On computingINV block preconditionings for the conjugate gradient method

The INV(k) and MINV(k) block preconditionings for the conjugate gradient method require generation of selected elements of the inverses of symmetric matrices of bandwidth 2k+1. Generalizing the previously describedk=1 (tridiagonal) case tok=2, explicit expressions for the inverse elements of a symmetric pentadiagonal matrix in terms of Green's matrix of rank two are given. These expressions are found to be seriously ill-conditioned; hence alternative computational algorithms for the inverse elements must be used. Behavior of thek=1 andk=2 preconditionings are compared for some discretized elliptic partial differential equation test problems in two dimensions.Presented by the first author at the Joint U.S.-Scandinavian Symposium on Scientific Computing and Mathematical Modeling, January 1985, in honor of Germund Dahlquist on the occasion of his 60th birthday. This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under contract DE AC03-76SF00098.     

A modified SSOR iterative method for augmented systems

Golub, Wu and Yuan [G.H. Golub, X. Wu, J.Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] have presented the SOR-like algorithm to solve augmented systems. In this paper, we present the modified symmetric successive overrelaxation (MSSOR) method for solving augmented systems, which is based on Darvishi and Hessari’s work above. We derive its convergence under suitable restrictions on the iteration parameter, determine its optimal iteration parameter and the corresponding optimal convergence factor under certain conditions. Finally, we apply the MSSOR method to solve augmented systems.     

Circulant preconditioners for second order hyperbolic equations

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO() orO(m), where is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.     

A total least squares method for Toeplitz systems of equations

A Newton method to solve total least squares problems for Toeplitz systems of equations is considered. When coupled with a bisection scheme, which is based on an efficient algorithm for factoring Toeplitz matrices, global convergence can be guaranteed. Circulant and approximate factorization preconditioners are proposed to speed convergence when a conjugate gradient method is used to solve linear systems arising during the Newton iterations. The work of the second author was partially supported by a National Science Foundation Postdoctoral Research Fellowship.     

Convergent improvement of SSOR multisplitting method for an H-matrix

Recently, Yun [Jae Heon Yun, Convergence of SSOR multisplitting method for an H$H$-matrix, J. Comput. Appl. Math. 217 (2008) 252–258] studied the convergence of the relaxed multisplitting method associated with SSOR multisplitting for solving a linear system whose coefficient matrix is an H$H$-matrix. In this paper, we improve the main results of Yun’s. Moreover, theoretical analysis and numerical examples clearly show that our new convergent domain is wider.     

Preconditioned conjugate gradient methods for large-scale fluid flow applications

The simulation of large-scale fluid flow applications often requires the efficient solution of extremely large nonsymmetric linear and nonlinear sparse systems of equations arising from the discretization of systems of partial differential equations. While preconditioned conjugate gradient methods work well for symmetric, positive-definite matrices, other methods are necessary to treat large, nonsymmetric matrices. The applications may also involve highly localized phenomena which can be addressed via local and adaptive grid refinement techniques. These local refinement methods usually cause non-standard grid connections which destroy the bandedness of the matrices and the associated ease of solution and vectorization of the algorithms. The use of preconditioned conjugate gradient or conjugate-gradient-like iterative methods in large-scale reservoir simulation applications is briefly surveyed. Then, some block preconditioning methods for adaptive grid refinement via domain decomposition techniques are presented and compared. These techniques are being used efficiently in existing large-scale simulation codes.     

Block SSOR preconditionings for high-order 3D FE systems. III: Incomplete BSSOR preconditionings based on p-partitionings

This paper considers the construction of high quality IBSSOR preconditionings for p-adaptive hierarchical high-order 3D FE systems. Instead of the commonly-adopted domain decomposition approach to the selection of a block partitioning underlying the corresponding block SSOR preconditioning, we introduce the so-called p-partitionings based on subdividing the degrees of freedom in accordance with the levels of p-refinement. A theoretical analysis of IBSSOR preconditionings is presented. The results of numerical experiments with 3D FE systems arising from the hierarchical approximations of order p, 3 ≤ p ≤ 5, of the 3D equilibrium equations for linear elastic orthotropic materials show that the IBSSOR preconditionings suggested are more efficient than the IBSSOR preconditionings based on domain decomposition partitionings. Moreover, we demonstrate that the new preconditionings are more reliable when solving extremely ill-conditioned 3D FE systems. © 1997 John Wiley & Sons, Ltd.     

Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations

In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.     

A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations

A class of modified block SSOR preconditioners is presented for solving symmetric positive definite systems of linear equations, which arise in the hierarchical basis finite element discretizations of the second order self‐adjoint elliptic boundary value problems. This class of methods is strongly related to two level methods, standard multigrid methods, and Jacobi‐like hierarchical basis methods. The optimal relaxation factors and optimal condition numbers are estimated in detail. Theoretical analyses show that these methods are very robust, and especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Local refinement techniques for elliptic problems on cell-centered grids

Summary Algebraic multilevel analogues of the BEPS preconditioner designed for solving discrete elliptic problems on grids with local refinement are formulated, and bounds on their relative condition numbers, with respect to the composite-grid matrix, are derived. TheV-cycle and, more generally,v-foldV-cycle multilevel BEPS preconditioners are presented and studied. It is proved that for 2-D problems theV-cycle multilevel BEPS is almost optimal, whereas thev-foldV-cycle algebraic multilevel BEPS is optimal under a mild restriction on the composite cell-centered grid. For thev-fold multilevel BEPS, the variational relation between the finite difference matrix and the corresponding matrix on the next-coarser level is not necessarily required. Since they are purely algebraically derived, thev-fold (v>1) multilevel BEPS preconditioners perform without any restrictionson the shape of subregions, unless the refinement is too fast. For theV-cycle BEPS preconditioner (2-D problem), a variational relation between the matrices on two consecutive grids is required, but there is no restriction on the method of refinement on the shape, or on the size of the subdomains.     

A parallel implementation of the restarted GMRES iterative algorithm for nonsymmetric systems of linear equations

We describe the parallelisation of the GMRES(c) algorithm and its implementation on distributed-memory architectures, using both networks of transputers and networks of workstations under the PVM message-passing system. The test systems of linear equations considered are those derived from five-point finite-difference discretisations of partial differential equations. A theoretical model of the computation and communication phases is presented which allows us to decide for which values of the parameterc our implementation executes efficiently. The results show that for reasonably large discretisation grids the implementations are effective on a large number of processors.     

Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems

A preconditioned conjugate gradient method is applied to finite element discretizations of some nonsymmetric elliptic systems. Mesh independent superlinear convergence is proved, which is an extension of a similar earlier result from a single equation to systems. The proposed preconditioning method involves decoupled preconditioners, which yields small and parallelizable auxiliary problems.     

A high order direct method for solving Poisson's equation in a disc

Summary. A direct method is developed for solving Poisson's equation on an annulus region with Hermite bicubic collocation approximation. In terms of FFT, the operation count is on an mesh. Received May 11, 1990 / Revised version received August 19, 1994     

Parallel preconditioners for large scale partial difference equation systems

We propose a new preconditioner DASP (discrete approximate spectral preconditioner), based on the existing well-known preconditioners and our computational experience. Parallel preconditioning strategies for large scale partial difference equation systems arising from partial differential equations are investigated. Numerical results are given to show the efficiency and effectiveness of the new preconditioners for both model problems and real applications in petroleum reservoir simulation.     

An analysis of the composite step biconjugate gradient method

Summary The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2×2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. We also prove a best approximation result for the method. Some numerical illustrations showing the effect of roundoff error are given.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.The work of this author was supported by the Office of Naval Research under contracts N00014-90-J-1695 and N00014-92-J-1890, the Department of Energy under, contract DE-FG03-87ER25307, the National Science Foundation under contracts ASC 90-03002 and ASC 92-01266, and the Army Research Office under contract DAAL03-91-G-0150. Part of this work was completed during a visit to the Computer Science Dept. The Chinese University of Hong Kong.     

An alternating direction implicit method for orthogonal spline collocation linear systems

Summary An Alternating Direction Implicit method is analyzed for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to some separable, second order, linear, elliptic partial differential equations in rectangles. On anNxN partition, with Jordan's selection of the acceleration parameters, the method requiresO(N ² ln ² N) arithmetic operations to produce an approximation whose accuracy, in theH ¹-norm, is that of the collocation solution.     

On algebraic multi-level methods for non-symmetric systems - Comparison results

We establish theoretical comparison results for algebraic multi-level methods applied to non-singular non-symmetric M-matrices. We consider two types of multi-level approximate block factorizations or AMG methods, the AMLI and the MAMLI method. We compare the spectral radii of the iteration matrices of these methods. This comparison shows, that the spectral radius of the MAMLI method is less than or equal to the spectral radius of the AMLI method. Moreover, we establish how the quality of the approximations in the block factorization effects the spectral radii of the iteration matrices. We prove comparisons results for different approximations of the fine grid block as well as for the used Schur complement. We also establish a theoretical comparison between the AMG methods and the classical block Jacobi and block Gauss-Seidel methods.     

Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems

Two classes of incomplete factorization preconditioners are considered for nonsymmetric linear systems arising from second order finite difference discretizations of non-self-adjoint elliptic partial differential equations. Analytic and experimental results show that relaxed incomplete factorization methods exhibit numerical instabilities of the type observed with other incomplete factorizations, and the effects of instability are characterized in terms of the relaxation parameter. Several stabilized incomplete factorizations are introduced that are designed to avoid numerically unstable computations. In experiments with two-dimensional problems with variable coefficients and on nonuniform meshes, the stabilized methods are shown to be much more robust than standard incomplete factorizations.The work presented in this paper was supported by the National Science Foundation under grants DMS-8607478, CCR-8818340, and ASC-8958544, and by the U.S. Army Research Office under grant DAAL-0389-K-0016.     

Block and asynchronous two-stage methods for mildly nonlinear systems

Block parallel iterative methods for the solution of mildly nonlinear systems of equations of the form are studied. Two-stage methods, where the solution of each block is approximated by an inner iteration, are treated. Both synchronous and asynchronous versions are analyzed, and both pointwise and blockwise convergence theorems provided. The case where there are overlapping blocks is also considered. The analysis of the asynchronous method when applied to linear systems includes cases not treated before in the literature. Received June 5, 1997 / Revised version received December 29, 1997     

A survey of multilevel preconditioned iterative methods

We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.On leave from the Institute of Mathematics, and Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria. The research of the second author reported here was partly supported by the Stichting Mathematisch Centrum, Amsterdam.