Stationary Iterative Methods Research Articles

A generalized Poisson solver for first-principles device simulations

Electronic structure calculations of atomistic systems based on density functional theory involve solving the Poisson equation. In this paper, we present a plane-wave based algorithm for solving the generalized Poisson equation subject to periodic or homogeneous Neumann conditions on the boundaries of the simulation cell and Dirichlet type conditions imposed at arbitrary subdomains. In this way, source, drain, and gate voltages can be imposed across atomistic models of electronic devices. Dirichlet conditions are enforced as constraints in a variational framework giving rise to a saddle point problem. The resulting system of equations is then solved using a stationary iterative method in which the generalized Poisson operator is preconditioned with the standard Laplace operator. The solver can make use of any sufficiently smooth function modelling the dielectric constant, including density dependent dielectric continuum models. For all the boundary conditions, consistent derivatives are available and molecular dynamics simulations can be performed. The convergence behaviour of the scheme is investigated and its capabilities are demonstrated.

Numerical radiative transfer with state-of-the-art iterative methods made easy

This article presents an on-line tool and its accompanying software resources for the numerical solution of basic radiation transfer out of local thermodynamic equilibrium (LTE). State-of-the-art stationary iterative methods such as Accelerated Λ-iteration and Gauss–Seidel schemes, using a short characteristics-based formal solver are used. We also comment on typical numerical experiments associated to the basic non-LTE radiation problem. These resources are intended for the largest use and benefit, in support to more classical radiation transfer lectures usually given at the Master level.

A new nonlocal thermodynamical equilibrium radiative transfer method for cool stars

Context: The solution of the nonlocal thermodynamical equilibrium (non-LTE) radiative transfer equation usually relies on stationary iterative methods, which may falsely converge in some cases. Furthermore, these methods are often unable to handle large-scale systems, such as molecular spectra emerging from, for example, cool stellar atmospheres. Aims: Our objective is to develop a new method, which aims to circumvent these problems, using nonstationary numerical techniques and taking advantage of parallel computers. Methods: The technique we develop may be seen as a generalization of the coupled escape probability method. It solves the statistical equilibrium equations in all layers of a discretized model simultaneously. The numerical scheme adopted is based on the generalized minimum residual method. Result:. The code has already been applied to the special case of the water spectrum in a red supergiant stellar atmosphere. This demonstrates the fast convergence of this method, and opens the way to a wide variety of astrophysical problems.

An Acceleration Method for Stationary Iterative Solution to Linear System of Equations

An acceleration scheme based on stationary iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method which requires accurate estimation of the bounds for iterative matrix eigenvalues, we use a wide range of Chebyshev-like polynomials for the accelerating process without estimating the bounds of the iterative matrix. A detailed error analysis is presented and convergence rates are obtained. Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods are provided. AMS subject classifications: 65F10, 15A06

On the generalized shift-splitting preconditioner for saddle point problems

In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1, 1)-block and symmetric positive semidefinite (2, 2)-block. The proposed preconditioner is extracted from a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some properties of the eigenvalues distribution of the corresponding preconditioned matrix are studied. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioners.

Convergence of Inner-Iteration GMRES Methods for Rank-Deficient Least Squares Problems

We develop a general convergence theory for the generalized minimal residual method preconditioned by inner iterations for solving least squares problems. The inner iterations are performed by stationary iterative methods. We also present theoretical justifications for using specific inner iterations such as the Jacobi and SOR-type methods. The theory improves previous work [K. Morikuni and K. Hayami, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1--22], particularly in the rank-deficient case. We also characterize the spectrum of the preconditioned coefficient matrix by the spectral radius of the iteration matrix for the inner iterations and give a convergence bound for the proposed methods. Finally, numerical experiments show that the proposed methods are more robust and efficient compared to previous methods for some rank-deficient problems.

Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems

In this paper we propose a new method to apply the Generalized Cross-Validation (GCV) as a stopping rule for the Conjugate Gradient (CG). In general, to apply GCV to an iterative method, one must estimate the trace of the so-called influence matrix which appears in the denominator of the GCV function. In the case of CG, unlike what happens with stationary iterative methods, the regularized solution has a nonlinear dependence on the noise which affects the data of the problem. This fact is often pointed out as a cause of poor performance of GCV. To overcome this drawback, our proposal linearizes the dependence by computing the derivatives through iterative formulas. We compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation on both 1D and 2D test problems.

On necessary conditions for convergence of stationary iterative methods for Hermitian semidefinite linear systems

In an earlier paper (Frommer et al. (2008) [5]) we gave sufficient conditions in terms of an energy seminorm for the convergence of stationary iterations for solving linear systems whose coefficient matrix is Hermitian and positive semidefinite. In this paper we show in which cases these conditions are also necessary, and show that they are not necessary in others.

Acceleration of reinforcement learning by policy evaluation using nonstationary iterative method.

Typical methods for solving reinforcement learning problems iterate two steps, policy evaluation and policy improvement. This paper proposes algorithms for the policy evaluation to improve learning efficiency. The proposed algorithms are based on the Krylov Subspace Method (KSM), which is a nonstationary iterative method. The algorithms based on KSM are tens to hundreds times more efficient than existing algorithms based on the stationary iterative methods. Algorithms based on KSM are far more efficient than they have been generally expected. This paper clarifies what makes algorithms based on KSM makes more efficient with numerical examples and theoretical discussions.

A fast algorithm for matrix balancing

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration can give quadratic convergence at low cost.

A fast stationary iterative method for a partial integro-differential equation in pricing options

A jump-diffusion model for the pricing of options leads to a partial integro-differential equation (PIDE). Discretizing the PIDE by certain method, we get a sequence of systems of linear equations, where the coefficient matrices are Toeplitz matrices. In this paper, we decompose the coefficient matrix as the sum of a tridiagonal matrix and a near low-rank matrix, and approximate the near low-rank matrix by low-rank matrices. Then we introduce a stationary iterative method for the approximate systems of linear equations. Comparison of the performance of our algorithm to that proposed in Pang et al. (Linear Algebra Appl. 434:2325–2342, 2011) is presented.

An Acceleration Method for Stationary Iterative Solution to Linear System of Equations

AbstractAn acceleration scheme based on stationary iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method which requires accurate estimation of the bounds for iterative matrix eigenvalues, we use a wide range of Chebyshev-like polynomials for the accelerating process without estimating the bounds of the iterative matrix. A detailed error analysis is presented and convergence rates are obtained. Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods are provided.

Some results on convergence of two-stage iterative methods for symmetric positive definite linear systems

Parallel iterative algorithms for linear systems have become an important area in high performance computation. This paper studies two-stage iterative methods for symmetric positive definite linear systems. An estimate of minimum inner iterative number is obtained for convergence of stationary iterative methods. The convergent range of relaxation factor of relaxed two-stage iterative methods is decided by the minimums of eigenvalues of inner and outer iteration matrices.

Convergence Analysis of Preconditioned AOR Iterative Method for Linear Systems

M-(H-)matrices appear in many areas of science and engineering, for example, in the solution of the linear complementarity problem (LCP) in optimization theory and in the solution of large systems for real-time changes of data in fluid analysis in car industry. Classical (stationary) iterative methods used for the solution of linear systems have been shown to convergence for this class of matrices. In this paper, we present some comparison theorems on the preconditioned AOR iterative method for solving the linear system. Comparison results show that the rate of convergence of the preconditioned iterative method is faster than the rate of convergence of the classical iterative method. Meanwhile, we apply the preconditioner toH-matrices and obtain the convergence result. Numerical examples are given to illustrate our results.

On generalized stationary iterative method for solving the saddle point problems

In this paper, the generalized stationary iterative (GSI) method is studied for solving the saddle point problems. The convergence of this method is studied under suitable restrictions on iteration parameters. Numerical example for solving Stokes equation is presented to show the superiority of GSI method.

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

AbstractGeneral stationary iterative methods with a singular matrix M for solving range‐Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega–Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range‐Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M. Copyright © 2009 John Wiley & Sons, Ltd.

Projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport

We investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport. The matrices of the corresponding linear systems are symmetric with a positive semi-definite real part and an imaginary part with a compatible nullspace. We discuss well posedness, the symmetry of generalized inverses and Cholesky methods. We investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field.

Convergence and quotient convergence of iterative methods for solving singular linear equations with index one

Singular systems with index one arise in many applications, such as Markov chain modelling. In this paper, we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one. We give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems. Next we show that for the stationary iterative method, the convergence and the quotient convergence are equivalent.

On the nonlinear matrix equation [formula omitted

Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, we prove that the nonlinear matrix equation X - ∑ i = 1 m A i ∗ X δ i A i = Q ( 0 < | δ i | < 1 ) always has a unique positive definite solution. A conjecture which is proposed in [X.G. Liu, H. Gao, On the positive definite solutions of the matrix equation X s ± A T X - t A = I n , Linear Algebra Appl. 368 (2003) 83–97] is solved. Multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective.

Convergence of Stationary Iterative Methods for Hermitian Semidefinite Linear Systems and Applications to Schwarz Methods

A simple proof is presented of a quite general theorem on the convergence of stationary iterations for solving singular linear systems whose coefficient matrix is Hermitian and positive semidefinite. In this manner, elegant proofs are obtained of some known convergence results, including the necessity of the P-regular splitting result due to Keller, as well as recent results involving generalized inverses. Other generalizations are also presented. These results are then used to analyze the convergence of several versions of algebraic additive and multiplicative Schwarz methods for Hermitian positive semidefinite systems.