Closer to the solution: restarted GMRES with adaptive preconditioning

The generalized minimum residual (GMRES) method proposed by Saad and Schultz in 1986 is one of the most popular iterative algorithms for solving systems of linear equations Ax = b for non-hermitian matrices A. The method computes the optimal approximation for which the 2-norm of the residual is minimal over a corresponding Krylov subspace. One drawback of the conventional GMRES method is its linearly increasing costs per iterations, which may become prohibitively large in practical applications. Ordinary restarted GMRES overcomes this difficulty at the cost of loss of the optimality property. This thesis provides variants of restarted GMRES that try to recover the optimality property by recycling approximate spectral information gathered during the iterations, to build adaptive preconditioners. This may result in better clustering of the eigenvalues around point one of the spectrum, and consequently less iteration steps than the ordinary restarted GMRES method. The theoretical background of the the new family of solvers is presented, and its numerical properties are illustrated by Matlab experiments on realistic matrix problems arising from different fields of application.