Automatic stopping rule for iterative methods in discrete ill-posed problems

Abstract

The numerical treatment of large-scale discrete ill-posed problems is often accomplished iteratively by projecting the original problem onto a $k$-dimensional subspace with $k$ acting as regularization parameter. Hence, to filter out the contribution of noise in the computed solution, the iterative process must be stopped early. In this paper, we analyze in detail a stopping rule for LSQR proposed recently by the authors, and show how to extend it to Krylov subspace methods such as GMRES, MINRES, etc. Like the original rule, the extended version works well without requiring a priori knowledge about the error norm and stops automatically after $\widetilde{k}+1$ steps where $\widetilde{k}$ is the computed regularization parameter. The performance of the stopping rule on several test problems is illustrated numerically.