This paper discusses some relationships between Incomplete LU (ILU) factoriza-tion techniques and factored sparse approximate inverse (AINV) techniques. While ILU factorizations compute approximate LU factors of the coeecient matrix A, AINV techniques aim at building triangular matrices Z and W such that W > AZ is approximately diagonal. The paper shows that certain forms of approximate inverse...

We extend graph embedding techniques for bounding the spectral condition number of preconditioned systems involving symmetric, irreducibly diagonally dominant M-matrices to systems where the preconditioner is not diagonally dominant. In particular, this allows us to bound the spectral condition number when the preconditioner is based on an incomplete factorization. We provide a review of previo...

A new preconditioner is proposed for the solution of an N x N Toeplitz system TNX = b, where TN can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner FN is obtained based on factorizing the generating function T(z) into the product of two terms corresponding, respectively, to minimum-phase causal and anticausal systems and therefore called the mini...

Incomplete LDL∗ factorizations sometimes produce an inde nite preconditioner even when the input matrix is Hermitian positive de nite. The two most popular iterative solvers for Hermitian systems, MINRES and CG, cannot use such preconditioners; they require a positive de nite preconditioner. We present two new Krylov-subspace solvers, a variant of MINRES and a variant of CG, both of which can b...

The analysis of preconditioners based on incomplete Cholesky factorization in which the neglected (dropped) components are orthogonal to the approximations being kept is presented. General estimate for the condition number of the preconditioned system is given which only depends on the accuracy of individual approximations. The estimate is further improved if, for instance, only the newly compu...

This paper addresses the problem of computing preconditioners for solving linear systems of equations with a sequence of slowly varying matrices. This problem arises in many important applications. For example, a common situation in computational fluid dynamics, is when the equations change only slightly, possibly in some parts of the physical domain. In such situations it is wasteful to recomp...

1. Sparse Approximate Inverses and Linear Equations We consider the problem of solving a system of linear equations Ax = b in a parallel environment. Here, the n n-matrix A is large, sparse, unstructured, nonsymmetric, and ill-conditioned. The solution method should be robust, easy to parallelize, and applicable as a black box solver. Direct solution methods like the Gaussian Elimination are no...

We show that a novel class of preconditioners, designed by Pravin Vaidya in 1991 but never before implemented, is remarkably robust and can outperform incomplete-Cholesky preconditioners. Our test suite includes problems arising from finitedifferences discretizations of elliptic PDEs in two and three dimensions. On 2D problems, Vaidya’s preconditioners often outperform drop-tolerance incomplete...

Parallel preconditioners are considered for improving the convergence rate of the conjugate gradient method for solving sparse symmetric positive deenite systems generated by nite element models of subsurface ow. The diiculties of adapting eeective sequential preconditioners to the parallel environment are illustrated by our treatment of incomplete Cholesky preconditioning. These diiculties are...

Two Krylov subspace iterative methods, GMRES and QMR, were studied in conjunction with several preconditioning techniques for solving the linear system raised from the finite element discretisation of incompressible Navier-Stokes equations for hydrodynamic problems. The preconditioning methods under investigation were the incomplete factorisation methods such as ILU(0) and MILU, the Stokes prec...