SSOR Research Articles

Numerical study of chemically reacting flows using a lower-upper symmetric successive overrelaxation scheme

A new computational fluid dynamics code has been developed for the study of mixing and chemical reactions in the flow fields of ramjets and scramjets. The code employs an implicit finite-volume, lower-upper symmetric successive overrelaxation (LU-SSOR) scheme for solving the complete two-dimensional Navier-Stokes equations and species transport equations in a fully coupled and very efficient manner. The combustion processes are modeled by an 8-species, 14-step, finite-rate chemistry model, and the turbulence is simulated by a Baldwin-Lomax algebraic model

On a class of preconditioned iterative methods on parallel computers

AbstractThe standard implementations of iterative solvers for finite element and finite difference methods frequently use a diagonal (Jacobi) preconditioner, particularly for element‐by‐element schemes. However, for such methods the actual order of the condition number with respect to mesh size is not reduced by the preconditioner. In the present paper we describe an iterative method where, in addition, the condition number is reduced by an order of magnitude. Moreover, the scheme may also be implemented as an element‐by‐element method. The method uses a generalized SSOR preconditioner and a wave front or multi‐frontal ordering of the mesh nodes. For a general irregular finite element mesh a striped irregular wave front ordering may be used. The performance of the method as well as various iterative acceleration techniques for a parallel computer are examined in the numerical studies.

Block SSOR preconditionings for high order 3d fe systems

For solving 3D high order hierarchical FE systems the block SSOR preconditioned CG algorithms based on new stripwise block two-color orderings of degrees of freedom and providing for efficient concurrent/vector implementation are suggested. As demonstrated by numerical results for the 3D Navier equations approximated using hierarchical orderp, 2 ≤p ≤ 5, FE's the convergence rate of such BSSOR-CG algorithms is only slightly dependent onp and mesh nonunformity.

Multisplitting with Different Weighting Schemes

Parallel algorithms generated by multisplittings are considered. A parallel algorithm may be formed, first, by concurrently executing the iteration associated with each splitting, and second, by forming a weighted sum of these computations. However, it is not imperative that the weighting be done last. Convergence results are obtained for a variety of other weighting schemes. In particular, it is shown that preweighting is in some cases more desirable than the traditional postweighting. Furthermore, we indicate how one can use a symmetric weighting scheme to obtain a good multisplitting version of the SSOR preconditioner. These algorithms are illustrated by computations done on an Alliant $\text{FX} /8$ .

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 PRECONDITIONED CONJUGATE GRADIENT METHOD ON THE CONNECTION MACHINE

This paper presents the results of the Connection Machine implementation of a number of preconditioners for the preconditioned conjugate gradient method. The preconditioners implemented include those based on the incomplete LU factorization, the modified incomplete LU factorization, the symmetric successive overrelaxation, and others such as several polynomial preconditioners and the hierarchical basis preconditioner. Results based on numerical experiments show that both the degree of parallelism inherent in a preconditioner and its convergence rate improvement play important roles on the overall execution time performance on parallel computers. Factors that affect the performance of the preconditioners will also be discussed. We conclude that to search for the best preconditioner on a parallel machine, we have to consider the tradeoffs between fast convergence rate and high degree of parallelism as well as the architecture of the target parallel computer.

An overview of NSPCG: A nonsymmetric preconditioned conjugate gradient package

The most recent research-oriented software package developed as part of the ITPACK Project is called “NSPCG” since it contains many nonsymmetric preconditioned conjugate gradient procedures. It is designed to solve large sparse systems of linear algebraic equations by a variety of different iterative methods. One of the main purposes for the development of the package is to provide a common modular structure for research on iterative methods for nonsymmetric matrices. Another purpose for the development of the package is to investigate the suitability of several iterative methods for vector computers. Since the vectorizability of an iterative method depends greatly on the matrix structure, NSPCG allows great flexibility in the operator representation. The coefficient matrix can be passed in one of several different matrix data storage schemes. These sparse data formats allow matrices with a wide range of structures from highly structured ones such as those with all nonzeros along a relatively small number of diagonals to completely unstructured sparse matrices. Alternatively, the package allows the user to call the accelerators directly with user-supplied routines for performing certain matrix operations. In this case, one can use the data format from an application program and not be required to copy the matrix into one of the package formats. This is particularly advantageous when memory space is limited. Some of the basic preconditioners that are available are point methods such as Jacobi, Incomplete LU Decomposition and Symmetric Successive Overrelaxation as well as block and multicolor preconditioners. The user can select from a large collection of accelerators such as Conjugate Gradient (CG), Chebyshev (SI, for semi-iterative), Generalized Minimal Residual (GMRES), Biconjugate Gradient Squared (BCGS) and many others. The package is modular so that almost any accelerator can be used with almost any preconditioner.

Parallel elliptic preconditioners: Fourier analysis and performance on the connection machine

We study the performance of several widely used preconditioners for 2D elliptic partial differential equations (SSOR, ILU, MILU and polynomial preconditioners) with the natural and red-black orderings implemented on the Connection Machine (CM). Their performance is primarily influenced by two factors: the rate of convergence and the ease of parallelization. The convergence rate is analyzed by Fourier analysis and confirmed with experimental results. Although the naturally ordered SSOR and MILU preconditioners have convergence rates one order faster than the other preconditioners, the experiments show that the red-black ordered SSOR, ILU, MILU, polynomial preconditioners takes less execution time than their naturally ordered counterparts. This is partially due to the fact that the red-blavk ordering provides more parallelism than the natural ordering.

Additive polynomial preconditioners for parallel computers

We present a polynomial preconditioner that can be used with the conjugate gradient method to solve symmetric and positive definite systems of linear equations. Each step of the preconditioning is achieved by simultaneously taking an iteration of the SOR method and an iteration of the reverse SOR method (equations taken in reverse order) and averaging the results. This yields a symmetric preconditioner that can be implemented on parallel computers by performing the forward and reverse SOR iterations simultaneously. We give necessary and sufficient conditions for additive preconditioners to be positive definite. We find an optimal parameter, ω, for the SOR-Additive linear stationary iterative method applied to 2-cyclic matrices. We show this method is asymptotically twice as fast as SSOR when the optimal ω is used. We compare our preconditioner to the SSOR polynomial preconditioner for a model problem. With the optimal ω, our preconditioner was found to be as effective as the SSOR polynomial preconditioner in reducing the number of conjugate gradient iterations. Parallel implementations of both methods are discussed for vector and multiple processors. Results show that if the same number of processors are used for both preconditioners, the SSOR preconditioner is more effective. If twice as many processors are used for the SOR-Additive preconditioner, it becomes more efficient than the SSOR preconditioner when the number of equations assigned to a processor is small. These results are confirmed by the Blue Chip emulator at the University of Washington.

A note on the SSOR and USSOR iterative methods applied top-cyclic matrices

The purpose of this note is threefold:i) to derive the new functional equation, $$\begin{gathered} \left[ {\lambda - \left( {1 - \omega } \right)\left( {1 - \hat \omega } \right)} \right]^p = \lambda ^k \left[ {\lambda \omega + \hat \omega - \omega \hat \omega } \right]^{\left| {\zeta _L } \right| - k} \left[ {\lambda \hat \omega + \omega - \omega \hat \omega } \right]^{\left| {\zeta _U } \right| - k} \hfill \\ \cdot \left( {\omega + \hat \omega - \omega \hat \omega } \right)^{2k} \mu ^p , \hfill \\ \end{gathered} $$ which couples the nonzero eigenvalues of the USSOR iteration matrix $$T_{\omega ,\hat \omega } $$ with the eigenvalues μ of the associated block Jacobi matrixB in thep-cyclic case,ii) to interpret the exponentk in this equation by means of graph theory, andiii) to connect the above equation with known results in the literature.

Optimum m-step SSOR preconditioning

This paper shows that if A is a symmetric positive definite matrix in red-black form, then the optimum relaxation factor for SSOR preconditioning of the conjugate gradient method is unity. This is true for any number of preconditioning steps.

Efficient Implementations of Certain Iterative Methods

We compare procedures given by Bank and Douglas [Appl. Numer. Math., 1 (1985), pp. 489–492], Conrad and Wallach [Numer. Math., 27 (1979), pp. 371–372], and Eisenstat [SIAMJ. Sci. Statist. Comput., 2 (1981), pp. 1–4] for reducing computation in certain preconditioned conjugate gradient or other iterative methods, for example, the SSOR iteration and SSOR preconditioning. The procedures are first formulated in a common framework and it is shown that, although different, they are more closely related than it first appears. Operation counts are given which show that the Bank–Douglas and Eisenstat procedures are essentially comparable, and are better than the Conrad–Wallach procedure for preconditioning.

A note on the SSOR convergence domain due to Neumaier and Varga

In a recent paper A. Neumaier and R. S. Varga show that if A is an n ×ncomplex H-matrix with 12 <ϱ(∣JA∣) < 1, then ϱ(SAω) < 1 for all 0 < ω <g̃w:=2/ (1 + √2ϱ(∣JA∣) − 1), where SAω and JA are, respectively, the symmetric successive overrelaxation (SSOR) iteration matrix and the Jacobi iteration matrix associated with A. We show that ϱ(SAg̃w) < 1. This allows us to also sharpen further results on the SSOR method for H-matrices due to Varga, Niethammer, and Cai.

On Vectorizing Incomplete Factorization and SSOR Preconditioners

We consider the problem of vectorizing the recursive calculations found in modified incomplete factorizations and SSOR preconditioners for the conjugate gradient method. We examine matrix problems derived from partial differential equations which are discretized on regular 2-D and 3-D grids, where the grid nodes are ordered in the natural ordering. By performing data dependency analyses of the computations, we show that there is concurrency in both the factorization and the forward and backsolves. The computations may be performed with an average vector length of $O(n)$ on an $n^2 $ or $n^3 $ grid in two and three dimensions. Numerical studies on four model problems show that the conjugate gradient method using these vectorized implementations of the modified incomplete factorizations and SSOR preconditioners achieves overall speeds approaching 100 megaflops on a Cray X-MP/24 vector computer. Furthermore, these methods require considerably less overall execution time than diagonal scaling and no-fill red-black incomplete factorization preconditioners, both of which allow full vectorization but are not as convergent.

A partial preconditioned conjugate gradient method for large eigenproblems

Abstract In this paper a preconditioned iterative method suitable for the solution of the generalized eigenvalue problem Ax = λBx is presented. The proposed method is suitable for the determination of extreme eigenvalues and their corresponding eigenvectors of large sparse eigenproblems derived from the finite element discretization method. The solution is obtained through the minimization of the Rayleigh quotient by a preconditioned conjugate gradient (CG) method. The proposed triangular splitting preconditioner combines Evans' SSOR preconditioner with a drop-off tolerance criterion and is called partial preconditioner (PPR). The PPR is attractive in a large FE framework because it is simple and can be implemented at the element level, as opposed to incomplete Choleski preconditioners, which require a sparse assembly. Because of the renewed interest in CG techniques for FE work on microprocessors and parallel computers it is believed that this improved approach to the generalized eigenvalue problem, through the minimization of the Rayleigh quotient, is likely to be very promising.

Neighborhoods of Dominant Convergence for the SSOR Method

Let A be an $n \times n$ nonsingular irreducible 3-cyclic H-matrix and let $J^A$, $L_\omega ^A$ and $S_\omega ^A $ denote, respectively, the Jacobi, the SOR, and the SSOR iteration matrices associated with A. In this paper we show that if the spectral radius $\rho ( | J^A | ) \in ( 0,r_0 )$, where $r_0$ is the unique root of the cubic $17r^3 + r^2 - r - 1$ in the interval (0, 1), then there exists a neighborhood $\Omega _{\omega ( A )} $ of $\omega ( A ) : = 2 /( 1 + \rho ( | J^A | )$ such that \[ \rho ( S_\omega ^A ) < | \omega - 1 | \leqq \rho ( L_\omega ^A )\quad \forall \omega \in \Omega _{\omega ( A )} . \]

Accelerating Vector Iteration Methods

This paper describes a technique for accelerating the convergence properties of iterative methods for the solution of large sparse symmetric linear systems that arise from the application of finite element method. The technique is called partial preconditioning process (PPR) and can be combined with pure vector iteration methods, such as the conjugate gradient, the dynamic relaxation, and the Chebyshev semi-iterative methods. The proposed triangular splitting preconditioner combines Evans’ SSOR preconditioner with a drop-off tolerance criterion. The (PPR) is attractive in a FE framework because it is simple and can be implemented at the element level as opposed to incomplete Cholesky preconditioners, which require a sparse assembly. The method, despite its simplicity, is shown to be more efficient on a set of test problems for certain values of the drop-off tolerance parameter than the partial elimination method.

An efficient implementation for SSOR and incomplete factorization preconditionings

We investigate methods for efficiently implementing a class of incomplete factorization preconditioners which includes Symmetric Gauss Seidel [9]. SSOR [9], generalized SSOR [1]. Dupont Kendall Rachford [4]. ICCG(0) [7], and MICCG(0) [6]. Our techniques can be extended to similar methods for nonsymmetric matrices.

Experiments in numerical methods for a problem in combustion modeling

In this paper, we have shown that the numerical method of lines can be used effectively to solve time dependent combustion models in one spatial dimension. By the numerical method of lines (NMOL), we mean the reduction of a system of partial differential equations to a system of ordinary differential equations (ODE's), followed by the solution of this ODE system with an appropriate ODE solver. We used finite differences for the spatial discretization and a variant of the GEAR package for the ODE's. We have presented various solution methods of interest for the nonlinear algebraic system in this setting; that is, in the corrector iteration section of the GEAR package applied to combustion models. These methods include Newton/block SOR (SOR denotes successive over-relaxation), block SOR/Newton, Newton/block-diagonal Jacobian, Newton/kinetics-only Jacobian, and Newton/block symmetric SOR. These methods have in common their lack of frequent use in ODE software and their eady applicability to partial differential equations in more than one spatial dimension. Finally, we have given the results of numerical tests, run on the CDC-7600 and Cray-1 computers. By so doing, we indicate the more promising nonlinear system solvers for the NMOL solution of combustion models.