Nonsymmetric Linear Systems Research Articles

Solution of Linear Equation Systems with a Dominant Skew-symmetric Part Using the Product Triangular Iterative Method

AbstractA new product triangular iterative methods for solving nonsymmetric linear systems of equations with positive real coefficient matrices based on the skewsymmetric part of the initial matrix are proposed. The convergence analysis for the new methods is presented.

Circulant-block preconditioners for solving ordinary differential equations

Boundary value methods for solving ordinary differential equations require the solution of non-symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A circulant-block preconditioner is proposed to speed up the convergence rate of the GMRES method. Theoretical and practical arguments are given to show that this preconditioner is more efficient than some other circulant-type preconditioners in some cases. A class of waveform relaxation methods is also proposed to solve the linear systems.

A class of product-type Krylov-subspace methods for solving nonsymmetric linear systems

In the present paper, a class of product-type Krylov-subspace methods for solving nonsymmetric linear systems is discussed. A characteristic of this class is the relationship r n=H n(A) r n BCG where r n is the residual vector corresponding to the nth iterate x n , and r n BCG is the nth residual generated in bi-conjugate gradient method. The polynomial H n is chosen to speed up and stabilize convergence, while satisfying standard three-term recurrence relations. These product-type methods can be regarded as unifications and generalizations of CGS and Bi-CGSTAB.

GRSIM: A FORTRAN subroutine for the solution of non‐symmetric linear systems

AbstractA FORTRAN subroutine called GRSIM subroutine is presented for the iterative solution of a set of non‐symmetric linear equations, Ax = b, where the coefficient matrix A is a sparse nearly symmetric structured M‐matrix. These matrices occur repeatedly in the finite‐difference solution of partial differential equations. The method solves non‐symmetric systems of linear equations, but uses highly developed techniques for the solution of symmetric systems of linear equations. A general description of the method, which is based on particular class of regular splitting, is given. The GRSIM subroutine uses a regular splitting and the convergence is, therefore, guaranteed. Copyright © 2002 John Wiley & Sons, Ltd.

Parallel algorithms for solving tridiagonal and near-circulant systems

Many problems in mathematics and applied science lead to the solution of linear systems having circulant coefficient matrices. This paper presents a new stable method for the exact solution of non-symmetric tridiagonal circulant linear systems of equations. The method presented in this paper is quite competitive with Gaussian elimination both in terms of arithmetic operations and storage requirements. It is also competitive with the modified double sweep method. This method can be applied to solve the near-circulant tridiagonal system. In addition, the method is modified to allow for parallel processing.

A Seed Method for Solving Nonsymmetric Linear Systems with Multiple Right-hand Sides

We present a seed method for solving large nonsymmetric linear systems with multiple right-hand sides. The method uses a single augmented Krylov subspace corresponding to a seed system as a generator of approximations to the nonseed systems. The residual evaluate of the method is shown, and a new strategy to form a seed system which could supply information shareable among the right-hand sides is given. Numerical experiments indicate that our seed selection strategy is more efficient than two existing strategies and our method has significant time saving compared with the block GMRES method and the GMRES method with a projection process.

SCALABILITY ANALYSIS OF PARALLEL GMRES IMPLEMENTATIONS

Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k). Two parallel versions of GMRES(k) based on different data distributions and using Householder reflections in the orthogonalization phase are analyzed with respect to scalability (their ability to maintain fixed efficiency with an increase in problem size and number of processors). A theoretical algorithm-machine model for scalability of GMRES(k) with fixed k is derived and validated by experiments on three parallel computers, each with different machine characteristics. The analysis for an adaptive version of GMRES(k), in which the restart value k is adapted to the problem, is also presented and scalability results for this case are briefly discussed.

LU-decomposition and numerical structure for solving large sparse nonsymmetric linear systems

In this work, the solution of a large sparse linear system of equations with an arbitrary sparsity pattern is obtained by using LU-decomposition method as well as numerical structure approach. The LU-decomposition method is based on Doolittle's method while the numerical structure approach is based on Cramer's rule. The numerical structure approach produces direct solution without facing fill-in problems as encountered in LU-decomposition. In order to reduce the ‘fill-ins’ in the decomposition, the powers of a Boolean matrix, obtained from the coefficient matrix A are taken so that the ‘fill-ins’ in the structure of A can be known in advance. The position of fill-ins in A are thus determined in the best choice manner, that is, it is very effective and memory-wise cheap. We also outline a method by using numerical structure with reduced computation efforts. Finally, experiments are performed on eight examples to compare the efficiency of the proposed methods. The results obtained are reported in a table. It is found that the LU-decomposition method is much better than numerical structure. The usefulness of numerical structure approach is also discussed.

A class of optimized row projection methods for solving large nonsymmetric linear systems

We present in this paper optimal and accelerated row projection algorithms arising from the use of quadratic programming, that allow us to define the iterate x k+1 as the projection of x k onto a hyperplane which minimizes its distance to the solution x ∗ . These algorithms also use a novel partition strategy into blocks based on sequential estimations of their condition numbers.

Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations

A new class of triangular iterative methods for solving nonsymmetric linear systems of equations with strongly nonsymmetric positive real matrices is proposed. The triangular operator of these iterative methods uses the skew-symmetric part of the initial matrix. For the new methods, a convergence analysis, a technique for choosing the optimal parameter, and an accelerating procedure are presented. Several numerical experiments include the solution of the strongly nonsymmetric linear systems arising from a central finite-difference approximation of the steady convection–diffusion equation with the Peclet numbers Pe=10 3, 10 4, and 10 5. The relative performance of these methods is compared to the popular SOR procedure.

On performance of SOR method for solving nonsymmetric linear systems

The main subject of the paper is the study of the performance of SOR algorithms for solving linear systems of the type arising from the difference approximation of nonself-adjoint two-dimensional elliptic partial differential equations. A special attention is paid to the development of efficient techniques for determining the optimum relaxation parameter providing the maximum rate of convergence. Four models of the behaviour of the spectral radius of the SOR matrix as a function of relaxation parameter are analyzed. Numerical experiments are performed for several problems with nonsymmetric coefficient matrices taken from the literature. A comparison of results of the line-SOR method with the results obtained from different GMRES algorithms shows that with the computational work comparable for both methods, the line-SOR method provides the solutions of considered problems with the second norm of the error vector a few orders lesser in the magnitude.

Is a Chebyshev method optimal for an elliptic region also optimal for a nearly elliptic region?

The asymptotic rate of convergence of an optimal Chebyshev semiiterative method for solving a real and nonsymmetric linear system x =T x + c can be improved by the related (2,2)-step iterative methods under certain conditions. The condition for which a Chebyshev method asymptotically optimal for an elliptic region is also asymptotically optimal for a nearly elliptic region is presented. Thus a (2,2)-step method is asymptotically superior to the Chebyshev method asymtotically optimal for a nearly elliptic region under certain conditions. A numerical example illustrates our results.

Preconditioners for domain decomposition methods

Within the FETI domain decomposition method applied to nonsymmetric linear systems, a generic set of algebraic parallel global preconditioners is presented. They can be applied to most problems with a saddle-point formulation. An industrial test on a fluid mechanic matrix shows the independence of these preconditioners with the number of subdomains.

DGMRES: A GMRES-type algorithm for Drazin-inverse solution of singular non-symmetric linear systems

In a recent work by the author [Linear Algebra Appl. 298 (1999) 99] Krylov subspace methods were derived for Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b, where A∈ C N×N is a singular and in general non-hermitian matrix that has arbitrary index. One of these methods, modeled after the Generalized Conjugate Residual method (GCR) and denoted DGCR, is considered in the present work again. It is shown that all of the approximations produced by DGCR exist, and a GMRES like algorithm, denoted DGMRES, for its implementation is derived. Like GMRES, DGMRES too is economical computationally and storagewise.

Can SOR be an efficient method for solving nonsymmetric linear systems?

Can SOR be an efficient method for solving nonsymmetric linear systems?

Variational principle for two-dimensional high Hartmann number flow

A variational principle for the stream function-vorticity form of the linearised two-dimensional MHD equations which incorporates the no-slip condition as a natural boundary condition has been used to solve a range of problems. Finite element discretisations leading to sparse non-symmetric linear systems have been solved by a corrected version of a standard non-symmetric frontal solver. The numerical solutions obtained for flows over an unbounded cylinder at the larger Hartmann numbers agree with recently obtained theoretical asymptotic estimates. Other calculations for bounded flows in channels with or without obstructions are also in agreement with previous results obtained by matched asymptotic expansion methods.

A block GMRES method augmented with eigenvectors

The GMRES method augmented with some eigenvectors is extended to a block version for solving large nonsymmetric linear systems with multiple right-hand sides. Residual bound is established which has an improved convergence rate on the standard block GMRES method. This improvement displays a good efficiency when restarting is used. Also an improved residual norm estimate is derived by using matrix-valued polynomials. Numerical experiments show that the new algorithm is more efficient, compared with the block GMRES method.

Free surface flow through rock-fill dams analyzed by FEM with level set approach

A stabilized-finite element formulation is coupled with a level set technique for computations of incompressible non-linear flow with interfaces between two immiscible fluids. An interface capturing formulation (ICF) for non-linear, free surface, seepage flow in rock-fill dams is proposed. The formulation is derived for two- and three-dimensional flow within a fixed mesh domain. The resulting formulation is general and applicable for various steady and transient two-phase flow problems. FE-refinement is processed for the entire fixed mesh domains. A general solver is also reviewed for large and non-symmetric non-positive definite linear system of equations with the GMRES-update technique based on a Newton-iterative method. The computational procedure has been implemented in MATLAB. A comparison is performed between the 2-D computed test problem for coarse and refined meshes together with some proposed analytical solutions for nonlinear seepage flow with free surface in rock-fill dams. An expansion of the 2-D program code to a 3-D one for a rectangular rock-fill dam is also developed and simulated in MATLAB. The performance of the computations in 3-D is very promising and its opening the future for possible industrial applications using the same simple technique. Computations for a simple 3-D seepage flow problem with free surface in rock-fill dam are included in present paper. A general mesh generator and solver for large scale and complex 3-D flow problems in a real embankment dam is also under construction in C++.

High Performance Computations for Large Scale Simulations of Subsurface Multiphase Fluid and Heat Flow

TOUGH2 is a widely used reservoir simulator for solving subsurface flow related problems such as nuclear waste geologic isolation, environmental remediation of soil and groundwater contamination, and geothermal reservoir engineering. It solves a set of coupled mass and energy balance equations using a finite volume method. This contribution presents the design and analysis of a parallel version of TOUGH2. The parallel implementation first partitions the unstructured computational domain. For each time step, a set of coupled non-linear equations is solved with Newton iteration. In each Newton step, a Jacobian matrix is calculated and an ill-conditioned non-symmetric linear system is solved using a preconditioned iterative solver. Communication is required for convergence tests and data exchange across partitioning borders. Parallel performance results on Cray T3E-900 are presented for two real application problems arising in the Yucca Mountain nuclear waste site study. The execution time is reduced from 7504 seconds on two processors to 126 seconds on 128 processors for a 2D problem involving 52,752 equations. For a larger 3D problem with 293,928 equations the time decreases from 10,055 seconds on 16 processors to 329 seconds on 512 processors.

The global Hessenberg and CMRH methods for linear systems with multiple right-hand sides

In this paper, we introduce two new methods for solving large sparse nonsymmetric linear systems with several right-hand sides. These methods are the global Hessenberg and global CMRH methods. Using the global Hessenberg process, these methods are less expensive than the global FOM and global GMRES methods [9]. Theoretical results about the new methods are given, and experimental results that show good performances of these new methods are presented.