On regularizing effects of MINRES and MR-II for large scale symmetric discrete ill-posed problems

Abstract. For large-scale symmetric discrete ill-posed problems, MINRES and MR-II are commonly used iterative solvers. In this paper, we analyze their regularizing effects. We first prove that the regularized solutions by MINRES have filtered SVD forms. Then we show that (i) a hybrid MINRES that uses explicit regularization within projected problems is generally needed to compute a best possible regularized solution to a given ill-posed problem and (ii) the kth iterate by MINRES is more accurate than the (k− 1)th iterate by MR-II until the semi-convergence of MINRES, though MR-II has globally better regularizing effects than MINRES. Here a best possible regularized solution means that it essentially has the minimum 2-norm error when standard-form Tikhonov regularization is used, or from the other perspective, it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition (TSVD) method. Moreover, we establish bounds for the distance between an underlying k-dimensional Krylov subspace and the k-dimensional dominant eigenspace. They show that MR-II has better regularizing effects for severely and moderately ill-posed problems than for mildly ill-posed problems, indicating that a hybrid MR-II is needed to get a best possible regularized solution for mildly ill-posed problems. Numerical experiments confirm our assertions. Stronger than our theory, MR-II is experimentally good enough to compute best possible regularized solutions for severely and moderately ill-posed problems. We also justify that MR-II is as equally effective as and twice as efficient as LSQR for symmetric ill-posed problems.