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.     

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.     

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 general finite element preconditioning for the conjugate gradient method

Discretizing a symmetric elliptic boundary value problem by a finite element method results in a system of linear equations with a symmetric positive definite coefficient matrix. This system can be solved iteratively by a preconditioned conjugate gradient method. In this paper a preconditioning matrix is proposed that can be constructed for all finite element methods if a mild condition for the node numbering is fulfilled. Such a numbering can be constructed using a variant of the reverse Cuthill-McKee algorithm.     

A practical termination criterion for the conjugate gradient method

The conjugate gradient method for the iterative solution of a set of linear equationsAx=b is essentially equivalent to the Lanczos method, which implies that approximations to certain eigen-values ofA can be obtained at low cost. In this paper it is shown how the smallest active eigenvalue ofA can be cheaply approximated, and the usefulness of this approximation for a practical termination criterion for the conjugate gradient method is studied. It is proved that this termination criterion is reliable in many relevant situations.     

An optimum iterative method for solving any linear system with a square matrix

A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues ofA. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.     

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 the convergence of the conjugate gradient method for singular capacitance matrix equations from the Neumann problem of the Poisson equation

Summary It is shown analytically in this work that the conjugate gradient method is an efficient means of solving the singular capacitance matrix equations arising from the Neumann problem of the Poisson equation. The total operation count of the algorithm does not exceed constant timesN ²logN(N=1/h) for any bounded domain with sufficiently smooth boundary.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MCS75-17385. Also partially supported by the Energy Research and Development Administration     

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.     

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.     

A generalized successive overrelaxation method for least squares problems

In this paper a new iterative method is given for solving large sparse least squares problems and computing the minimum norm solution to underdetermined consistent linear systems. The new scheme is called the generalized successive overrelaxation (GSOR) method and is shown to be convergent ifA is full column rank. The GSOR method involves a parameter ρ and an auxiliary matrixP. One can choose matrix P so that the GSOR method only involves matrix and vector operations; therefore the GSOR method is suitable for parallel computations. Besides, the GSOR method can be combined with preconditioning techniques, and therefore can be expected to be more effective. This author's work was supported by Natural Science Foundation of Liaoning Province, China.     

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.     

The convergence rate of the minimal residual method for the Stokes problem

Summary. Discretisation of the classical Stokes problem gives rise to symmetric indefinite matrices with eigenvalues which, in a precise way, are not symmetric about the origin, but which do depend on a mesh size parameter. Convergence estimates for the Conjugate Residual or Minimum Residual iterative solution of such systems are given by best minimax polynomial approximations on an inclusion set for the eigenvalues. In this paper, an analytic convergence estimate for such problems is given in terms of an asymptotically small mesh size parameter. Received November 16, 1993 / Revised version received August 2, 1994     

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.     

Global aspects of the continuous and discrete Newton method: A case study

Newton's method has recently become one of the paradigms in the revival of Julia set theory and complex dynamical systems. This paper, to a large extent experimental in nature, investigates Newton's method for some particular model problems as a real dynamical system of several simultaneous equations guided by the Julia set theory.     

An asymptotically optimal Schur complement reduction for the Stokes equation

Summary. In this paper we develop an efficient Schur complement method for solving the 2D Stokes equation. As a basic algorithm, we apply a decomposition approach with respect to the trace of the pressure. The alternative stream function-vorticity reduction is also discussed. The original problem is reduced to solving the equivalent boundary (interface) equation with symmetric and positive definite operator in the appropriate trace space. We apply a mixed finite element approximation to the interface operator by iso triangular elements and prove the optimal error estimates in the presence of stabilizing bubble functions. The norm equivalences for the corresponding discrete operators are established. Then we propose an asymptotically optimal compression technique for the related stiffness matrix (in the absence of bubble functions) providing a sparse factorized approximation to the Schur complement. In this case, the algorithm is shown to have an optimal complexity of the order , q = 2 or q = 3, depending on the geometry, where N is the number of degrees of freedom on the interface. In the presence of bubble functions, our method has the complexity arithmetical operations. The Schur complement interface equation is resolved by the PCG iterations with an optimal preconditioner. Received March 20, 1996 / Revised version received October 28, 1997     

An algorithm for coarsening unstructured meshes

Summary. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order , where is the number of hierarchical basis levels. Received December 5, 1994     

The hierarchical basis multigrid method

Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

Convergence rate estimate for a domain decomposition method

Summary We provide a convergence rate analysis for a variant of the domain decomposition method introduced by Gropp and Keyes for solving the algebraic equations that arise from finite element discretization of nonsymmetric and indefinite elliptic problems with Dirichlet boundary conditions in ². We show that the convergence rate of the preconditioned GMRES method is nearly optimal in the sense that the rate of convergence depends only logarithmically on the mesh size and the number of substructures, if the global coarse mesh is fine enough.This author was supported by the National Science Foundation under contract numbers DCR-8521451 and ECS-8957475, by the IBM Corporation, and by the 3M Company, while in residence at Yale UniversityThis author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under Contract W-31-109-Eng-38This author was supported by the National Science Foundation under contract number ECS-8957475, by the IBM Corporation, and by the 3M Company     

Convergence analysis of a multigrid method for convection–diffusion equations

Summary. This paper is concerned with the convergence analysis of robust multigrid methods for convection-diffusion problems. We consider a finite difference discretization of a 2D model convection-diffusion problem with constant coefficients and Dirichlet boundary conditions. For the approximate solution of this discrete problem a multigrid method based on semicoarsening, matrix-dependent prolongation and restriction and line smoothers is applied. For a multigrid W-cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion/convection ratio. For the contraction number of a multigrid V-cycle a bound is proved which is uniform for a class of convection-dominated problems. The analysis is based on linear algebra arguments only. Received April 26, 2000 / Published online June 20, 2001