An Adaptive MultiPreconditioned Conjugate Gradient Algorithm

Abstract

This article introduces and analyzes a new adaptive algorithm for solving symmetric positive definite linear systems in cases where several preconditioners are available or the usual preconditioner is a sum of contributions. A new theoretical result allows us to select, at each iteration, whether a classical preconditioned conjugate gradient (CG) iteration is sufficient (i.e., the error decreases by a factor of at least some chosen ratio) or whether convergence needs to be accelerated by performing an iteration of multipreconditioned CG [4]. This is first presented in an abstract framework with the one strong assumption being that a bound for the smallest eigenvalue of the preconditioned operator is available. Then, the algorithm is applied to the balancing domain decomposition method and its behavior is illustrated numerically. In particular, it is observed to be optimal in terms of local solves, for both well-conditioned and ill-conditioned test cases, which makes it a good candidate to be a default par...