An improved preconditioned LSQR for discrete ill-posed problems

We present a modified version of the two-level iterative method proposed in [M. Hanke, R. Vogel, Two-level preconditioners for regularized inverse problems. I: Theory, Numerische Mathematik 83 (1999) 385–402]. Here, we propose the application of the two-level Schur complement CG on the unregularized problem and the introduction of the regularization process for solving only one of the linear systems produced by the algorithm. The modified algorithm is substantially cheaper and numerical examples show similar approximations in both cases. A novel basis for the coarse subspace is incorporated in the analysis. Numerical experiments for some test problems and a practical scattering problem are presented.