We outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of Navier-Stokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid V-cycle for the scalar convection-diffusion operator, and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid, robust with respect to the time-step, and only deteriorates very slowly as the Reynolds number is increased.

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2\Theta2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3 \Theta 3-block systems to symmetric 2 \Theta 2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefini...

The design of a parallel implementation of multilevel recursive spectral bisection on the Cray T3D is described. The code is intended to be fast enough to enable dynamic repartitioning of adaptive meshes and to partition meshes that are too large for workstations. Two innovations in the implementation are recursive asynchronous task teams and a parallel version of the multilevel accelerator. A performance improvement of a factor of 140 over the best available serial implementation is demonstrated. 1 Introduction The efficient implementation of unstructured problems on distributed memory parallel computers requires that the data be distributed across the memories in a way that simultaneously balances the work load of the processors and minimizes interprocessor communication. This paper describes a tool called PMRSB, which is implemented on the Cray T3D, that performs this function. Currently, it is restricted I would like to thank Horst Simon for his support and encouragement. I am ...

The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, ...) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor ae 1 can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.

Many important applications involve the solution of large linear systems with symmetric, but indefinite coefficient matrices. For example, such systems arise in incompressible flow computations and as subproblems in optimization algorithms for linear and nonlinear programs. Existing Krylov-subspace iterations for symmetric indefinite systems, such as SYMMLQ and MINRES, require the use of symmetric positive definite preconditioners, which is a rather unnatural restriction when the matrix itself is highly indefinite with both many positive and many negative eigenvalues. In this note, we describe a new Krylov-subspace iteration for solving symmetric indefinite linear systems that can be combined with arbitrary symmetric preconditioners. The algorithm can be interpreted as a special case of the quasi-minimal residual method for general non-Hermitian linear systems, and like the latter, it produces iterates defined by a quasi-minimal residual property. The proposed method has the same work ...

The nonsymmetric Lanczos process simplifies when applied to J-symmetric and J-Hermitian matrices, and work and storage requirements are roughly halved compared to the general case. In this paper, we describe FORTRAN-77 implementations of simplified versions of the look-ahead Lanczos algorithm and of the quasi-minimal residual (QMR) method, which is a Lanczos-based iterative procedure for the solution of linear systems. These implementations of the simplified algorithms complete our software package QMRPACK, which so far contained only codes for Lanczos and QMR algorithms for general matrices. We describe in some detail the use of two routines, one for the solution of linear systems, and the other for eigenvalue computations. We present examples that lead to J-symmetric and J-Hermitian matrices. Results of numerical experiments are reported. Keywords. Lanczos process; quasi-minimal residual iteration; linear system; eigenvalue computation; J-symmetric matrix; J-Hermitian matrix; look-...

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

We consider conjugate gradient type methods for the solution of large linear systems Az = b with complex coefficient matrices of the type A = T + io1 where T is Hermitian and u a real scalar. Three different conjugate gradient type approaches with it-erates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunders’s SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure of A can be preserved by using polynomial precon-ditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.

We discuss approaches for an efficient handling of the correction equation in the Jacobi-Davidson method. The correction equation is effective in a subspace orthogonal to the current eigenvector approximation. The operator in the correction equation is a dense matrix, but it is composed from three factors that allow for a sparse representation. If the given matrix eigenproblem is sparse then one often aims for the construction of a preconditioner for that matrix. We discuss how to restrict this preconditioner effectively to the subspace orthogonal to the current eigenvector. The correction equation itself is formulated in terms of approximations for an eigenpair. In order to avoid misconvergence one has to make the right selection for the approximations, and this aspect will be discussed as well.

The recent development of Krylov subspace methods for the solution of operator equations has shown that two basic construction principles, the orthogonal residual (OR) and minimal residual (MR) approaches, underlie the most commonly used algorithms. It is shown that these can both be formulated as techniques for solving an approximation problem on a sequence of nested subspaces of a Hilbert space, a problem not necessarily related to an operator equation. Most of the familiar Krylov subspace algorithms result when these subspaces form a Krylov sequence. The well-known relations among the iterates and residuals of OR/MR pairs are shown to hold also in this rather general setting. We further show that a common error analysis for these methods involving the canonical angles between subspaces allows many of the recently developed error bounds to be derived in a simple manner. An application of this analysis to compact perturbations of the identity shows that OR/MR pairs of Krylov subspace methods converge q-superlinearly when applied to such operator equations.