This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient–type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell–Boeing collection and from Tim Davis’s collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.

Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an M-matrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using A-norms when the matrix A is symmetric positive definite. A new theorem concerning P -regular splittings is presented, which provides a useful tool for the A-norm bounds. Furthermore, a theory of splittings is developed to represent Algebraic Additive Schwarz Iterations. This representation makes a connection with multisplitting methods. With this representation, and using a comparison theorem, it is shown that a coarse grid correction improves the convergence of Additive Schwarz Iterations when measured in weighted max norm.

This paper describes and compares several methods for computing stationary probability distributions of Markov chains. The main linear algebra problem consists of computing an eigenvector of a sparse, non-symmetric, matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous, singular linear system. We present several methods based on combinations of Krylov subspace techniques, single vector power iteration/relaxation procedures and acceleration techniques. We compare the performance of these methods on some realistic problems. Key words: Markov chain models; Homogeneous linear systems; Direct methods; Successive Overrelaxation; Preconditioned power iterations; Arnoldi's method; GMRES. y IRISA, Rennes, France. Research supported by CNRS (87:N 920070). Research Institute for Advanced Computer Science, NASA Ames Research Center. Moffett Field CA 94035. Research supported by Cooperative Agreement NCC 2-387 between the National Aeronautics and S...

We describe the implementation and performance of a novel class of preconditioners. These preconditioners were proposed and theoretically analyzed by Pravin Vaidya in 1991, but no report on their implementation or performance in practice has ever been published. We show experimentally that these preconditioners have some remarkable properties. We show that within the class of diagonally-dominant symmetric matrices, the cost and convergence of these preconditioners depends almost only on the nonzero structure of the matrix, but not on its numerical values. In particular, this property leads to robust convergence behavior on di#cult 3-dimensional problems that cause stagnation in incomplete-Cholesky preconditioners (more specifically, in drop-tolerance incomplete Cholesky without diagonal modification, with diagonal modification, and with relaxed diagonal modification). On such problems, we have observed cases in which a Vaidya-preconditioned solver is more than 6 times faster than an incomplete-Cholesky-preconditioned solver, when we allow similar amounts of fill in the factors of both preconditioners. We also show that Vaidya's preconditioners perform and scale similarly or better than drop-tolerance relaxed-modified incomplete Cholesky preconditioners on a wide range of 2-dimensional problems. In particular, on anisotropic 2D problems, Vaidya delivers robust convergence independently of the direction of anisotropy and the ordering of the unknowns. However, on many 3D problems in which incomplete-Choleskypreconditioned solvers converge without stagnating, Vaidya-preconditioned solvers are much slower. We also show how the insights gained from this study can be used to design faster and more robust solvers for some di#cult problems. 1.

Incomplete factorization preconditioners such as ILU, ILUT and MILU are well-known robust general-purpose techniques for solving linear systems on serial computers. However, they are difficult to parallelize efficiently. Various techniques have been used to parallelize these preconditioners, such as multicolor orderings and subdomain preconditioning. These techniques may degrade the performance and robustness of ILU preconditionings. The purpose of this paper is to perform numerical experiments to compare these techniques in order to assess what are the most effective ways to use ILU preconditioning for practical problems on serial and parallel computers.

Abstract. A new preconditioner for symmetric positive definite systems is proposed, analyzed, and tested. The preconditioner, compressed incomplete modified Gram–Schmidt (CIMGS), is based on an incomplete orthogonal factorization. CIMGS is robust both theoretically and empirically, existing (in exact arithmetic) for any full rank matrix. Numerically it is more robust than an incomplete Cholesky factorization preconditioner (IC) and a complete Cholesky factorization of the normal equations. Theoretical results show that the CIMGS factorization has better backward error properties than complete Cholesky factorization. For symmetric positive definite M-matrices, CIMGS induces a regular splitting and better estimates the complete Cholesky factor as the set of dropped positions gets smaller. CIMGS lies between complete Cholesky factorization and incomplete Cholesky factorization in its approximation properties. These theoretical properties usually hold numerically, even when the matrix is not an M-matrix. When the drop set satisfies a mild and easily verified (or enforced) property, the upper triangular factor CIMGS generates is the same as that generated by incomplete Cholesky factorization. This allows the existence of the IC factorization to be guaranteed, based solely on the target sparsity pattern.

We study the solution of large sparse nonsingular and unsymmetric systems of linear equations. We present a class of incomplete orthogonal factorization methods based on Givens rotations. These methods include: Incomplete Givens Orthogonalization (IGO-method) and Generalized Incomplete Givens Orthogonalization (GIGO-method), which drop entries from the incomplete orthogonal and upper triangular factors by position; Threshold Incomplete Givens Orthogonalization (TIGO()-method), which drops entries dynamically by their magnitudes; and Generalized Threshold Incomplete Givens Orthogonalization (GTIGO(; p)-method), which drops entries dynamically by both their magnitudes and positions. Theoretical analyses show that these methods can produce a nonsingular sparse incomplete upper triangular factor and either a complete orthogonal factor or a sparse nonsingular incomplete orthogonal factor for a general nonsingular matrix. Therefore, these methods can potentially generate efficient preconditi...

Abstract. The development of robust and efficient algorithms for both steady-state simulations and fully-implicit time integration of the Navier–Stokes equations is an active research topic. To be effective, the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence. In particular, they should be insensitive to parameters in the problem such as mesh size, time step, and Reynolds number. In this context, we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an Oseen nonlinear iteration for the primitive variable formulation of the system. The key to this preconditioner is the approximation of a certain Schur complement operator by a technique first proposed by Kay, Loghin, and Wathen [26] and Silvester, Elman, Kay, and Wathen [46]. The resulting operator entails subsidiary computations (solutions of pressure Poisson and convection–diffusion subproblems) that are similar to those required for decoupled solution methods; however, in this case these solutions are applied as preconditioners to the coupled Oseen system. One important aspect of this approach is that the convection–diffusion and Poisson subproblems are significantly easier to solve than the entire coupled system, and a solver can be built using tools developed for the subproblems. In this paper, we apply smoothed aggregation algebraic multigrid to both subproblems. Previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial, two-dimensional, steady-state computations employing geometric multi-grid methods; we focus on extending these methods to large-scale, parallel, three-dimensional, transient and steadystate simulations employing algebraic multigrid (AMG) methods. Our results display nearly optimal convergence rates for steady-state solutions as well as for transient solutions over a wide range of CFL numbers on the two-dimensional and three-dimensional lid-driven cavity problem. 1. Introduction. Recently

IFISS is a graphical Matlab package for the interactive numerical study of incompressible flow problems. It includes algorithms for discretisation by mixed finite element methods and a posteriori error estimation of the computed solutions. The package can also be used as a computational laboratory for experimenting with state-of-the-art preconditioned iterative solvers for the discrete linear equation systems that arise in incompressible flow modelling. A unique feature of the package is its comprehensive nature; for each problem addressed, it enables the study of both discretisation and iterative solution algorithms as well as the interaction between the two and the resulting effect on overall efficiency.

We present an efficient implicit time stepping method for Discontinuous Galerkin discretizations of the compressible Navier-Stokes equations on unstructured meshes. The Local Discontinuous Galerkin method is used for the discretization of the viscous terms. For unstructured meshes, the Local Discontinuous Galerkin method is known to produce non-compact discretizations. In order to circumvent the difficulties accociated with this non-compactness, we represent the irregular matrices arising from the discretization algorithm as a product of matrices with a more structured pattern. Time integration is carried out using backward difference formulas. This leads to a non-linear system of equations to be solved at each timestep. In this paper, we study various iterative solvers for the linear systems of equations that arise in the Newton algorithm. We show that a two-level preconditioner with incomplete LU as a pre-smoother is highly efficient yet inexpensive to compute and to store. It performs particularly well for low Mach number flows, where it is more than a magnitude more efficient than pure two-level or ILU preconditioning. Our methods are demonstrated using three typical test problems with various parameters and timesteps. I.