Improved error bounds for the deflated multi-preconditioned CG method

Abstract

Preconditioning and deflation are well-known techniques to speed up the convergence of the CG method. The concept of multiple-preconditioning however is introduced in the last decade. Recently, in [21], a new adaptive preconditioned CG method is established that combines all these techniques. The main tool of the adaptive method is a new error bound for the deflated preconditioned CG method. Using this bound it is decided in each iteration if the deflated preconditioned CG method is sufficient in reducing the error or whether an acceleration by performing iterations of the multi-preconditioned CG method is needed. Here we improve this error bound. This new bound contributes to the theory of deflation methods but can also lead to new decision rules for the adaptive multi-preconditioned CG method.