Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method

We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdown-free and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly ill-conditioned linear systems. We also investigate an alternative approach to a stable approximate inverse algorithm, based on the idea of diagonally compensated reduction of matrix entries. The results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented.