Inexact preconditioned conjugate gradient method with inner-outer iteration

An important variation of preconditioned conjugate gradient algorithms is i nexact preconditioner implemented with inner-outer iterations [G. H. Golub and M. L. Overton, Numerical Analysis, Lecture Notes in Math. 912, Springer , Berlin, New York, 1982], where the preconditioner is solved by an inner i teration to a prescribed precision. In this paper, we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive defini te system and analyze its convergence property. We establish a linear conve rgence result using a local relation of residual norms. We also analyze the algorithm using a global equation and show that the algorithm may have the superlinear convergence property when the inner iteration is solved to hig h accuracy. The analysis is in agreement with observed numerical behavior o f the algorithm. In particular, it suggests a heuristic choice of the stopp ing threshold for the inner iteration. Numerical examples are given to show the effectiveness of this choice and to compare the convergence bound.