Abstract not found

This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.

Abstract not found

Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutationsand scalingsaimed at placing large entrieson the diagonal in the context of preconditioning for general sparse matrices. The permutations and scalings are those developed by Olschowka and Neumaier [Linear Algebra Appl., 240 (1996), pp. 131–151] and by Duff and

The influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered. Some theoretical results on the effect of orderings on the fill-in and decay behavior of the inverse factors of a sparse matrix are presented. It is shown experimentally that certain reorderings, like minimum degree and nested dissection, can be very beneficial. The benefit consists of a reduction in the storage and time required for constructing the preconditioner, and of faster convergence of the preconditioned iteration in many cases of practical interest.

Abstract. A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph-based strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILU threshold preconditioner.

Abstract. We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible Navier-Stokes equations. The spatial discretization is carried out using a Discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear system of equations which is solved at each timestep. This is accomplished using Newton’s method. The resulting linear systems are solved using a preconditioned GMRES iterative algorithm. We consider several existing preconditioners such as block-Jacobi and Gauss-Seidel combined with multi-level schemes which have been developed and tested for specific applications. While our results are consistent with the claims reported, we find that these preconditioners lack robustness when used in more challenging situations involving low Mach numbers, stretched grids or high Reynolds number turbulent flows. We propose a preconditioner based on a coarse scale correction with post-smoothing based on a block incomplete LU factorization with zero fill-in (ILU0) of the Jacobian matrix. The performance of the ILU0 smoother is found to depend critically on the element numbering. We propose a numbering strategy based on minimizing the discarded fill-in in a greedy fashion. The coarse scale correction scheme is found to be important for diffusion dominated

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.

. We describe an algebraic multilevel multigraph algorithm. Many of the multilevel components are generalizations of algorithms originally applied to general sparse Gaussian elimination. Indeed, general sparse Gaussian elimination with minimum degree ordering is a limiting case of our algorithm. Our goal is to develop a procedure which has the robustness and simplicity of use of sparse direct methods, yet oers the opportunity to obtain the optimal or near-optimal complexity typical of classical multigrid methods. Key words. algebraic multigrid, incomplete LU factorization, multigraph methods. AMS subject classications. 65M55, 65N55 1. Introduction. In this work, we develop a multilevel-multigraph algorithm. Algebraic multigrid methods are currently a topic of intense research interest [17, 18, 20, 46, 12, 48, 38, 11, 44, 3, 4, 1, 2, 5, 16, 7, 29, 28, 27, 42, 41, 21]. An excellent recent survey is given in Wagner [49]. In many \real world" calculations, direct methods are still wid...

This research presents new preconditioners for linear systems. We proceed from the most general case to the very specific problem area of sparse optimal control. In the first most general approach, we assume only that the coefficient matrix is nonsingular. We target highly indefinite, nonsymmetric problems that cause difficulties for preconditioned iterative solvers, and where standard preconditioners, like incomplete factorizations, often fail. We experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of precon-ditioning for general sparse matrices. Our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing. Secondly, we present two new preconditioners for KKT systems. KKT systems arise in areas such as quadratic programming, sparse optimal control, and mixed finite element formulations. Our preconditioners approximate a constraint precon-ditioner with incomplete factorizations for the normal equations. Numerical experiments compare these two preconditioners with exact constraint preconditioning and the approach described above of permuting large entries to the diagonal. Finally, we turn to a specific problem area: sparse optimal control. Many optimal control problems are broken into several phases, and within a phase, most variables and constraints depend only on nearby variables and constraints. However, free initial and final times and time-independent parameters impact variables and constraints throughout a phase, resulting in dense factored blocks in the KKT matrix. We drop fill due to these variables to reduce density within each phase. The resulting preconditioner is tightly banded and nearly block tridiagonal. Numerical experiments demonstrate that the preconditioners are effective, with very little fill in the factorization.