gmres(m) is a Krylov subspace method for solving nonsymmetric linear systems of equations. The difficulty of this method lies in choosing the appropriate restart cycle m. We propose a new strategy for the adaptive restart for gmres(m) which is based on using the difference of the Ritz and harmonic Ritz values. We also report on numerical experiments which show that this new approach is both effective and robust.

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The gmres method is an iterative method that provides better solutions when dealing with large linear systems of equations with a nonsymmetric coefficient matrix. The gmres method generates a Krylov subspace for the solution, and the augmented gmres method allows augmentation of the Krylov subspaces by a user supplied subspace which represents certain known features of the desired solution. The augmented gmres method performs well with suitable augmentation, but performs poorly with unsuitable augmentation. The adaptive augmented gmres method automatically selects a suitable subspace from a set of candidates supplied by the user. This study shows that this method maintains the performance level of augmented gmres and lightens the burden it puts on its users. Numerical experiments compare robustness as well as the efficiency of various heuristic strategies.

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