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.

This paper presents a general preconditioning method based on a multilevel partial solution approach. The basic step in constructing the preconditioner is to separate the initial points into two subsets. The first subset which can be termed "coarse" is obtained by using "block" independent sets, or "aggregates". Two aggregates have no coupling between them, but nodes in the same aggregate may be coupled. The nodes not in the coarse set are part of what might be called the "Fringe" set. The idea of the methods is to form the Schur complement related to the fringe set. This leads to a natural block LU factorization which can be used as a preconditioner for the system. This system is then solver recursively using as preconditioner the factorization that could be obtained from the next level. Unlike other multilevel preconditioners available, iterations between levels are allowed. One interesting aspect of the method is that it provides a common framework for many other technique...

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

We conduct experimental study on the behavior of several preconditioned iterative methods to solve nonsymmetric matrices arising from computational fluid dynamics (CFD) applications. The preconditioned iterative methods consist of Krylov subspace accelerators and a powerful general purpose multilevel block ILU (BILUM) preconditioner. The BILUM preconditioner and an enhanced version of it are slightly modified versions of the originally proposed preconditioners. They will be used in combination with different Krylov subspace methods. We choose to test three popular transposefree Krylov subspace methods: BiCGSTAB, GMRES and TFQMR. Numerical experiments, using several sets of test matrices arising from various relevant CFD applications, are reported. Key words: Multilevel preconditioner, Krylov subspace methods, nonsymmetric matrices, CFD applications. AMS subject classifications: 65F10, 65F50, 65N06, 65N55. 1 Introduction A challenging problem in computational fluid dynamics (...

A parallel version of the Algebraic Recursive Multilevel Solver (ARMS) is developed for distributed computing environments. The method adopts the general framework of distributed sparse matrices and relies on solving the resulting distributed Schur complement system. Numerical experiments are presented which compare these approaches on regularly and irregularly structured problems.

Domain-decomposition and multi-level techniques are often formulated for linear systems that arise from the solution of elliptic-type Partial Differential Equations. In this paper, generalizations of these techniques for irregularly structured sparse linear systems are considered. An interesting common approach used to derive successful preconditioners is to resort to Schur complements. In particular, we discuss a multi-level domain decompositiontype algorithm for iterative solution of large sparse linear systems based on independent subsets of nodes. We also discuss a Schur complement technique that utilizes incomplete LU factorizations of local matrices. Key words: Schur complement techniques; Incomplete LU factorization; Schwarz iterations; Multielimination; Multi-level ILU preconditioners; Krylov subspace methods. 1 Introduction A recent trend in parallel preconditioning techniques for general sparse linear systems is to exploit ideas from domain decomposition concepts an...

We present a preconditioning method for the iterative solution of large sparse systems of equations. The preconditioner is based on ideas both from ILU preconditioning and from multigrid. The resulting preconditioning technique requires the matrix only. A multilevel structure is obtained by using maximal independent sets for graph coarsening. A Schur complement approximation is constructed using a sequence of point Gaussian elimination steps. The resulting preconditioner has a transparant modular structure similar to the algoritmic structure of a multigrid V-cycle.

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.

This paper introduces techniques based on diagonal threshold tolerance when developing multi-elimination and multi-level incomplete LU (ILUM) factorization preconditioners for solving general sparse linear systems. Existing heuristics solely based on the adjacency graph of the matrices have been used to find independent sets and are not robust for matrices arising from certain applications in which the matrices may have small or zero diagonals. New heuristic strategies based on the adjacency graph and the diagonal values of the matrices for finding independent sets are introduced. Analytical bounds for the factorization and preconditioned errors are obtained for the case of a two-level analysis. These bounds provide useful information in designing robust ILUM preconditioners. Extensive numerical experiments are conducted in order to compare robustness and efficiency of various heuristic strategies.

A sparse approximate inverse technique is introduced to solve general sparse linear systems. The sparse approximate inverse is computed as a factored form and used as a preconditioner to work with some Krylov subspace methods. The new technique is derived from a matrix decomposition algorithm for inverting dense nonsymmetric matrices. Several strategies and special data structures are proposed to implement the algorithm efficiently. Sparsity patterns of the the factored inverse are exploited to reduce computational cost. The computation of the factored sparse approximate inverse is relatively cheaper than the techniques based on norm minimization techniques. The new preconditioner possesses much greater inherent parallelism than traditional preconditioners based on incomplete LU factorizations. Numerical experiments are used to show the effectiveness and efficiency of the new sparse approximate inverse preconditioner.