GMRES with multiple preconditioners

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.