Optimal Iteration Parameters Research Articles

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.

An estimation method for the state-of-charge of lithium-ion battery based on PSO-LSTM

The accuracy of state-of-charge (SOC) estimation will affect the performance of the battery management system. The higher the accuracy the better the performance. To improve the accuracy of SOC estimation, a particle swarm optimization (PSO) based method is proposed to optimize the long short term memory. First, a PSO-Long Short Term Memory (LSTM) estimation model is established by the PSO algorithm, thereby achieving optimal iteration parameters of the model. Then, the PSO-LSTM estimation model is simulated under different working conditions and temperatures. Finally, the voltage, current, and other discharge data of the lithium-ion battery are input into the PSO-LSTM neural network model to compare with the LSTM algorithm. The results show that the estimation accuracy of the optimized PSO-LSTM algorithm model and extended Kalman filter is 2.1% and 1.5%, respectively. The accuracy is improved.

On the Alternative SOR-like Iteration Method for Solving Absolute Value Equations

In this paper, by equivalently reformulating the absolute value equation (AVE) into an alternative two-by-two block nonlinear equation, we put forward an alternative SOR-like (ASOR-like) iteration method to solve the AVE. The convergence of the ASOR-like iteration method is established, subjecting to specific restrictions placed on the associated parameter. The selection of the optimal iteration parameter is investigated theoretically. Numerical experiments are given to validate the feasibility and effectiveness of the ASOR-like iteration method.

Assessing the robustness and scalability of the accelerated pseudo-transient method

Abstract. The development of highly efficient, robust and scalable numerical algorithms lags behind the rapid increase in massive parallelism of modern hardware. We address this challenge with the accelerated pseudo-transient (PT) iterative method and present a physically motivated derivation. We analytically determine optimal iteration parameters for a variety of basic physical processes and confirm the validity of theoretical predictions with numerical experiments. We provide an efficient numerical implementation of PT solvers on graphical processing units (GPUs) using the Julia language. We achieve a parallel efficiency of more than 96 % on 2197 GPUs in distributed-memory parallelisation weak-scaling benchmarks. The 2197 GPUs allow for unprecedented tera-scale solutions of 3D variable viscosity Stokes flow on 49953 grid cells involving over 1.2 trillion degrees of freedom (DoFs). We verify the robustness of the method by handling contrasts up to 9 orders of magnitude in material parameters such as viscosity and arbitrary distribution of viscous inclusions for different flow configurations. Moreover, we show that this method is well suited to tackle strongly nonlinear problems such as shear-banding in a visco-elasto-plastic medium. A GPU-based implementation can outperform direct-iterative solvers based on central processing units (CPUs) in terms of wall time, even at relatively low spatial resolution. We additionally motivate the accessibility of the method by its conciseness, flexibility, physically motivated derivation and ease of implementation. This solution strategy thus has a great potential for future high-performance computing (HPC) applications, and for paving the road to exascale in the geosciences and beyond.

A new two-step iteration method for discrete ill-posed problems and image restoration

In this study, for the augmented linear system of discrete ill-posed problems we establish a new two-step (NTS) iteration method containing a parameter and a parameter matrix, which is based on the Hermitian and skew-Hermitian splitting (HSS) and the upper and lower triangular splitting (ULT) of the coefficient matrix. Then, we theoretically study its convergence properties and determine its optimal iteration parameters. It is seen that the NTS method converges faster when the parameters are chosen properly. Experimental examples are carried out to further validate the effectiveness and accuracy of the new method compared to the newly developed methods in terms of the numerical performance and image recovering quality.

A modified SOR-like method for absolute value equations associated with second order cones

In this paper, we propose a modified SOR-like method for solving absolute value equations associated with second order cones (SOCAVE in short), which is obtained by reformulating the SOCAVE as a two-by-two block nonlinear equation. The convergence analysis and error estimation of this method are established under mild assumptions on system matrix and iteration parameters. And, the optimal iteration parameters and the corresponding optimal convergence factor are studied. In particular, we present the approximate optimal iteration parameters which are independent of the number of iterations. Numerical results are given to show the efficiency of the proposed iteration method with suitable parameters.

Modified special HSS method for discrete ill-posed problems and image restoration

ABSTRACT For solving ill-posed inverse problems, we establish a modified version of special Hermitian and skew-Hermitian splitting (SHSS) iteration method, called the modified SHSS (MSHSS) method in this paper. Based on an augmented system formulation, we apply the modified splitting to solve ill-posed inverse problems, and theoretically study its convergence and determine its optimal iteration parameters. Furthermore, the SOR acceleration scheme for MSHSS iterative method is discussed. Finally, numerical experiments arising from a Fredholm integral equation of the first kind and image restoration are used to further examine the effectiveness of the new method.

Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems

In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al. (2015), we establish the preconditioned AGSOR (PAGSOR) iteration method for solving a class of complex symmetric linear systems. The convergence conditions, optimal iteration parameters and corresponding optimal convergence factor of the PAGSOR iteration method are determined. Besides, we prove that the spectral radius of the PAGSOR iteration method is smaller than that of the AGSOR one under proper restrictions, and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting (PSBTS) one put forward by Zhang et al. (2018) recently. The spectral properties of the preconditioned PAGSOR matrix are also proposed. Numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual (GMRES) method.

Preconditioned Richardson iteration for augmented linear systems

For solving a class of augmented linear systems, we propose a new efficient iteration method, which is called preconditioned Richardson iteration (PR). Under suitable restrictions on the iteration parameters, we prove that the iterative sequences converge to the unique solution of the augmented linear system. Moreover, the optimal iteration parameters and the corresponding optimal convergence factor are discussed in detail. Numerical results show that the PR iteration method has an advantage over several other iteration methods by computing with the preconditioned GMRES methods from the point of view of iteration steps and CPU times.

Optimizing and improving of the C-to-R method for solving complex symmetric linear systems

For complex symmetric linear systems, Axelsson et al. (2014) proposed the C-to-R method. In this paper, by further studying the C-to-R method with W and T being symmetric positive semidefinite, the optimal iteration parameter for the C-to-R method α o p t = 2 2 2 is obtained and the C-to-R method is optimized. Furthermore, based on the optimized C-to-R method, we further propose an optimized preconditioner. Eigenvalue properties of the optimized preconditioned matrix are analyzed, which show that all the eigenvalues of the preconditioned matrix are located in tighter interval. Numerical results are presented, not only confirm the validity of the theoretical analysis, but also demonstrate the feasibility and effectiveness of the proposed optimized C-to-R method.

The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems

For the second-order cone linear complementarity problems, abbreviated as SOCLCPs, we establish two classes of modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently the SOCLCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The convergence of these modulus-based matrix splitting iteration methods has been established and the optimal iteration parameters of these methods are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving the SOCLCPs.

Modified PHSS iterative methods for solving nonsingular and singular saddle point problems

For large sparse saddle point problems, we establish a new version of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method, called the modified PHSS (MPHSS) method in this paper. Then, we theoretically study its convergence and semi-convergence properties and determine its optimal iteration parameter and corresponding optimal convergence factor. Furthermore, the spectral properties of the MPHSS preconditioned matrix are discussed in detail. Numerical experiments show that the MPHSS iteration method is effective and robust when it is used either as a solver or as a matrix splitting preconditioner for the generalized minimal residual (GMRES) method.

Parameterized approximate block LU preconditioners for generalized saddle point problems

In this paper, we are concerned with the iteration solution of generalized saddle point problems. Based on the exact block LU factorization of the coefficient matrix, we construct a class of parameterized approximate block LU factorization preconditioners, which rely on suitable approximations of the Schur complement of the (1,1) block of the coefficient matrix. Convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced. Algebraic characteristics of the related preconditioned matrices are discussed, including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial. The established results extend those of the approximate factorization and variants of the Hermitian and skew-Hermitian splitting and positive and skew-Hermitian splitting preconditioners for saddle point problems. Numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners.

The Semi-convergence of GSOR-like Methods for Singular Saddle Point Problems

In this paper, we prove the semi-convergence of the class of GSOR-like methods with two real functions applied to solve the singular saddle point problems, and discuss the optimal iteration parameters and the corresponding optimal semi-convergence factor for methods in the class.

A class of accelerated parameterized inexact Uzawa algorithms for complex symmetric linear systems

We establish a class of accelerated parameterized inexact Uzawa (APIU) algorithms for solving the complex symmetric linear systems. Our main contribution is accelerating the convergence of the PIU algorithm by making use of the extrapolation technique which is based on the eigenvalues of the iterative matrix. These accelerated parameterized inexact Uzawa algorithms involve two iteration parameters whose special choices can recover the parameterized inexact Uzawa algorithm and some other methods. First, the accelerated model for the PIU algorithm is established and the accelerated PIU algorithm is presented. Then we study the convergence of the corrected PIU algorithm. Moreover, we present the optimal iteration parameter and the corresponding optimal convergence factor for the PIU method. We also consider acceleration of the PIU iteraton by Krylov subspace methods. Numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method.

A class of upper and lower triangular splitting iteration methods for image restoration

• Two kinds of upper and lower triangular (ULT) splitting iteration methods are established. • The necessary and sufficient conditions for convergence of the ULT methods are derived. • The optimal iteration parameters are determined and corresponding convergence factors are described. • The ULT methods can outperform some recently proposed methods for image restoration. • The SOR acceleration scheme for the ULT iteration method is discussed. Based on the augmented linear system, a class of upper and lower triangular (ULT) splitting iteration methods are established for solving the linear systems arising from image restoration problem. The convergence analysis of the ULT methods is presented for image restoration problem. Moreover, the optimal iteration parameters which minimize the spectral radius of the iteration matrix of these ULT methods and corresponding convergence factors for some special cases are given. In addition, numerical examples from image restoration are employed to validate the theoretical analysis and examine the effectiveness and competitiveness of the proposed methods. Experimental results show that these ULT methods considerably outperform the newly developed methods such as SHSS and RGHSS methods in terms of the numerical performance and image recovering quality. Finally, the SOR acceleration scheme for the ULT iteration method is discussed.

Double-step scale splitting iteration method for a class of complex symmetric linear systems

In this paper, a double-step scale splitting (DSS) iteration method is proposed to solve a class of complex symmetric of linear equation. Unconditional convergence result is established under appropriate restrictions. Two reciprocal optimal iteration parameters and the corresponding optimal convergence factor are also determined. Numerical experiments on a few model problems are used to verify the effectiveness of the DSS iteration method.

On semi-convergence of the parameterized generalized MHSS method for singular complex linear systems

For solving a class of complex symmetric singular linear systems, we propose a parameterized generalized MHSS (PGMHSS) iteration method and investigate the semi-convergence conditions by analyzing the spectrum of the iteration matrix. Then we analyze the optimal iteration parameters that minimize the upper bound of the semi-convergence factor. Numerical experiments are used to test the feasibility and the effectiveness of the PGMHSS iteration method.

The parameterized upper and lower triangular splitting methods for saddle point problems

In this paper, we propose a class of parameterized upper and lower triangular splitting (denoted by PULTS) methods for solving nonsingular saddle point problems. The eigenvalues and eigenvectors of iteration matrix of the proposed iteration methods are analyzed. It is shown that the proposed methods converge to the unique solution of linear equations under certain conditions. Besides, the optimal iteration parameters and corresponding convergence factors are obtained with some special cases of the PULTS methods. Numerical experiments are presented to confirm the theoretical results, which implies that PULTS methods are effective and feasible for saddle point problems.

Two new variants of the HSS preconditioner for regularized saddle point problems

Two new preconditioners, which can be viewed as improved variants of the Hermitian and skew-Hermitian splitting (HSS) preconditioner, are presented for regularized saddle point problems. The unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed. Moreover, eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained. Compared with the HSS preconditioner, they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions. Numerical experiments from the Stokes problem are presented, which illustrates the advantages of the two preconditioners over the HSS preconditioner and some other preconditioners to accelerate the convergence rate of GMRES.