Iterative regularization and MINRES

In this paper we present three theorems which give insight into the regularizingproperties of MINRES. While our theory does not completely characterize the regularizing behavior
of the algorithm, it provides a partial explanation of the observed behavior of the method. Unlike
traditional attempts to explain the regularizing properties of Krylov subspace methods, our approach
focuses on convergence properties of the residual rather than on convergence analysis of the harmonic
Ritz values. The import of our analysis is illustrated by two examples. In particular, our theoret-
ical and numerical results support the following important observation: in some circumstances the
dimension of the optimal Krylov subspace can be much smaller than the number of the components
of the truncated spectral solution that must be computed to attain comparable accuracy.