Iterative regularization and MINRES

In this paper we present three theorems which give insight into the regular izing properties of MINRES. While our theory does not completely characteri ze the regularizing behavior of the algorithm, it provides a partial explan ation of the observed behavior of the method. Unlike traditional attempts t o explain the regularizing properties of Krylov subspace methods, our appro ach focuses on convergence properties of the residual rather than on conver gence analysis of the harmonic Ritz values. The import of our analysis is i llustrated by two examples. In particular, our theoretical and numerical re sults support the following important observation: in some circumstances th e dimension of the optimal Krylov subspace can be much smaller than the num ber of the components of the truncated spectral solution that must be compu ted to attain comparable accuracy.