Abstract

. Iterative methods based on Lanczos bidiagonalization with full reorthogonalization (LBDR) are considered for solving large scale discrete ill-posed linear least squares problems of the form min x kAx \Gamma bk 2 . Methods for regularization in the Krylov subspaces are discussed which use generalized cross validation (GCV) for determining the regularization parameter. These methods have the advantage that no a priori information about the noise level is required. To improve convergence of the Lanczos process we apply a variant of the implicitly restarted Lanczos algorithm by Sorenson using zero shifts. Although this restarted method simply corresponds to using LBDR with a starting vector (AA T ) p b, it is shown that carrying out the process implicitly is essential for numerical stability. An LBDR algorithm is presented which incorporates implicit restarts to ensure that the global minimum of the CGV curve corresponds to a minimum on the curve for the truncated SVD solution. Nume...

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