Stationary Iterative Methods Research Articles

Topology optimization of nonlinear heat conduction problems involving large temperature gradient

In this work, we develop a topology optimization method for steady-state nonlinear heat conduction problems involving large temperature gradient to minimize the maximum structural temperature. Temperature-dependent material properties (TDMPs) are taken into account to break the widely-used assumption of constant material properties (CMPs) in conventional topology optimization. The Kreisselmeier–Steinhauser function is adopted as an aggregated measure of the maximum temperature over a specific region and the adjoint method is used to derive the sensitivity expressions. To effectively solve the well-known unsymmetric adjoint problem caused by material nonlinearity, an engineering-oriented stationary iterative method (ESIM) is constructed to transform the unsymmetric adjoint problem into a symmetric one with a series of right-hand sides that can be efficiently solved with mature linear system solvers. Both 2D and 3D numerical examples are provided to illustrate the validity and utility of the proposed method, including the accuracy and convergence of the ESIM. The results show that the assumption of CMPs can result in significant inaccuracy in the analysis and design of heat conduction systems working under large temperature gradient. On the contrary, by considering TDMPs and employing the proposed method, reasonable designs are achieved and thermal performance is greatly improved.

Convergence of a Class of Stationary Iterative Methods for Saddle Point Problems

A unified convergence theory is derived for a class of stationary iterative methods for solving linear equality constrained quadratic programs or saddle point problems. This class is constructed from essentially all possible splittings of the submatrix residing in the (1,1)-block of the augmented saddle point matrix that would produce non-expansive iterations. The classic augmented Lagrangian method and alternating direction method of multipliers are two special members of this class.

Approximate Schur-Block ILU Preconditioners for Regularized Solution of Discrete Ill-Posed Problems

High order iterative methods with a recurrence formula for approximate matrix inversion are proposed such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p=4k+3, where k≥1 is integer, are reduced through factorizations and nested loops in which the iterations are defined using a recurrence formula. Therefore, the computational cost is lowered from κ=4k+3 to κ=k+4 matrix multiplications per step. An algorithm is proposed to obtain regularized solution of ill-posed discrete problems with noisy data by constructing approximate Schur-Block Incomplete LU (Schur-BILU) preconditioner and by preconditioning the one step stationary iterative method. From the proposed methods of approximate matrix inversion, the methods of orders p=7,11,15,19 are applied for approximating the Schur complement matrices. This algorithm is applied to solve two problems of Fredholm integral equation of first kind. The first example is the harmonic continuation problem and the second example is Phillip’s problem. Furthermore, experimental study on some nonsymmetric linear systems of coefficient matrices with strong indefinite symmetric components from Harwell-Boeing collection is also given. Numerical analysis for the regularized solutions of the considered problems is given and numerical comparisons with methods from the literature are provided through tables and figures.

Shift-splitting preconditioners for a class of block three-by-three saddle point problems

In this paper, shift-splitting preconditioners are studied for a special class of block three-by-three saddle point problems, which arise from many practical problems and are different from the traditional saddle point problems. It is proved that the block three-by-three saddle point matrix is positive stable and the corresponding shift-splitting stationary iteration method is unconditionally convergent, which leads to a nice clustering property of the eigenvalues of the shift-splitting preconditioned matrix. Numerical results show that the proposed shift-splitting preconditioners outperform much better than some existing block diagonal preconditioners studied recently.

An effective stationary iterative method via double splittings of matrices

In this paper, a new stationary iterative method based on two double splittings of coefficient matrix is proposed to solve linear system Ax=b. Some convergence results and comparison theorems of the new method are provided. Theoretical analysis shows that this method is superior to some existing ones under certain conditions. Numerical examples are given to illustrate the theoretical results and examine the effectiveness of the proposed method.

An effective approach to implement the Maxwellian and non-Maxwellian distributions in the fluid simulation of solitary waves in plasmas

Solitary waves are commonly observed in both thermal and nonthermal plasmas. Thermal plasmas are generally represented by the Maxwellian distribution, whereas the nonthermal plasmas represented by the non-Maxwellian distributions, such as Kappa or Cairns distributions. Numerous theoretical fluid models have used these distributions to model the waves in space and laboratory plasmas. However, apart from our recent studies, there are no attempts to include these distributions in a fluid simulation of waves in plasma. In this paper, we present an efficient approach to deal with the stability and convergence of the Poisson solver in the fluid simulation of plasma that follows the Kappa, Maxwell, and Cairns distributions. We used the stationary iterative methods, namely, Jacobi (JA), Gauss–Seidel (GS) and Successive Over Relaxation (SOR) in the development of the Poisson solver. Our results show that the SOR method significantly improves the performance towards the stability and convergence of the Poisson solver for the plasma with all three-velocity distributions. The new fluid code with SOR Poisson solver is applied to examine the evolutionary characteristics of the ion acoustic solitary waves in a plasma and they are found to be in agreement with the fluid theory.

Regularized DPSS preconditioners for non-Hermitian saddle point problems

Based on a new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) of the coefficient matrix, a RDPSS preconditioner is introduced for non-Hermitian saddle point problems. From implementation aspects, the RDPSS preconditioner has better computing efficiency than the regularized Hermitian and skew-Hermitian preconditioner studied recently (Bai and Benzi, 2017). It is proved that the corresponding RDPSS stationary iteration method is convergent unconditionally. In addition, clustering property of the eigenvalues of the RDPSS preconditioned matrix is carefully studied.

Two-stage iterations based on composite splittings for rectangular linear systems

In this paper, we introduce a two-stage method to solve rectangular linear systems that exhibits faster convergence than typical stationary iterative methods. Under suitable conditions, we prove convergence of the new method. The number of outer iterations can be reduced by using a few significant number of inner iterations for efficient computations. Further, we perform a comparison analysis, and establish that a higher number of inner iterations ensures a smaller spectral radius of the global iteration matrix. We also discuss the uniqueness of a proper splitting, and illustrate different comparison theorems for different subclasses of proper splittings.

Efficient parameterized rotated shift-splitting preconditioner for a class of complex symmetric linear systems

By utilizing the equivalent real block two-by-two linear systems and the shift-splitting techniques, we establish an efficient parameterized rotated shift-splitting (PRSS) preconditioner for solving a class of complex symmetric linear systems. The proposed preconditioner is extracted from a stationary iteration method which is unconditionally convergent. Moreover, some spectral properties of the corresponding preconditioned matrix are studied in detail. Finally, numerical results are presented to show the feasibility and effectiveness of the proposed preconditioner.

Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations

Recently in [Journal of Computational Physics, 321 (2016), 829-907], an approach has been developed for solving linear system of equations with nonsingular coefficient matrix. The method is derived by using a delayed over-relaxation step (DORS) in a generic (convergent) basic stationary iterative method. In this paper, we first prove semi-convergence of iterative methods with DORS to solve singular linear system of equations. In particular, we propose applying the DORS in the Modified HSS (MHSS) to solve singular complex symmetric systems and in the Richardson method to solve normal equations. Moreover, based on the obtained results, an algorithm is developed for solving coupled matrix equations. It is seen that the proposed method outperforms the relaxed gradient-based (RGB) method [Comput. Math. Appl. 74 (2017), no. 3, 597-604] for solving coupled matrix equations. Numerical results are examined to illustrate the validity of the established results and applicability of the presented algorithms.

A new relaxed PSS preconditioner for nonsymmetric saddle point problems

A new relaxed PSS-like iteration scheme for the nonsymmetric saddle point problem is proposed. As a stationary iterative method, the new variant is proved to converge unconditionally. When used for preconditioning, the preconditioner differs from the coefficient matrix only in the upper-right components. The theoretical analysis shows that the preconditioned matrix has a well-clustered eigenvalues around (1, 0) with a reasonable choice of the relaxation parameter. This sound property is desirable in that the related Krylov subspace method can converge much faster, which is validated by numerical examples.

On the iterative refinement of the solution of ill-conditioned linear system of equations

ABSTRACTRecently, Salkuyeh and Fahim [A new iterative refinement of the solution of ill-conditioned linear system of equations, Int. Comput. Math. 88(5) (2011), pp. 950–956] have proposed a two-step iterative refinement of the solution of an ill-conditioned linear system of equations. In this paper, we first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations and study its convergence properties. Afterward, it is shown that the idea of an orthogonal projection technique together with a basic stationary iterative method can be utilized to construct a new efficient and neat hybrid algorithm for solving the mentioned problem. The convergence of the offered hybrid approach is also established. Numerical examples are examined to demonstrate the feasibility of proposed algorithms and their superiority to some of existing approaches for solving ill-conditioned linear system of equations.

On equivalence of optimal relaxed block iterative methods for the singular nonsymmetric saddle point problem

There exist many classes of relaxed block iterative methods for the solution of the nonsingular and singular saddle point problems. Recently, the singular nonsymmetric saddle point problem has been optimally solved by means of a stationary linear second-order iterative method using the Manteuffel algorithm [Hadjidimos (2016) [19]]. The main purpose of this work is to extend, analyze and study a number of classes of stationary iterative methods based on generalizations of SOR-like methods, determine their optimal parameters, via the optimal parameters in the aforementioned work, and show the equivalence of the optimal methods studied. Finally, a computational comparison of the performances of the above optimal methods and their nonstationary counterparts shows the superiority of the latter methods.

Iterative Methods for Solving the Nonlinear Matrix Equation <i>X</i>-<i>A</i>*<i>X</i><sup>p</sup><i>A</i>-<i>B</i>*<i>X</i><sup>-q</sup><i>B</i>=<i>I</i> (0<<i>p</i>,<i>q</i><1)

Consider the nonlinear matrix equation X-A*XpA-B*X-qB=I (0<p,q<1). By using the fixed point theorem for mixed monotone operator in a normal cone, we prove that the equation with 0<p,q<1 always has the unique positive definite solution. Two different iterative methods are given, including the basic fixed point iterative method and the multi-step stationary iterative method. Numerical examples show that the iterative methods are feasible and effective.

A shift-splitting preconditioner for a class of block two-by-two linear systems

In this paper, we construct a shift-splitting preconditioner for a class of block two-by-two linear systems. The proposed preconditioner is extracted from a stationary iterative method which is unconditionally convergent. Moreover, the eigenvalue distribution of the corresponding preconditioned matrix is studied. Numerical experiments are presented to show that our new preconditioner can be quite competitive when used to precondition Krylov subspace iterative methods such as GMRES.

A new relaxed HSS preconditioner for saddle point problems

We present a preconditioner for saddle point problems. The proposed preconditioner is extracted from a stationary iterative method which is convergent under a mild condition. Some properties of the preconditioner as well as the eigenvalues distribution of the preconditioned matrix are presented. The preconditioned system is solved by a Krylov subspace method like restarted GMRES. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioner.

The saddle point problem and the Manteuffel algorithm

In the last two decades, the augmented linear systems and the saddle point problems have been solved by many researchers who have used the conjugate gradient method or the generalized SOR iterative method and variants of them. In the latter class of methods, when the block \(A \in {\mathrm{I\!R}}^{m\times m}\) of the matrix coefficient \(\mathcal {A} = \left[ \begin{array}{cc} A &{} B \\ -B^T &{} 0\end{array}\right] \in {\mathrm{I\!R}}^{(m+n) \times (m+n)}\), \(m \ge n,\) of the linear system to be solved, is symmetric positive definite and \({\mathrm{rank}}(B) =r \le n\), convergence regions and optimal values of the parameters involved have been determined. In this work, we consider the block A to be nonsymmetric positive definite, \({\mathrm{rank}}(B) =r < n \), and use a two-level stationary iterative method whose main step is the linear second-order stationary iterative method for the solution of this class of problems. This method leads to the singular Manteuffel algorithm and the determination of its optimal parameters. As a byproduct, the optimal parameters of the Generalized Modified SSOR method in a particular case are also determined. Numerical examples verify our theoretical findings.

A new iterative method for solving a class of complex symmetric system of linear equations

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761---776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523---2538, 2013), when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.

Fast solution of elliptic partial differential equations using linear combinations of plane waves.

Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations Ax=b, where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O(NlogN) memory and executing an iteration in O(Nlog(2)N) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

Solutioin of Poisson's Equation in Electrostatic Particle-on-cell Simulation

In electrostatic Particle-in-Cell simulations of the HEMP-DM3a ion thruster the role of different solution strategies for Poisson?s equation was investigated. The direct solution method of LU decomposition is compared to a stationary iterative method, the successive over-relaxation solver. Results and runtime of solvers were compared, and an outlook on further improvements and developments is presented.