An Adaptive MultiPreconditioned Conjugate Gradient Algorithm

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 to select, at each iteration, whether a classical preconditioned 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 behaviour is illustrated numerically. In particular, it is observed to be optimal in terms of local solves, both for well-conditioned and ill-conditioned test cases, which makes it a good candidate to be a default parallel linear solver.