Symmetric Successive Overrelaxation Iteration Research Articles

Conjugate gradient method preconditioned with modified block SSOR iteration for multiplicative half-quadratic image restoration

Image restoration problem is often solved by minimizing a cost function which consists of data-fidelity terms and regularization terms. Half-quadratic regularization has the advantage that it can preserve image details well in the recovered images. In this paper, we consider solving the image restoration model which involves multiplicative half-quadratic regularization term. Newton method is employed to solve the nonlinear system of equations resulted from the optimization problem for image restoration. At each Newton iteration step, a linear system of equations with symmetric positive definite coefficient matrix arises. The preconditioned conjugate gradient method with the proposed modified block SSOR (symmetric successive over-relaxation) preconditioner is applied to solve this linear system of equations. The condition number of the preconditioned matrix is estimated and numerical experiments are also implemented for image restoration. Both theoretical and numerical results show that the modified block SSOR preconditioned PCG methods can greatly improve the computation efficiency when solving the multiplicative half-quadratic regularized image restoration problem.

A preconditioned SSOR iteration method for solving complex symmetric system of linear equations

We present a preconditioned version of the symmetric successive overrelaxation (SSOR) iteration method for a class of complex symmetric linear systems. The convergence results of the proposed method are established and conditions under which the spectral radius of the iteration matrix of the method is smaller than that of the SSOR method are analyzed. Numerical experiments illustrate the theoretical results and depict the efficiency of the new iteration method.

On SSOR iteration method for a class of block two-by-two linear systems

In this paper, the optimal iteration parameters of the symmetric successive overrelaxation (SSOR) method for a class of block two-by-two linear systems are obtained, which result in optimal convergence factor. An accelerated variant of the SSOR (ASSOR) method is presented, which significantly improves the convergence rate of the SSOR method. Furthermore, a more practical way to choose iteration parameters for the ASSOR method has also been proposed. Numerical experiments demonstrate the efficiency of the SSOR and ASSOR methods for solving a class of block two-by-two linear systems with the optimal parameters.

Convergence in Variance of Chebyshev Accelerated Gibbs Samplers

A stochastic version of a stationary linear iterative solver may be designed to converge in distribution to a probability distribution with a specified mean $\mu$ and covariance matrix $A^{-1}$. A common example is Gibbs sampling applied to a multivariate Gaussian distribution which is a stochastic version of the Gauss--Seidel linear solver. The iteration operator that acts on the error in mean and covariance in the stochastic iteration is the same iteration operator that acts on the solution error in the linear solver, and thus both the stationary sampler and the stationary solver have the same error polynomial and geometric convergence rate. The polynomial acceleration techniques that are well known in numerical analysis for accelerating the linear solver may also be used to accelerate the stochastic iteration. We derive first-order and second-order Chebyshev polynomial acceleration for the stochastic iteration to accelerate convergence in the mean and covariance by mimicking the derivation for the linear solver. In particular, we show that the error polynomials are identical and hence so are the convergence rates. Thus, optimality of the Chebyshev accelerated solver implies optimality of the Chebyshev accelerated sampler. We give an algorithm for the stochastic version of the second-order Chebyshev accelerated SSOR (symmetric successive overrelaxation) iteration and provide numerical examples of sampling from multivariate Gaussian distributions to confirm that the desired convergence properties are achieved in finite precision.

A new modified SSOR iteration method for solving augmented linear systems

In this paper, we present a new symmetric successive over-relaxation method to find solution of the large sparse augmented linear systems, which is the extension of the symmetric successive overrelaxation iteration method. The convergence analysis of our method is also studied. Furthermore, the functional equation between the parameters and the eigenvalues of the relevant iteration matrix is obtained. Finally, numerical computations are presented to show the effectiveness of our algorithm.

Symmetric successive overrelaxation methods for solving the rank deficient linear least squares problem

Symmetric successive overrelaxation (SSOR) methods are developed for finding the least squares solution of minimal norm to the linear system Ax = b , where A ∈ C m × n is a matrix of rank r and b ∈ C m . By first augmenting the system to a block 4 × 4 consistent system and then splitting the augmented coefficient matrix by subproper SSOR splittings, we obtain intervals for the relaxation parameter in which the subproper SSOR iteration matrix is semiconvergent and a solution for the augmented system. Subsequently, a method for transforming such a solution to the least squares solution of minimal 2-norm is presented.

Power control algorithm in CDMA systems using symmetric successive overrelaxation iteration

System capacity is the paramount requirement in the design of code division multiple access (CDMA) cellular systems. Power control to mobiles is an essential tool used to maximise the system capacity by minimising interference while at the same time maintaining transmission quality. The convergence speed of power control algorithm is critical in determining its practical application especially when the propagation and traffic condition are rapidly changing. In this letter, a new centralised power control algorithm is proposed to solve the power control problem by using symmetric successive overrelaxation (SSOR) preconditioned generalised product-type bi-conjugate gradient (GPBi-CG) iteration method. The simulation results show that our proposed algorithm has much faster convergent rate than the distributive constrained second-order power control (CSOPC) and constrained Gauss-Seidel (CGS) algorithm. The fast convergent speed can increase the overall network capacity. Copyright © 2005 AEIT.

Calculating Flux Perturbations with a Davidson Algorithm

In perturbation calculations, obtaining an accurate flux shape of a perturbed core is more difficult than the multiplication factor. Generalized Davidson algorithms using a symmetric successive overrelaxation preconditioner are developed to solve the unperturbed eigenvalue problem and the related perturbed eigenvalue problem of large sparse matrices. The bases of the subspace obtained from the sequence of solving the unperturbed problem through the algorithm can be used in the perturbed problem to save computational time. One- and two-dimensional test problems indicate that by incorporating symmetric successive overrelaxation iteration, the optimized relaxation factor, and the newly developed shifted form-function vector method for a large perturbation, a considerable amount of computational time can be saved in the perturbed calculations with accuracy comparable to the existing CITATION code. This method also provides an efficient means for survey calculations where the requirement of accuracy is not stringent.

Superior convergence domains for the p-cyclic SSOR majorizer

Thus far for an n × n complex nonsingular matrix A, the symmetric successive overrelaxation (SSOR) majorizing operator has been used to establish convergence properties of the SSOR method mostly in the case where A is an H-matrix. In this paper we use (actually a similarity transformation of) the SSOR majorizer to investigate convergence properties of the block SSOR method when A is a block p-cyclic matrix. Let J A denote the block Jacobi method and let ν = ϱ(¦J A¦) . We establish regions in the ( ν, ω)-plane where ϱ( S ω A) ⩽ ϱ( Q ω A) < ¦ω − 1¦ [⩽ ϱ( L ω A)] . Here S ω A is the block SSOR iteration operator associated with A, L ω A is the block successive overrelaxation (SOR) iteration operator associated with A, and Q ω A is a convenient similarity transformation of the majorizing operator for S ω A . Of special interest to us are the values of ν for which the above inequality holds for the corresponding values of the relaxation parameter ω(A) = 2 (1 + ν) , the latter being an important quantity in the SOR-SSOR theory for H-matrices.

Convergence domains of the SSOR method for a class of generalized consistently ordered matrices

The functional equation relating the eigenvalues of the block Symmetric Successive Overrelaxation (SSOR) iteration matrix with those of the block Jacobi iteration matrix found by Chong and Cai (1985) is used in order to obtain precise domains of convergence of the block SSOR iteration method associated with a class of generalized consistently ordered (GCO) ( k, p− k)-matrices A( p⩾2, k=1, …, p−1). We show that the domain of convergence depends only on the relaxation parameter ω, the spectral radius of the block Jacobi iteration matrix and the value of l = k p . Unlike the case ( k, p− k) = (1, p−1), p⩾3, which was studied in an earlier paper by the authors (1989), beyond certain critical values of l the domain of convergence does not grow monotonically with l and manifests various sorts of behavior. However, as in the case ( k, p− k) = (1, p−1), it is shown that the intersection of the convergence domains taken over all pairs ( p, k) coincides with the exact domain of convergence of the point SSOR iteration method associated with nonsingular H-matrices A, that is, matrices which can be diagonally scaled to being strictly diagonally dominant.

Precise domains of convergence for the block SSOR method associated with p-Cyclic Matrices

We apply Rouche's theorem to the functional equation relating the eigenvalues of theblock symmetric successive overrelaxation (SSOR) matrix with those of the block Jacobi iteration matrix found by Varga, Niethammer, and Cai, in order to obtain precise domains of convergence for the block SSOR iteration method associated withp-cyclic matricesA, p≥3. The intersection of these domains, taken over all suchp's, is shown to coincide with the exact domain of convergence of thepoint SSOR iteration method associated withH-matricesA. The latter domain was essentially discovered by Neumaier and Varga, but was recently sharpened by Hadjidimos and Neumann.

Precise domains of convergence for the block SSOR method associated withp-cyclic matrices

We apply Rouche's theorem to the functional equation relating the eigenvalues of theblock symmetric successive overrelaxation (SSOR) matrix with those of the block Jacobi iteration matrix found by Varga, Niethammer, and Cai, in order to obtain precise domains of convergence for the block SSOR iteration method associated withp-cyclic matricesA, p≥3. The intersection of these domains, taken over all suchp's, is shown to coincide with the exact domain of convergence of thepoint SSOR iteration method associated withH-matricesA. The latter domain was essentially discovered by Neumaier and Varga, but was recently sharpened by Hadjidimos and Neumann.

The S.S.O.R. Iteration Scheme for Equations with 1 Ordering

The symmetric successive over-relaxation iteration scheme is compared with successive over-relaxation for equations with σ1 ordering.