Splitting Iteration 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 Relaxation Two-Step Newton-Based Matrix Splitting Iteration Method for Generalized Absolute Value Equations Associated with Circular Cones

A Relaxation Two-Step Newton-Based Matrix Splitting Iteration Method for Generalized Absolute Value Equations Associated with Circular Cones

Two-Step Simplified Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Two-Step Simplified Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Accelerated iterative splitting methods on Hadamard manifolds

This paper aims to solve the monotone inclusion problem, minimization problem of multiple summands and the generalized Heron problem. We present an innovative approach, the modified normal S-iteration method, designed to approximate common fixed points of nearly nonexpansive sequences and families of operators via the property ( A ) . Some deductions of our results improve some existing results in the literature. To show the applicability of our result, we give application to the inclusion problem via forward–backward splitting method version of our algorithm and minimization problem via Douglas–Rachford splitting method version of our algorithm. To demonstrate the practical utility of the algorithm, we apply it to the generalized Heron problem.

Unconditionally convergent [formula omitted] splitting iterative methods for variable coefficient Riesz space fractional diffusion equations

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a τ matrix from the coefficient matrix, and use their sum to construct a class of τ splitting iterative methods. Additionally, we design a preconditioner for the conjugate gradient method. Theoretical analyses show that the proposed τ splitting iterative methods are unconditionally convergent with convergence rates independent of step-sizes. Numerical results are provided to demonstrate the effectiveness of the proposed iterative methods.

Improved modulus-based matrix splitting iteration methods for a class of horizontal implicit complementarity problems

Recently, the modulus-based matrix splitting (MMS) iteration method has been successfully extended to solve the horizontal implicit complementarity problem (HICP). However, due to the existence of two system matrices and one nonlinear term, in addition to an equivalent fixed-point equation of the HICP, a system of linear equations has to be solved at each iteration to satisfy both the nonnegative constraint and the orthogonal constraint. This costs very expensive. In this paper, by introducing an additional judgement criterion after obtaining the approximate solution, a class of improved MMS (IMMS) iteration methods are proposed. By suitably choosing the judgement criterion, the IMMS iteration methods do not need to solve such system of linear equations at each iteration and thus greatly improves the computing efficiency. Convergence conditions of the IMMS iteration methods are studied when the system matrices are positive definite matrices and H+-matrices, respectively. Finally, two numerical examples are presented to show the effectiveness of the proposed IMMS iteration methods and their advantages over the existing MMS iteration methods when solving the HICP.

Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem

Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem

Modified two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

Modified two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

On the splitting iteration method for Pareto eigenvalue complementarity problems of H +-matrices

We present a class of inexact splitting-modulus iteration methods for solving the Pareto Eigenvalue Complementarity Problem (EiCP) when the system matrix A is an H + -matrix. Our method first employs the matrix splitting technique to transform the original eigenvalue complementarity problem (EiCP) into an easily solvable Linear Complementarity Problem (LCP), and the inner iterative process corresponds to the inexact split iterative process when the modulus iteration method is used, where we employed an effective and fast inner iterative stopping criterion. We then provide a convergence proof for inexact matrix splitting iteration methods. Numerical experiments demonstrate that our proposed algorithms exhibit significant superiority over the classical matrix splitting iteration method in terms of computational time, especially for solving large-scale symmetric and asymmetric eigenvalue complementarity problems of H + -matrices.

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

Two Block Splitting Iteration Methods for Solving Complex Symmetric Linear Systems from Complex Helmholtz Equation

In this paper, we study the improved block splitting (IBS) iteration method and its accelerated variant, the accelerated improved block splitting (AIBS) iteration method, for solving linear systems of equations stemming from the discretization of the complex Helmholtz equation. We conduct a comprehensive convergence analysis and derive optimal iteration parameters aimed at minimizing the spectral radius of the iteration matrix. Through numerical experiments, we validate the efficiency of both iteration methods.

Minimum residual NDSS iteration method for a class of complex symmetric linear systems

The double-step scale splitting and the minimum residual modified HSS iteration method are effective methods for a class of large sparse complex symmetric systems of linear equations. In this paper, we give an efficient method to solve the linear equations related to complex symmetric systems. Firstly, we modify the double-step scale splitting (DSS) iteration methods, obtaining the new double-step scale splitting (NDSS) iteration methods. Furthermore, we improve the algorithm by taking use of minimum residual technique. We call the modified algorithm minimum residual NDSS (MRNDSS) iteration method. For the new presented method, we study some properties of the algorithm carefully. Compared with the NDSS method, MRNDSS has two different parameters in each alternating iteration step and two additional parameters depending on the residual error from previous step. In our numerical experiments, our new method performs more efficient than four other iteration methods which are usually applied to solve the same problems.

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The projected splitting iterative methods based on tensor splitting and its majorization matrix splitting for the tensor complementarity problem

The projected splitting iterative methods based on tensor splitting and its majorization matrix splitting for the tensor complementarity problem

Modified CRI iteration methods for complex symmetric indefinite linear systems

This work investigates the iterative solution of complex symmetric linear systems with indefinite matrix term. Based on a technical equivalent reformulation of the original indefinite systems, an efficient stationary splitting iteration method is established by combining with its real and imaginary parts. In the view of the structure, the new method can be viewed as a modification of the combination method of the real and imaginary parts (CRI) in [Wang & Lu, J Comput Appl Math. 2017;325:188–197], which is originally designed for solving complex symmetric linear systems with positive semi-definite coefficient matrices. Despite that it is devoted to more difficult indefinite problems, the new method keeps the merits of unconditional convergence and parameter free implementation of the original method. Moreover, Chebyshev polynomial acceleration is considered to further improve the convergence rate of the new method. Numerical experiments on a set of model problems are tested to illustrate the efficiency of the proposed methods compared with some existing ones.

Two-parameter modified matrix splitting iteration method for Helmholtz equation

For effectively solving the nonsymmetric and indefinite linear system originating from the discretized Helmholtz equation with complex wavenumber, we propose a two-parameter modified matrix splitting iteration method. We establish the asymptotic convergence theory for this method and demonstrate its convergence under certain conditions. The proposed iteration method leads to a new preconditioner that can be efficiently inverted by using the discrete sine transform. Numerical experiments are carried out to show that the new preconditioner significantly improves the convergence rate of the GMRES method, which results in effective preconditioned GMRES method for solving the Helmholtz equation of high wavenumber.

The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

<abstract><p>In this paper, the simplified modulus-based matrix splitting iteration method was extended to solve the nonlinear complementarity problem, and the convergence conditions were presented from the spectral radius and the matrix norm. Then, for the special cases of this method, we provided some concrete convergence conditions as well as the quasi-optimal parameter matrix. Moreover, some numerical examples were illustrated to show the validity of the convergence results.</p></abstract>

Splitting iteration methods to solve non-symmetric algebraic Riccati matrix equation $$YAY-YB-CY+D=0$$

Splitting iteration methods to solve non-symmetric algebraic Riccati matrix equation $$YAY-YB-CY+D=0$$

A modified Newton-based matrix splitting iteration method for generalized absolute value equations

Many problems in scientific computing and engineering fields lead to the solution of generalized absolute value equations (GAVE). In this paper, by simultaneously splitting both coefficient matrices in the differential and non-differential parts of the GAVE, a modified Newton-based matrix splitting (MNMS) iteration method is proposed. The MNMS iteration method not only covers the well-known generalized Newton iteration method as well as the recent proposed modified Newton-based iteration method and the Newton-based matrix splitting iteration method for solving the GAVE, but also results in a series of relaxation versions that are very flexible in real applications. Convergence properties of the MNMS iteration method are studied in detail when the coefficient matrices are positive definite matrices and H+-matrices. Finally, three numerical examples are presented to show the feasibility and effectiveness of the proposed MNMS iteration method.

A two-step new modulus-based matrix splitting method for vertical linear complementarity problem

In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516).