MPGMRES: a generalized minimum residual method with multiple preconditioners

We propose a variant of GMRES which we call MPGMRES, whereby 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 starting with multiple right hand sides, we start with the initial residual and grow the space by applying each of the preconditioners to all current search directions and minimizing the residual norm over the resulting larger subspace. To alleviate the difficulty of rapidly increasing storage requirements and make the method practical, we further propose a selective algorithm that uses limited memory, and show theoretically and experimentally that this approach is highly effective. Numerical results for problems in domain decomposition, PDE-constrained optimization, and fluid flow problems are presented, illustrating the viability and the potential of the proposed method.