Error Estimation in Preconditioned Conjugate Gradients

Abstract

In practical problems, iterative methods can hardly be used without some acceleration of convergence, commonly called preconditioning, which is typically achieved by incorporation of some (incomplete or modified) direct algorithm as a part of the iteration. Effectiveness of preconditioned iterative methods increases with possibility of stopping the iteration when the desired accuracy is reached. This requires, however, incorporating a proper measure of achieved accuracy as a part of computation.The goal of this paper is to describe a simple and numerically reliable estimation of the size of the error in the preconditioned conjugate gradient method. In this way this paper extends results from [Z. Strakoš and P. Tichý, ETNA, 13 (2002), pp. 56-80] and communicates them to practical users of the preconditioned conjugate gradient method. (2000): 15A06, 65F10, 65F25, 65G50. AMS subject classification

An affine covariant composite step method for optimization with PDEs as equality constraints

“…Moreover, the additional quantity q = P d is required anyway for termination criteria based on the Pnorm [25,34]. We will not employ such a norm here, but, when considering the inexact solution of normal steps, this norm is favorable in the computation of the tangential step as it allows a proper matching of inaccuracies.…”

“…We choose to truncate the iteration in the case that a prescribed minimal decrease in the quantity underlying the termination criterion has been achieved. As termination criteria for the energy norm or the norm induced by preconditioner require some look-ahead parameter n, see [25,34], we can not estimate the error before the (n + 1) st iteration. Thus, in this case we will always regularize our problem as we can not decide if truncation makes sense.…”

“…For Tab. 1 we used a termination criterion based on an estimate for the relative energy error from [34], see (61), which is admissible as we only work on the subspaces where H + θP is positive definite. , α = 10 −6 ( †: not convergent after 500 iterations).…”

“…The computation of δx should be done in such a way as to satisfy the acceptance criteria (34) and (38). Motivated by (30), we introduce a parametrized model for Θ, equipped with an algorithmic parameter [ω c ] > 0:…”

“…Thus we slightly modify our update rules, i. e. in each inner loop whenever i) (34) has failed at least once, we do not allow decrease in [ω c ] after failure of (38), ii) (38) has failed at least once, we do not allow decrease in [ω f ] after failure of (34).…”