A Spectral Projection Preconditioner for Solving Ill Conditioned Linear   Systems
  We present a preconditioner based on spectral projection that is combined with a deflated Krylov subspace method for solving ill conditioned linear systems of equations. Our results show that the proposed algorithm requires many fewer iterations to achieve the convergence criterion for solving an ill conditioned problem than a Krylov subspace solver. In our numerical experiments, the solution obtained by the proposed algorithm is more accurate in terms of the norm of the distance to the exact solution of the linear system of equations. 
Man-Chung Yeung, Craig C. Douglas, Long Lee
math.NA