Augmented GMRES-type methods

GMRES is a popular iterative method for the solution of large linear systems of equations with a square nonsymmetric matrix. The method generates a Krylov subspace in which an approximate solution is determined. We present modifications of the GMRES and the closely related RRGMRES methods that allow augmentation of the Krylov subspaces generated by these methods by a user-supplied subspace. We choose this subspace to enable the representation of certain known nonsmooth features of the desired solution, such as jumps, or to make it possible to represent certain smooth functions, such as constants or linear functions. The latter choice of augmenting subspace appears to be new. Applications to the solution of both well-posed and ill-posed problems are presented. Copyright c © 2006 John Wiley & Sons, Ltd.