The adaptive augmented GMRES method for solving ill-posed problems

The GMRES method is an iterative method that provides better solutions when dealing with large linear systems of equations with a non-symmetric 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.