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  1. ; ; ; ; (, Numerical Algorithms)
    Abstract Tikhonov regularization is commonly used in the solution of linear discrete ill-posed problems. It is known that iterated Tikhonov regularization often produces approximate solutions of higher quality than (standard) Tikhonov regularization. This paper discusses iterated Tikhonov regularization for large-scale problems with a general regularization matrix. Specifically, the original problem is reduced to small size by application of a fairly small number of steps of the Arnoldi or Golub-Kahan processes, and iterated Tikhonov is applied to the reduced problem. The regularization parameter is determined by using an extension of a technique first described by Donatelli and Hanke for quite special coefficient matrices. Convergence of the method is established and computed examples illustrate its performance. 
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    Free, publicly-accessible full text available May 6, 2026
  2. ; ; ; (, Numerical Linear Algebra with Applications)
    ABSTRACT Several iterative soft‐thresholding algorithms, such as FISTA, have been proposed in the literature for solving regularized linear discrete inverse problems that arise in various applications in science and engineering. These algorithms are easy to implement, but their rates of convergence may be slow. This paper describes novel approaches to reduce the computations required for each iteration by using Krylov subspace techniques. Specifically, we propose to impose sparsity on the coefficients in the representation of the computed solution in terms of a Krylov subspace basis. Several numerical examples from image deblurring and computerized tomography are used to illustrate the efficiency and accuracy of the proposed methods. 
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    Free, publicly-accessible full text available February 1, 2026