Iterative Method Research Articles

Nonlinear Regularized HSS Iteration Method for a Class of Nonlinear Saddle-Point Systems with Applications to Image Deblurring Problems

In this paper, we construct a nonlinear regularized Hermitian and skew-Hermitian splitting (NRHSS) iteration method for a class of nonlinear saddle-point systems. The NRHSS iteration method can be seen as a generalization of the Peaceman-Rachford splitting method with monotone plus skew-symmetric splitting (PRSM/MSS). Besides, if the nonlinear operator reduces to the linear case, the NRHSS iteration method reduces to the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration method. Hence, the NRHSS iteration method can also be seen as a generalization of the RHSS iteration method. To overcome the heavy workload when direct solution methods are used, we also propose an inexact version of the NRHSS iteration method. Detailed theoretical analysis regarding the convergence properties is given. Numerical experiments on image deblurring problems are carried out to illustrate the feasibility and efficiency of the NRHSS iteration method.

A Single-Variable Method for Solving the Min–Max Programming Problem with Addition–Overlap Function Composition

Min–max programming problems with addition–min constraints have been studied in the literature to model data transfer in BitTorrent-like peer-to-peer file-sharing systems. It is well known that the class of overlap functions contains various operators, including the “min” operator. The aim of this paper is to generalize the above min–max programming problem with addition–overlap function constraints. We demonstrate that this new optimization problem can be transformed into a simplified single-variable optimization problem, which makes it easier to find an optimal solution. The bisection method will be used to find this optimal solution. In addition, when the overlap function is explicitly specified, an iterative method is given to compute the optimal objective value with a polynomial time complexity. A numerical example is provided to illustrate the procedures.

An improved radial basis reproducing kernel particle method for geometrically nonlinear problem analysis of SMAs

In this paper, the radial basis function (RBF) without shaped parameter is utilized in the radial basis reproducing kernel particle method (RRKPM), and an improved radial basis reproducing kernel particle method (IRRKPM) is proposed. Compared with traditional RKPM, the IRRKPM effectively reduces the impact of different kernel functions on calculation precision, and is further employed to examine geometrically nonlinear problems associated with shape memory alloys (SMAs). The displacement boundary condition is enforced via the penalty function method, while the Galerkin integration method in its weak form, along with the total Lagrangian (TL) approach, is utilized to derive the geometrically nonlinear equations for SMAs within the IRRKPM framework. The equilibrium equations are then solved using the Newton Raphson (N-R) iterative method. The impact of the different penalty factor and the radius control parameter of influence domain on errors is analyzed, the computational precision of the IRRKPM is compared with the RRKPM, and the computational stability is evaluated. Finally, the suitability of the IRRKPM for the analysis of geometrically nonlinearity problems in SMAs are confirmed through specific numerical examples.

Correction to: Survey of a class of iterative row-action methods: The Kaczmarz method

Correction to: Survey of a class of iterative row-action methods: The Kaczmarz method

Strong Convergence of an Iterative Method for Solving Generalized Mixed Equilibrium Problems and Split Feasibility Problems

In this paper, we investigate iterative methods for solving generalized mixed equilibrium problems, split feasibility problems, and fixed point problems in Banach spaces. We introduce a new extragradient algorithm using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions to the split feasibility problem, the set of fixed points of a resolvent operator, and the set of solutions of the generalized mixed equilibrium problem. The algorithm is analyzed in a real 2-uniformly convex and uniformly smooth Banach space, taking into account computational errors. A numerical example is provided to illustrate the applicability and performance of the proposed method.

Parabolic logistic equation with harvesting involving the fractional Laplacian

This paper deals with a class of parabolic reaction-diffusion equations driven by the fractional Laplacian as the diffusion operator over a bounded domain with zero Dirichlet external condition. Using a comparison principle and monotone iteration method, we establish existence and uniqueness results. We apply the existence result to the logistic growth problems with constant yield harvesting by constructing an ordered pair of positive sub- and supersolution of the corresponding elliptic problem.

Research on wind turbines preventive maintenance strategies based on reliability and cost-effectiveness ratio

Purpose The purpose of this study is to optimize the operational efficiency of the entire system by developing a reasonable maintenance strategy for wind turbines that improves component reliability and safety while reducing maintenance costs. Design/methodology/approach A hybrid incomplete preventive maintenance (PM) model based on boundary intensity process is established to give dynamic PM intervals for wind turbines using an iterative method with reliability as a constraint; the selection method of PM and replacement is given based on the cost-effectiveness ratio, which in turn determines the optimal number of PM for wind turbines. Findings The reliability is used to obtain the components’ maintenance cycle, and the cost-effectiveness ratio is used to select the number of maintenance times, thus, getting the optimal maintenance strategy. The validity of this paper’s method is verified by arithmetic cases, which provides a new method for formulating a reasonable PM strategy for wind turbines. Practical implications The wind turbine preventive maintenance strategy for Boundary intensity process proposed in this paper can scientifically formulate the maintenance strategy, optimize the cost-effectiveness per unit of time of the wind power generation system, and solve the problems of difficulty in formulating a reasonable maintenance strategy for the wind turbine components and high operation and maintenance costs. Originality/value In this paper, the authors describe the failure pattern by a Boundary intensity process, establish a hybrid incomplete PM model by introducing a failure intensity increment factor and an age reduction factor and establish a maintenance strategy optimization model with comprehensive consideration of reliability and cost-effectiveness ratio. Finally, the validity of the model in this paper is verified by arithmetic case analysis. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-05-2024-0153/

Thermo-hydraulic performance of concentric tube heat exchangers with turbulent flow: Predictive correlations and iterative methods for pumping power and heat transfer

This research addresses the problem of predicting the thermo-hydraulic performance of concentric tube heat exchangers (CTHE) under turbulent flow conditions, a critical aspect in energy-efficient industrial systems such as HVAC, power generation, and chemical processing. Existing studies often lack accurate predictive methods for balancing heat transfer performance with pumping power requirements. To tackle this issue, novel correlations and an iterative Newton–Raphson method were developed for predicting pumping power and heat transfer rates. Three-dimensional CFD simulations of a water-to-water counter-flow CTHE were conducted, with Reynolds numbers ranging from 4000 to 8000 for both the hot and cold fluids. The simulations employed the Reynolds-Averaged Navier–Stokes (RANS) equations with the k−ω SST turbulence model. The results demonstrated that increasing the Reynolds number enhances both heat transfer rates and pumping power, with the cold fluid requiring consistently higher pumping power. New correlations were developed to predict pumping power, capturing the impact of both entry and fully developed flow regions. These correlations showed an average error of less than 2.33% when compared with the CFD data. The iterative Newton–Raphson method for predicting heat transfer rates demonstrated high accuracy, with an average error of 0.66% for heat transfer rate, 0.03% for hot fluid outlet temperature, and 0.01% for cold fluid outlet temperature. Additionally, we identified optimal operating conditions for efficient cooling and heating based on the heat capacity ratio (Cr). The novelty of this work lies in the development of new, highly accurate predictive correlations and iterative methods for optimizing CTHE performance, going beyond existing literature by providing comprehensive insights into the relationship between pumping power, heat transfer efficiency, and flow conditions.

A Modified Iteration Method for an Inverse Problem of Diffusion Equation with Laplace and Riesz-Feller Space Fractional Operators

A Modified Iteration Method for an Inverse Problem of Diffusion Equation with Laplace and Riesz-Feller Space Fractional Operators

Matching Tabular Data to Knowledge Graph with Effective Core Column Set Discovery.

Matching tabular data to a knowledge graph (KG) is critical for understanding the semantic column types, column relationships, and entities of a table. Existing matching approaches rely heavily on core columns that represent primary subject entities on which other columns in the table depend. However, discovering these core columns before understanding the table’s semantics is challenging. Most prior works use heuristic rules, such as the leftmost column, to discover a single core column, while an insightful discovery of the core column set that accurately captures the dependencies between columns is often overlooked. To address these challenges, we introduce Dependency-aware Core Column Set Discovery ( DaCo ), an iterative method that uses a novel rough matching strategy to identify both inter-column dependencies and the core column set. Additionally, DaCo can be seamlessly integrated with pre-trained language models, as proposed in the optimization module. Unlike other methods, DaCo does not require labeled data or contextual information, making it suitable for real-world scenarios. In addition, it can identify multiple core columns within a table, which is common in real-world tables. We conduct experiments on six datasets, including five datasets with single core columns and one dataset with multiple core columns. Our experimental results show that DaCo outperforms existing core column set detection methods, further improving the effectiveness of table understanding tasks.

Accuracy and efficiency of 2D/3D single-vertebra spine navigation registration method based on dual-view feature fusion

Objective: To investigate the accuracy and efficiency of spine 2D/3D preoperative CT and intraoperative X-ray registration through a framework for spine 2D/3D single-vertebra navigation registration based on the fusion of dual-position image features. Methods: The preoperative CT and intraoperative anteroposterior (AP) and lateral (LAT) X-ray images of 140 lumbar spine patients who visited Huashan Hospital Affiliated to Fudan University from January 2020 to December 2023 were selected. In order to achieve rapid and high-precision single vertebra registration in clinical orthopedic surgery, a designed transformation parameter feature extraction module combined with a lightweight module of channel and spatial attention (CBAM) was used to accurately extract the local single vertebra image transformation information. Subsequently, the fusion regression module was used to complement the features of the anterior posterior (AP) and lateral (LAT) images to improve the accuracy of the registration parameter regression. Two 1×1 convolutions were used to reduce the parameter calculation amount, improve computational efficiency, and accelerate intraoperative registration time. Finally, the regression module outputed the final transformation parameters. Comparative experiments were conducted using traditional iterative methods (Opt-MI, Opt-NCC, Opt-C2F) and existing deep learning methods convolutional neural network (CNN) as control group. The registration accuracy (mRPD), registration time, and registration success rate were compared among the iterative methods. Results: Through experiments on real CT data, the image-guided registration accuracy of the proposed method was verified. The method achieved a registration accuracy of (0.81±0.41) mm in the mRPD metric, a rotational angle error of 0.57°±0.24°, and a translation error of (0.41±0.21) mm. Through experimental comparisons on mainstream models, the selected DenseNet alignment accuracy was significantly better than ResNet as well as VGG (both P<0.05). Compared to existing deep learning methods [mRPD: (2.97±0.99) mm, rotational angle error: 2.64°±0.54°, translation error: (2.15±0.41) mm, registration time: (0.03±0.05) seconds], the proposed method significantly improved registration accuracy (all P<0.05). The registration success rate reached 97%, with an average single registration time of only (0.04±0.02) seconds. Compared to traditional iterative methods [mRPD: (0.78±0.26) mm, rotational angle error: 0.84°±0.57°, translation error: (1.05±0.28) mm, registration time: (35.5±10.5) seconds], registration efficiency of the proposed method was significantly improved (all P<0.05). The dual-position study also compensated for the limitations in the single-view perspective, and significantly outperforms both the front and side single-view perspectives in terms of positional transformation parameter errors (both P<0.05). Conclusion: Compared to existing methods, the proposed CT and X-ray registration method significantly reduces registration time while maintaining high registration accuracy, achieving efficient and precise single vertebra registration.

Iterative Methods for Vecchia-Laplace Approximations for Latent Gaussian Process Models

Latent Gaussian process (GP) models are flexible probabilistic non-parametric function models. Vecchia approximations are accurate approximations for GPs to overcome computational bottlenecks for large data, and the Laplace approximation is a fast method with asymptotic convergence guarantees to approximate marginal likelihoods and posterior predictive distributions for non-Gaussian likelihoods. Unfortunately, the computational complexity of combined Vecchia-Laplace approximations grows faster than linearly in the sample size when used in combination with direct solver methods such as the Cholesky decomposition. Computations with Vecchia-Laplace approximations can thus become prohibitively slow precisely when the approximations are usually the most accurate, i.e., on large data sets. In this article, we present iterative methods to overcome this drawback. Among other things, we introduce and analyze several preconditioners, derive new convergence results, and propose novel methods for accurately approximating predictive variances. We analyze our proposed methods theoretically and in experiments with simulated and real-world data. In particular, we obtain a speed-up of an order of magnitude compared to Cholesky-based calculations and a threefold increase in prediction accuracy in terms of the continuous ranked probability score compared to a state-of-the-art method on a large satellite data set. All methods are implemented in a free C++ software library with high-level Python and R packages. 4

Variational Iteration Method for Solving First Kind Bessel Differential Equation of Order Zero

The aim of this paper is to study the first kind Bessel differential equation of order zero. The variational iteration method (VIM) is used to obtain the approximate solutions of first kind Bessel differential equation (BDE) of order zero. The approximate solution of this equation is calculated in the form of a series which its components are computed, can be calculated rapidlyon computer systems and the convergence of this method is studied. Also two numerical examples are given . The findings, which illustrate the effectiveness and precision of the suggested strategy, are presented in tables and figures.

Mathematical Analysis of a Navier–Stokes Model with a Mittag–Leffler Kernel

In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order β is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of β and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order β becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution and observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation.

PET image reconstruction using weighted nuclear norm maximization and deep learning prior.

The ill-posed Positron emission tomography (PET) reconstruction problem usually results in limited resolution and significant noise. Recently, deep neural networks have been incorporated into PET iterative reconstruction framework to improve the image quality. In this paper, we propose a new neural network-based iterative reconstruction method by using weighted nuclear norm (WNN) maximization, which aims to recover the image details in the reconstruction process. The novelty of our method is the application of WNN maximization rather than WNN minimization in PET image reconstruction. Meanwhile, a neural network is used to control the noise originated from WNN maximization. Our method is evaluated on simulated and clinical datasets. The simulation results show that the proposed approach outperforms state-of-the-art neural network-based iterative methods by achieving the best contrast/noise tradeoff with a remarkable contrast improvement on the lesion contrast recovery. The study on clinical datasets also demonstrates that our method can recover lesions of different sizes while suppressing noise in various low-dose PET image reconstruction tasks. Our code is available at https://github.com/Kuangxd/PETReconstruction.

Linear and Nonlinear Optical Properties of Symmetric and Asymmetric Double Triangular Quantum Dots Withinside the Presence of Magnetic Field

AbstractLinear and third‐order nonlinear optical absorption coefficients and relative refractive index changes in symmetric and asymmetric double triangular quantum dots are examined theoretically. The dependence of these optical properties on the magnetic field is examined. After calculating energies and wave functions within the effective mass and parabolic band approaches, analytical expressions of linear and nonlinear optical properties are obtained using the compact density matrix approach and iterative method. Numerical calculations are presented for typical GaAs/AlGaAs material. The results show that the magnetic field causes different effects on the and transitions. Moreover, the calculated results also reveal that the resonance frequency and nonlinear contribution are different in symmetric and asymmetric structures. As a result, it is concluded that the magnetic field plays a vital and important role in the electronic and optical properties of the system and can be used to tune the inter‐subband transitions and change the corresponding optical sensitivities.

Particle tracking-aided digital volume correlation for clay–sand soil mixtures

In this study, a novel, interdisciplinary method is introduced that merges fundamental geomechanics with computer vision to develop an advanced hybrid feature-aided digital volume correlation (DVC) technique. This technique is specifically engineered to measure and compute the full-field strain distribution in fine-grained soil mixtures. A clay–sand mixture specimen composed of quartz sand particles and kaolinite was created. Its mechanical properties and deformation behaviour were then tested using a mini-triaxial apparatus, combined with micro-focus X-ray computed tomography (μCT). The CT slices underwent image processing for denoising, segmentation of distinct phases, reconstruction of sand particles and feature extraction within the soil specimen. The proposed approach incorporated a two-step particle tracking method, which initially uses particle volume and surface area features to establish a preliminary matching list for a reference particle and then use the iterative closest point method for precise target particle matching. The soil specimen's initial displacement field was then mapped onto the DVC method's grid, and further refined through sub-voxel registration by way of a three-dimensional inverse compositional Gauss–Newton algorithm. The proposed method's effectiveness and efficiency were validated by accurately calculating the displacement and strain fields of the soil mixture sample, and comparing the results with those from a traditional DVC method. Given the soil's compositional and microstructural characteristics, these image-matching techniques can be integrated to create a versatile, efficient, and robust DVC system, suitable for a variety of soil mixture types.

Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm for MIMO Detection

In addressing the multifaceted problem of multiple-input multiple-output (MIMO) detection in wireless communication systems, this work highlights the pressing need for enhanced detection reliability under variable channel conditions and MIMO antenna configurations. We propose a novel method that sets a new standard for deep unfolding in MIMO detection by integrating the iterative conjugate gradient method with Tikhonov regularization, combining the adaptability of modern deep learning techniques with the robustness of classical regularization. Unlike conventional techniques, our strategy treats the Tikhonov regularization parameter, as well as the step size and search direction coefficients of the conjugate gradient (CG) method, as trainable parameters within the deep learning framework. This enables dynamic adjustments based on varying channel conditions and MIMO antenna configurations. Detection performance is significantly improved by the proposed approach across a range of MIMO configurations and channel conditions, consistently achieving lower bit error rate (BER) and normalized minimum mean square error (NMSE) compared to well-known techniques like DetNet and CG. The proposed method has superior performance over CG and other model-based methods, especially with a small number of iterations. Consequently, the simulation results demonstrate the flexibility of the proposed approach, making it a viable choice for MIMO systems with a range of antenna configurations, modulation orders, and different channel conditions.

A novel iterative algorithm for nonlinear inelastic dynamic analysis of truss structures; MCB-Spline+Sp time integration method

The focus of this work is on the development of a novel algorithm for the nonlinear dynamic analysis of structures. In the sake of a general and easy-to-implement solution method, the tangential stiffness matrix is calculated by combining the Cauchy-Riemann equations derived from the differentiation of complex functions and using the nodal displacement perturbation vector. An implicit direct time integration method based on a cubic B-Spline is integrated with the incremental-iterative method using a quadrature rule based on the interpolation of spline functions, which is referred to as the MCB-Spline+Sp time integration method. A step-by-step algorithm for developing a computer code to implement this method is provided. The effectiveness of the proposed approach is evaluated through several nonlinear time history analyses, which demonstrate its accuracy and efficiency by observing less computational efforts and fast convergence rate. Moreover, a comparison is made between the developed method and well-established techniques such as the Newmark and Standard-Bathe time integration methods in combination with the iterative Newton-Raphson method.

Iterative Dissipativity of Partial Difference Equation Dynamics in Open-Loop Iterative Learning Control Mode

Complex physical processes, which could evolve in both spatial and temporal dimensions and be represented by partial difference equations, could also operate in a repetitive mode with iterative learning methods as suitable control laws. For these three-dimensional systems (of the spatial, temporal, and iterative dimensions), the stability in the iterative direction is critical for many applications, which can be analyzed and synthesized under the proposed concept of iterative dissipativity. The definition of iterative dissipativity, which is first introduced in this paper, encapsulates the dominant information in both the spatial and temporal dimensions, while also placing a particular emphasis on the iteration improvement. This property allows for the derivation of sufficient conditions for asymptotic stability in the iteration direction, which are represented by linear matrix inequality criteria that can be readily solved. Performance in both the spatial and temporal dimensions can also be satisfied under this iterative dissipativity concept, even in absence of real-time feedback. Moreover, the optimization solutions of the control parameters can be determined. Finally, a thermal process and a numeric example are presented to illustrate the effectiveness of the proposed iteratively dissipative learning control approach.