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 is the modified version of Gram-Schmidt orthogonalization with a rejection test applied right after the formation of the off-diagonal elements of the factor R. For a given rejection parameter 0 / 1, the rejection test is: if r ij ! /= k a

This paper proposes, analyzes, and numerically tests methods to assure the existence of incomplete Cholesky (IC) factorization preconditioners, based solely on the target sparsity pattern for the triangular factor R. If the sparsity pattern has a simple property (called property C+), then the IC factor exists in exact arithmetic. Two algorithms for modifying the target sparsity pattern to have property C+ are proposed, one based on adding elements into the set of retained elements and the other based on dropping elements. Tests show that the modifications do ensure the numerical existence of the IC factor, and the resulting preconditioners are effective in accelerating the conjugate gradient iteration method. 1 Introduction The incomplete Cholesky (IC) factorization is one of the most important and commonly used preconditioners for iterative methods of solving large sparse symmetric positive definite linear systems [4]. Its major weakness is a lack of robustness, by which we mean the ...

. To efficiently solve second order discrete elliptic PDEs, by Krylov subspace like methods, one needs to use some robust preconditioning techniques. Relaxed incompletefactorizations (RILU) are powerful candidates. Unfortunately, their efficiency critically depends on the choice of the relaxation parameter ! whose "optimal" value is not only hard to estimate, but also strongly varies from a problem to another. These methods interpolate between the popular ILU and its modified variant MILU. Concerning the pointwise schemes, a new variant of RILU that dynamically computes variable ! = ! i has been introduced recently. Like its ancestor RILU and unlike standard methods, it is robust with respect to both existence and performance. On top of that, it breaks the problem-dependence of "!opt ". A block version of this dynamically relaxed method is proposed, and compared with classical pointwise and blockwise methods as well as with some existing "dynamic" variants. Key words. Discretized part...

We address the hard question of efficient use on parallel platforms, of incomplete factorization preconditioning techniques for solving large and sparse linear systems by Krylov subspace methods. A novel parallelization strategy based on pseudooverlapped subdomains is explored. This results in efficient parallelizable preconditioners. Numerical results give evidence that high performance can be achieved. Key words: Large sparse linear systems; incomplete factorizations; preconditioned conjugate gradient; multiprocessor computers; domain decomposition 1 Introduction Combined with suitable preconditioners, Krylov subspace methods can be powerful (iterative) methods for solving the large sparse linear systems that arise in many scientific computations [6,23]. In particular, incomplete factorizations as preconditioning techniques are often efficient [31,32]. Their major drawback is that they are not easy to parallelize without seriously affecting the convergence. Several attempts have been...

A robust two-level incomplete