GMRES with multiple preconditioners

Abstract

Abstract. We propose a variant of GMRES, where multiple (two or more) preconditioners are applied simultaneously, while maintaining minimal residual optimality properties. To accomplish this, a block version of Flexible GMRES is used, but instead of considering blocks associated with multiple right hand sides, we consider a single right-hand side and grow the space by applying each of the preconditioners to all current search directions, minimizing the residual norm over the resulting larger subspace. To alleviate the difficulty of rapidly increasing storage requirements, we present a heuristic limited-memory selective algorithm, and demonstrate the effectiveness of this approach. Numerical results for problems in PDE-constrained optimization and fluid flow problems are presented, illustrating the viability and the potential of the proposed method.

Two-level adaptation for Adaptive Multipreconditioned FETI

“…In these methods, the contribution of each subdomain to (5) is considered as a separate preconditioner which, at each iteration, supplies one direction inside a search space of dimension N d . All these strategies rely on multipreconditioned Krylov solvers [16,3] and more specifically multipreconditioned conjugate gradient (MPCG) [7].…”

“…Multipreconditioning is well adapted to the FETI and BDD methods because of the additive nature of their classical preconditioners. Moreover, this technique has been successfully applied to solve non symmetric systems [16,3] for which finding the suitable augmentation is still an open question.…”

Preconditioners for Krylov subspace methods: An overview

“…We conclude this section by commenting on a technique that can be applied when a problem has two (or more) preconditioners that accelerate the convergence rate of a linear system. While it may be difficult to formulate a new preconditioner that captures the good behavior of each individual preconditioner, they can be combined within a multipreconditioned iterative solver [243,244]; this has been successfully used in the solution of parameter‐dependent problems [245] and two‐phase flow [246]. More generally, when dealing with sequences of problems it can make sense to reuse information from previously computed systems, including preconditioners [19, section 14] (see also Section 4.4).…”

Domain Decomposition Methods in Science and Engineering XXIII

“…This was further studied in Greif et al (2014) which considers Additive Schwarz preconditioners in the Multipreconditioned GMRES (MPGMRES) Greif et al (2016) setting. It is an iterative linear solver, adapted from the Preconditioned Conjugate Gradient (PCG) algorithm , which can be used in cases where several preconditioners are available or the usual preconditioner is a sum of operators.…”