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LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, andits mathematically equivalent CGLS applied to normal equations system, arecommonly used for large-scale discrete ill-posed problems. It is well knownthat LSQR and CGLS have regularizing effects, where the number of iterationsplays the role of the regularization parameter. However, it has long beenunknown whether the regularizing effects are good enough to find best possibleregularized solutions. Here a best possible regularized solution means that itis at least as accurate as the best regularized solution obtained by thetruncated singular value decomposition (TSVD) method. In this paper, weestablish bounds for the distance between the $k$-dimensional Krylov subspaceand the $k$-dimensional dominant right singular space. They show that theKrylov subspace captures the dominant right singular space better for severelyand moderately ill-posed problems than for mildly ill-posed problems. Ourgeneral conclusions are that LSQR has better regularizing effects for the firsttwo kinds of problems than for the third kind, and a hybrid LSQR withadditional regularization is generally needed for mildly ill-posed problems.Exploiting the established bounds, we derive an estimate for the accuracy ofthe rank $k$ approximation generated by Lanczos bidiagonalization. Numericalexperiments illustrate that the regularizing effects of LSQR are good enough tocompute best possible regularized solutions for severely and moderatelyill-posed problems, stronger than our theory predicts, but they are not formildly ill-posed problems and additional regularization is needed.

Some Results on the Regularization of LSQR for Large-Scale Discrete Ill-Posed Problems