General Iterative Method Research Articles

General double-relaxation two-sweep modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

General double-relaxation two-sweep modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

Signal detection algorithms for massive MIMO system

As 5G communication networks are maturing, we have higher and higher requirements for the detection of communication signals. In this paper, for the Massive MIMO system signal detection problem, we mainly summarize the detection algorithms that can be used to replace the traditional ZF and MMSE, so as to avoid large-scale matrix inverse and reduce the computational complexity. It mainly includes the general iterative method, typically represented by SSOR, which makes the transmit signal matrix constantly close to the ideal value by iterating; the other is the level expansion class solution method, which takes the order expansion of the level as the initial value of the iteration to accelerate the convergence rate of the algorithm, typically represented by the MLI algorithm. However, today where the demand for communication is gradually increasing and the number of users is constantly getting larger, the performance of the above algorithms may degrade seriously, so the AI signal detection algorithm is a good alternative, which learns autonomously through deep neural networks, including model-driven and data-driven schemes.

A new matrix splitting generalized iteration method for linear complementarity problems

The linear complementarity problems (LCPs) can be encountered in various scientific computing, management science, and operations research. In this study, we introduce and analyze a new generalized accelerated overrelaxation (NGAOR) method for solving LCPs, in which one special case reduces to a new generalized successive overrelaxation (NGSOR) method. Moreover, we prove the convergence of the proposed methods when the system matrix is an H-matrix (irreducible or strictly diagonally dominant matrix). Numerical results for several experiments are present to show the effectiveness and efficiency of the proposed methods.AMS classification: 65F10, 90C33

Iterative decoupling procedure for the coupled 1D problems on example of closed cylindrical shell at static loading

The general iterative method of formal decoupling of two coupled 1D problems is suggested. Its essence consists in the alternative consideration of one of two problems as the main one, the aim of which is to produce the eigenfunctions. Then the other problem is considered as an auxiliary one, which is aimed to obtain the particular solution. The latter is presented as a linear combination of already known eigenfunctions, and then is substituted again into the first problem to refine the constants of main governing differential equation and to get the corrected eigenfunctions. The convergent iterative procedure of gradual refinement of the above constants is elaborated. The accuracy and technique of application is considered in details for the closed cylindrical shell problem, which is expanded in Fourier series with respect to circumferential coordinate. This actually leads to coupled 1D problems – one is the plane elasticity problem, and other – the plate problem. The results of calculation show the high accuracy of the method, which allows to get any required number of significant digits in eigenvalues. The comparison with known theoretical and FEM results for radial displacement and axial bending moment distribution at the vicinity of point of concentrated radial force application demonstrates the efficiency and accuracy of the method.

On the complexity of a unified convergence analysis for iterative methods

A local and a semi-local convergence of general iterative methods for solving nonlinear operator equations in Banach spaces is developed under ω-continuity conditions. Our approach unifies existing results and provides a new way of studying iterative methods. The main idea is to find a more accurate domain containing the iterates. No extra effort is used to obtain this. Also, the results of the numerical experiments are given that confirm obtained theoretical estimates.

Fast iterative reconstruction for multi-spectral CT by a Schmidt orthogonal modification algorithm (SOMA)

Multi-spectral CT (MSCT) is increasingly used in industrial non-destructive testing and medical diagnosis because of its outstanding performance like material distinguishability. The process of obtaining MSCT data can be modeled as a nonlinear system and the basis material decomposition comes down to the inverse problem of the nonlinear system. For different spectra data, geometric inconsistent parameters cause geometrical inconsistent rays, which will lead to the mismatched nonlinear system. How to solve the mismatched nonlinear equations accurately and quickly is a hot issue. This paper proposes a general iterative method (SOMA) to invert the mismatched nonlinear equations. The SOMA method gives different equations different confidence and searches along the more accurate hyperplane by Schmidt orthogonalization, which can get the optimal solution quickly. The validity of the SOMA method is verified by MSCT basis material decomposition experiments. The results show that the SOMA method can decompose the basis material images accurately and improve the convergence speed greatly.

Fast geolocation solution for bistatic interferometric synthetic aperture radar configuration of inclined geosynchronous transmitter with low earth orbit receivers

AbstractSpaceborne interferometric synthetic aperture radar (InSAR) technology is an effective method to obtain digital elevation model (DEM) data. The bistatic InSAR configuration of the inclined‐geosynchronous (InGEO) transmitter with low earth orbit (LEO) receivers (InGEO‐LEO) is a novel InSAR system to acquire terrain information. This novel system is characterised with high resolution, wide swath and timeliness, but the Newton iterative method is time‐consuming to solve the bistatic InSAR equations for fast DEM generation. For the conventional LEO bistatic InSAR system, the closed‐form solution is an effective method to improve the efficiency of solving InSAR equations, which is invalid in the InGEO‐LEO InSAR system because of the significant geometry difference caused by the orbits of InGEO transmitter and LEO receivers. To address this issue, we analyse the bistatic InGEO‐LEO geometry in detail and exploit the bistatic InSAR equations to propose an approximate closed‐form solution (ACS) for the novel system. Compared with the general Newton iterative method, the ACS significantly improves the efficiency of geolocation for the bistatic InGEO‐LEO InSAR system with high precision. Simulation experiments are carried out to verify the effectiveness and superiority of ACS.

The general tensor regular splitting iterative method for multilinear PageRank problem

<abstract><p>The paper presents an iterative scheme called the general tensor regular splitting iterative (GTRS) method for solving the multilinear PageRank problem, which is based on a (weak) regular splitting technique and further accelerates the iterative process by introducing a parameter. The method yields familiar iterative schemes through the use of specific splitting strategies, including fixed-point, inner-outer, Jacobi, Gauss-Seidel and successive overrelaxation methods. The paper analyzes the convergence of these solvers in detail. Numerical results are provided to demonstrate the effectiveness of the proposed method in solving the multilinear PageRank problem.</p></abstract>

A preconditioned general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

A preconditioned general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

Split monotone variational inclusion problem involving Cayley operators

Abstract In this article, we introduce a new kind of split monotone variational inclusion problem involving Cayley operator in the setting of infinite-dimensional Hilbert spaces. We develop a general iterative method to approximate the solution of the split monotone variational inclusion problem involving Cayley operator. Under some suitable conditions, a convergence theorem for the sequences generated by the proposed iterative scheme is established, which also solves certain variational inequality problems related to strongly positive linear operators. Finally, a numerical example is presented to study the efficiency of the proposed algorithm through MATLAB programming.

A General Picard-Mann Iterative Method for Approximating Fixed Points of Nonexpansive Mappings with Applications

Fixed point theory provides an important structure for the study of symmetry in mathematics. In this article, a new iterative method (general Picard–Mann) to approximate fixed points of nonexpansive mappings is introduced and studied. We study the stability of this newly established method which we find to be summably almost stable for contractive mappings. A number of weak and strong convergence theorems of such iterative methods are established in the setting of Banach spaces under certain geometrical assumptions. Finally, we present a number of applications to address various important problems (zero of an accretive operator, mixed equilibrium problem, convex optimization problem, split feasibility problem, periodic solution of a nonlinear evolution equation) appearing in the field of nonlinear analysis.

GIBS: a general and efficient iterative method for computing the approximate inverse and Moore–Penrose inverse of sparse matrices based on the Schultz iterative method with applications

In this paper, an algorithm is proposed to compute the inverse of an invertible matrix. The new algorithm is a generalization of the algorithms based on the well-known Schultz-type iterative methods. We show that the convergence order of the new method is a linear combination of the Fibonacci sequence and also is powerful and efficient in finding and keeping sparsity of the obtained approximate inverse of sparse matrices. The convergence of the algorithm is analysed and some applications are studied. It is shown that the proposed algorithm can be used for computing an approximation of the Moore–Penrose inverse of matrices. Numerical examples are provided to verify the feasibility and effectiveness of the new method.

General semi-implicit approximations with errors for common fixed points of nonexpansive-type operators and applications to Stampacchia variational inequality

It is meaningful and valuable to find common fixed points of different nonexpansive-type operators, which are associated with variational inequalities, integral equations, image process and other optimization problems in real life. The purpose of this paper is to suggest and consider a class of general semi-implicit iterative methods involving semi-implicit rule and inaccurate computing errors, which extend the iterative algorithm introduced by Ali et al. in 2020. Using Liu’s lemma, we analyze convergence and stability of the new iterative approximations for common fixed points of three different nonexpansive-type operators. Furthermore, we provide convergence rates of the new iterations and some numerical examples to illustrate the efficiency and stability of the new iterative schemes. As an application of our main results presented in this paper, we use the proposed iterative schemes to solve the known Stampacchia variational inequality.

Modified general policy iteration based adaptive dynamic programming for unknown discrete‐time linear systems

AbstractIn this article, the general policy iteration (GPI) method for the optimal control of discrete‐time linear systems is studied. First, the existing result on the GPI method is recalled and some new properties are proposed. Based on these new properties, a model‐based modified GPI algorithm is proposed with its convergence proof. Moreover, the data‐driven implementation for the proposed method is introduced without using the information of the system matrices. Compared with the existing results, the condition to initiate the GPI based algorithm is further relaxed by the proposed algorithm. The simulation example verifies the effectiveness of the proposed modified GPI based algorithm.

Spectral Coarse Spaces for the Substructured Parallel Schwarz Method

The parallel Schwarz method (PSM) is an overlapping domain decomposition (DD) method to solve partial differential equations (PDEs). Similarly to classical nonoverlapping DD methods, the PSM admits a substructured formulation, that is, it can be formulated as an iteration acting on variables defined exclusively on the interfaces of the overlapping decomposition. In this manuscript, spectral coarse spaces are considered to improve the convergence and robustness of the substructured PSM. In this framework, the coarse space functions are defined exclusively on the interfaces. This is in contrast to classical two-level volume methods, where the coarse functions are defined in volume, though with local support. The approach presented in this work has several advantages. First, it allows one to use some of the well-known efficient coarse spaces proposed in the literature, and facilitates the numerical construction of efficient coarse spaces. Second, the computational work is comparable or lower than standard volume two-level methods. Third, it opens new interesting perspectives as the analysis of the new two-level substructured method requires the development of a new convergence analysis of general two-level iterative methods. The new analysis casts light on the optimality of coarse spaces: given a fixed dimension m, the spectral coarse space made by the first m dominant eigenvectors is not necessarily the minimizer of the asymptotic convergence factor. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

A relaxation general two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems

For the linear complementarity problem, we introduce a relaxation general two-sweep matrix splitting iteration method. Convergence analysis shows that the method converges to the exact solution of the linear complementarity problem when the system matrix is an H+-matrix. Numerical experiments show that the proposed method is more efficient than existing methods.

Fast Geolocation Solution and Accuracy Analysis for Bistatic InSAR Configuration of Geostationary Transmitter With LEO Receivers

To obtain digital elevation model (DEM) data through spaceborne interferometric synthetic aperture radar (InSAR) technology, the solution of nonlinear InSAR geolocation equations is one of the most important steps. On one hand, Newton iterative method is time-consuming to solve the equations. On the other hand, the conventional closed-form solution for low earth orbit (LEO) bistatic InSAR system is not suitable for the novel InSAR configuration of geostationary (GEO) transmitter with LEO receivers because of the significant geometry difference. To address these issues, this letter analyzes the bistatic GEO-LEO geometry in detail and exploits the bistatic InSAR equations to propose a GEO-LEO fast location method (GFLM), which obtains the effective closed-form solution for the novel system. Compared with the general Newton iterative method, the GFLM significantly improves the efficiency of geolocation for the bistatic GEO-LEO InSAR system with high precision. Moreover, the geolocation accuracy of the novel system under realistic parameters uncertainties is introduced. Finally, we carry out simulation experiments to verify the effectiveness and superiority of GFLM.

Automotive Rubber Product Design Using Response Surface Method

In rubber bumper design one of the most important technical properties of the product is the force-displacement curve under a compression load, which is a highly nonlinear behavior because of the large deformation, the rubber material and the contacts. Finite element analysis is a good way to evaluate the working characteristics of the rubber part. Fulfillment of customer needs requires a general iterative design method where the objective can be reached with the modification of the product shape. The determination of the optimum requires numerous iterations of finite element analysis which is computationally expensive. With the integration of the Response Surface Method (RSM) into the design process of a two-variable shape optimization task, the optimal design can be achieved more efficiently. Four different Design of Experiments (DOE) methods were used to intelligently chose design points. As a metamodeling technique, Genetic Aggregation was selected to predict the relation between the sampled geometric variables and the nonlinear objective function value. The Nonlinear Programming by Quadratic Lagrangian (NLPQL) and the Mixed-Integer Sequential Quadratic Programming (MISQP) algorithms with different settings were tested to find the optimum of the response surface. As a result, the most accurate and efficient DOE method and optimization algorithm were determined. The introduced Response Surface Method-based optimization is proved to be suitable to determine the shape of the rubber jounce bumper, which meets the technical requirements.

Forecast of Waterway Cargo Turnover Volume Based on Genetic Algorithm to Optimize Neural Network Parameters

The traditional BP neural network is difficult to achieve the target effect in the prediction of waterway cargo turnover. In order to improve the accuracy of waterway cargo turnover forecast, a waterway cargo turnover forecast model was created based on genetic algorithm to optimize neural network parameters. The genetic algorithm overcomes the trap that the general iterative method easily falls into, that is, the “endless loop” phenomenon that occurs when the local minimum is small, and the calculation time is small, and the robustness is high. Using genetic algorithm optimized BP neural network to predict waterway cargo turnover, and the empirical analysis of the waterway cargo turnover forecast is carried out. The results obtained show that the neural network waterway optimized by genetic algorithm has a higher accuracy than the traditional BP neural network for predicting waterway cargo turnover, and the optimization model can long-term analysis of the characteristics of waterway cargo turnover changes shows that the prediction effect is far better than traditional neural networks.

A general fast shift-splitting iteration method for nonsymmetric saddle point problems

A general fast shift-splitting iteration method for nonsymmetric saddle point problems