convert this problem into a non-smooth and non-linear unconstrained minimizing problem. The characterization of the minimizer satisfies the variational form of a mixed Dirilect-Neumann-Robin boundary value problem. So we apply CEM-GMsFEM combined with simismooth Newton method and introduce special boundary correctors along with multiscale spaces to achieve an optimal convergence rate. Numerical results are conducted for different highly heterogeneous permeability fields, validating the fast convergence of the CEM-GMsFEM iteration in handling the contact boundary and illustrating the stability of the proposed method with different sets of parameters. We also prove the fast convergence of the proposed iterative CEM-GMsFEM method and provide an error estimate of the multiscale solution under a mild assumption.

Effective Rates for Iterations Involving Bregman Strongly Nonexpansive Operators

We develop the theory of Bregman strongly nonexpansive maps for uniformly Fréchet differentiable Bregman functions from a quantitative perspective. In that vein, we provide moduli witnessing quantitative versions of the central assumptions commonly used in this field on the underlying Bregman function and the Bregman strongly nonexpansive maps. In terms of these moduli, we then compute explicit and effective rates for the asymptotic regularity of Picard iterations of Bregman strongly nonexpansive maps and of the method of cyclic Bregman projections. Further, we also provide similar rates for the asymptotic regularity and metastability of a strongly convergent Halpern-type iteration of a family of such mappings and we use these new results to derive rates for various special instantiations like a Halpern-type proximal point algorithm for monotone operators in Banach spaces as well as Halpern-Mann- and Tikhonov-Mann-type methods.

Iterative Convergence for Solving the Exit Plastic Zone and Friction Coefficient Model of Ultra-thin Strip Rolling Force

Iterative Convergence for Solving the Exit Plastic Zone and Friction Coefficient Model of Ultra-thin Strip Rolling Force

Fully implicit bound-preserving discontinuous Galerkin algorithm with unstructured block preconditioners for multiphase flows in porous media

Massively parallel simulation of multiphase flows in porous media is challenging due to the highly nonlinear governing equations with the complexity of various modeling features and the significant heterogeneity of material coefficients. In this paper, we introduce a fully implicit discontinuous Galerkin (DG) reservoir simulator designed for parallel computers to handle large-scale multiphase flow simulations. Our approach formulates a fully implicit DG scheme on 3D unstructured grids to discretize the nonlinear governing equations, which take into account capillary effects, permeability heterogeneity, gravity, and injection/production wells. A vertex-based slope limiter within the fully implicit DG framework is adopted to suppress oscillations near discontinuities and ensure the boundedness of the solution. We address the resulting nonlinear systems from the fully implicit discretization using a family of inexact Newton–Krylov algorithms with an analytic Jacobian. Moreover, we employ an unstructured block-type preconditioner with an efficient overlapping additive Schwarz approximation to accelerate the convergence of iterations and enhance the scalability of the simulator. Large-scale simulations of both benchmark and realistic reservoir problems on 3D unstructured grids are conducted to demonstrate the efficiency and scalability of the proposed reservoir simulator.

Efficient Brain Cancer Classification Based on Advanced CNN and Transformer Technologies

Abstract. This study addresses the challenge faced by primary medical staff in accurately distinguishing different types of brain cancer by comparing the training accuracy and iteration speed of various classification models. Specifically, thesis evaluated a basic Convolutional Neural Network (CNN) alongside two transfer learning models: EfficientNetV2 and Vision Transformer. The analysis is based on publicly available data from Kaggle, which provided a robust dataset for evaluation. The results indicate that both EfficientNetV2 and Vision Transformer models achieve significantly higher iteration convergence speeds and classification accuracy compared to the basic CNN model trained solely on the database. These advanced models demonstrate a capacity for efficient and rapid brain cancer differentiation, making them suitable for settings with limited computing resources. The study highlights that the transfer learning models can effectively support outpatient doctors by providing fast and accurate etiological screening without requiring extensive hardware. Overall, the findings suggest that EfficientNetV2 and Vision Transformer models offer a practical solution for improving brain cancer classification in primary healthcare environments, where rapid and precise diagnosis is crucial. This research underscores the potential of transfer learning techniques to enhance diagnostic capabilities while accommodating resource constraints.

Iterative algorithm for computing magnetic field considering remanent magnetization and demagnetization

SUMMARY Efficient and high-precision methods for performing large-scale numerical calculations under high magnetic susceptibility conditions are essential for magnetic exploration applications. This paper proposes an algorithm for magnetic field modelling considering both demagnetization and remanent magnetization. The 3-D partial differential equation of magnetic potential is reduced to a 1-D ordinary differential equation in the space-wavenumber domain through a 2-D Fourier transform in the horizontal direction. Using the quadratic interpolation finite element method to obtain the pentagonal equation, the chasing method is used to solve the equation efficiently, and a contraction operator based on electromagnetic integral equation method is introduced to ensure stable iteration convergence. The validity of the algorithm was confirmed by comparing it with analytical solutions. Numerical examples demonstrate the importance of considering demagnetization under high magnetization conditions. Test results reveal that magnetic susceptibility is the sole factor affecting convergence among the factors examined. Further analysis showed that, for the same magnetic susceptibility, the algorithm converges with the same number of iterations and higher precision is achieved with an increasing number of nodes. Under the same grid configuration, higher magnetic susceptibility requires more iterations to converge and results in lower algorithm precision. The algorithm presented in this paper shows higher efficiency compared to those based on integral equations. Moreover, we demonstrate the algorithm's high accuracy when applied to models with anomalies near boundaries. Finally, the adaptability of the algorithm to complex geological models and undulating topography is verified by combining model tests with DEM data. This algorithm presents an efficient and high-precision method for magnetic field calculations under conditions of high susceptibility and strong remanent magnetization, offering promising prospects for geoscience applications.

Negative flux fixups using positivity-preserving limiters for SN-discontinuous finite element scheme of neutron transport equation on unstructured triangular meshes

One significant challenge of spatial discretization for the SN transport equation is the appearance of negative solutions, resulting in numerical algorithms to be unstable or slow iterative convergence in some problems. The discontinuous Galerkin finite element (DGFEM) spatial scheme on unstructured meshes has attracted much attention in recent years, but the issue of negative solutions is still a long-standing problem. This paper studies a negative flux fixup method using positivity-preserving limiters for solving the SN-DGFEM neutron transport equation. This method uses the hierarchical basis functions and triangular reference elements to obtain arbitrary high order DGFEM scheme. In the element where appears negativities, a scaling or a rotation positivity-preserving limiter is used to scale or rotate the polynomial distribution to ensure positive solutions. Five different types of problems are selected to verify the accuracy and convergence of the method. Numerical results demonstrate that the method can produce non-negative solutions and maintain the convergence order of the original scheme, as well as not introduce too much additional computational effort.

Online Q-learning for stochastic linear systems with state and control dependent noise

For continuous-time (CT) systems that are characterized by stochastic differential equations (SDEs) with completely unknown dynamics parameters, a reinforcement learning (RL)-based optimal control framework is presented in this paper. To obtain the near-optimal control policy, an online Q-learning algorithm is proposed by learning the data sampled from the system state trajectory, while an integral reinforcement learning (IRL) approach is developed in stochastic situation so as to formulate the Q-learning iterative algorithm. In particular, an actor/critic neural network (NN) structure is applied in iteration, where a critic approximator is used for estimating the designed Q-function while an actor approximator is for estimating the optimal control policy online. To ensure the convergence of iteration, the tuning laws of two neural networks are designed, respectively, by a gradient descent scheme. Moreover, the mean-square stability of the closed-loop system is proved through rigorous analysis, and the convergence to optimal solution is guaranteed as well.

Reliability Analysis of Complex Structures Under Multi-Failure Mode Utilizing an Adaptive AdaBoost Algorithm

A reliability analysis can become intricate when addressing issues related to nonlinear implicit models of complex structures. To improve the accuracy and efficiency of such reliability analyses, this paper presents a surrogate model based on an adaptive AdaBoost algorithm. This model employs an adaptive method to determine the optimal training sample set, ensuring it is as evenly distributed as possible on both sides of the failure curve and fully contains the information it represents. Subsequently, with the integration and iterative characteristics of the AdaBoost algorithm, a simple binary classifier is iteratively applied to build a high-precision alternative model for complex structural fault diagnosis to cope with multiple failure modes. Then, the Monte Carlo simulation technique is employed to meticulously assess the failure probability. The accuracy and stability of the proposed method’s iterative convergence process are validated through three numerical examples. The findings of the study illuminate that the proposed method is not only remarkably precise but also exceptionally efficient, capable of addressing the challenges related to the reliability evaluation of complex structures under multi-failure mode. The method proposed in this paper enhances the application of mechanical structures and facilitates the utilization of complex mechanical designs.

Data-Driven Robust Finite-Iteration Learning Control for MIMO Nonrepetitive Uncertain Systems.

This work considers three main problems related to fast finite-iteration convergence (FIC), nonrepetitive uncertainty, and data-driven design. A data-driven robust finite-iteration learning control (DDRFILC) is proposed for a multiple-input-multiple-output (MIMO) nonrepetitive uncertain system. The proposed learning control has a tunable learning gain computed through the solution of a set of linear matrix inequalities (LMIs). It warrants a bounded convergence within the predesignated finite iterations. In the proposed DDRFILC, not only can the tracking error bound be determined in advance but also the convergence iteration number can be designated beforehand. To deal with nonrepetitive uncertainty, the MIMO uncertain system is reformulated as an iterative incremental linear model by defining a pseudo partitioned Jacobian matrix (PPJM), which is estimated iteratively by using a projection algorithm. Further, both the PPJM estimation and its estimation error bound are included in the LMIs to restrain their effects on the control performance. The proposed DDRFILC can guarantee both the iterative asymptotic convergence with increasing iterations and the FIC within the prespecified iteration number. Simulation results verify the proposed algorithm.

IGG(χ): A new and simple implicit gradient scheme on unstructured meshes

A new and simple implicit Green-Gauss gradient (IGG) scheme for unstructured meshes is proposed exploiting the ideas from the linearity-preserving U-MUSCL scheme to define values at cell faces. We construct an implicit one-parameter family of gradient schemes referred to as IGG(χ) where χ is a free-parameter. The computed gradients are at least first-order accurate on generic polygonal meshes and second-order accurate on uniform Cartesian meshes except when χ=1/3 for which fourth-order accuracy can be realised. A theoretical analysis is carried out to understand the effect of the control parameter χ on accuracy and resolution of the gradients and numerical experiments on various mesh topologies confirm the theoretical findings. Finite volume simulations of the Poisson and Euler equations on Cartesian and unstructured meshes further highlight that the IGG(χ) is a versatile gradient scheme that gives second-order accurate solutions with the iterative convergence of the solver dependent on the choice of the χ parameter. The framework described in this study can also be employed to devise an implicit least-squares gradient scheme that applies equally well to unstructured finite volume and meshfree solvers.

Application of Ant Colony Optimization Algorithm in Determining PID Parameters in AC Motor Control

Application of Ant Colony Optimization (ACO) Algorithm in determining PID (Proportional-Integral-Derivative) parameters to optimize AC motor control through simulation using MATLAB. AC motors are a critical component in a wide range of industrial applications requiring efficient control to ensure optimal stability and response. This research focuses on optimizing the motor's RPM control by fine-tuning PID parameters using the ACO algorithm. Precise RPM control is crucial for maintaining performance in dynamic industrial environments. The ACO algorithm is used to optimize the PID parameter by referring to the objective function of Integral Time Absolute Error (ITAE). The optimization results show that this algorithm can achieve optimal convergence in the 33rd iteration with a fitness value of 6269. The optimal PID parameters obtained were Kp of 164.98, Ki of 23.47, and Kd of 10.51. The simulation of the AC motor control system shows a significant improvement in performance compared to the Trial-and-Error method. The simulation results demonstrate that ACO reduces steady-state errors by up to 9%, while Trial-and-Error reaches 25%. The settling time is also faster with ACO, which is 0.7 seconds, compared to the Trial-and-Error method which takes longer. The use of the ACO method in PID tuning has been proven to be more efficient and accurate than conventional approaches, thus improving the RPM stability and response of the AC motor control system. This study concludes that the integration between ACO and PID can be the optimal solution in automated control applications in industries that require responsive and stable motor RPM control.