This paper presents a novel approach to the regularization of linear problems involving total variation (TV) penalization, with a particular emphasis on image deblurring applications. The starting point of the new strategy is an approximation of the nondifferentiable TV regularization term by a sequence of quadratic terms, expressed as iteratively reweighted 2-norms of the gradient of the solution. The resulting problem is then reformulated as a Tikhonov regularization problem in standard form, and solved by an efficient Krylov subspace method. Namely, flexible GMRES is considered in order to incorporate new weights into the solution subspace as soon as a new approximate solution is computed. The new method is dubbed TV-FGMRES. Theoretical insight is given, and computational details are carefully unfolded. Numerical experiments and comparisons with other algorithms for TV image deblurring, as well as other algorithms based on Krylov subspace methods, are provided to validate TV-FGMRES.

Large-scale linear systems coming from suitable discretizations of linear inverse problems are challenging to solve. Indeed, since they are inherently ill-posed, appropriate regularization should be applied; since they are large-scale, well-established direct regularization methods (such as Tikhonov regularization) cannot often be straightforwardly employed, and iterative linear solvers should be exploited. Moreover, every regularization method crucially depends on the choice of one or more regularization parameters, which should be suitably tuned. The aim of this paper is twofold: (a) survey some well-established regularizing projection methods based on Krylov subspace methods (with a particular emphasis on methods based on the Golub-Kahan bidiagonalization algorithm), and the so-called hybrid approaches (which combine Tikhonov regularization and projection onto Krylov subspaces of increasing dimension); (b) introduce a new principled and adaptive algorithmic approach for regularization similar to specific instances of hybrid methods. In particular, the new strategy provides reliable parameter choice rules by leveraging the framework of bilevel optimization, and the links between Gauss quadrature and Golub-Kahan bidiagonalization. Numerical tests modeling inverse problems in imaging illustrate the performance of existing regularizing Krylov methods, and validate the new algorithms. K E Y W O R D S hybrid methods, imaging problems, Krylov subspace methods, large-scale linear inverse problems, regularization parameter choice rules, Tikhonov regularization 1 INTRODUCTION This paper considers linear, large-scale, discrete ill-posed problems of the form Ax true + e = b true + e = b , (1) where the matrix A ∈ R m×n represents a forward mapping that is ill-conditioned with ill-determined rank (ie, the singular values of A decay and cluster at zero without an evident gap between two consecutive ones), x true ∈ R n is the desired solution, and e ∈ R m is some unknown noise that affects the data b ∈ R m. Systems like (1) typically stem from the discretization of first-kind Fredholm integral equations, and model inverse problems arising in a variety of applications, such This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

This paper presents two new algorithms to compute sparse solutions of large-scale linear discrete ill-posed problems. The proposed approach consists in constructing a sequence of quadratic problems approximating an 2 -1 regularization scheme (with additional smoothing to ensure differentiability at the origin) and partially solving each problem in the sequence using flexible Krylov-Tikhonov methods. These algorithms are built upon a new solid theoretical justification that guarantees that the sequence of approximate solutions to each problem in the sequence converges to the solution of the considered modified version of the 2 -1 problem. Compared to other traditional methods, the new algorithms have the advantage of building a single (flexible) approximation (Krylov) subspace that encodes regularization through variable "preconditioning" and that is expanded as soon as a new problem in the sequence is defined. Links between the new solvers and other well-established solvers based on augmenting Krylov subspaces are also established. The performance of these algorithms is shown through a variety of numerical examples modeling image deblurring and computed tomography.

This paper is concerned with the regularization of large-scale discrete inverse problems by means of inexact Krylov methods. Specifically, we derive two new inexact Krylov methods that can be efficiently applied to unregularized or Tikhonov-regularized least squares problems, and we study their theoretical properties, including links with their exact counterparts and strategies to monitor the amount of inexactness. We then apply the new methods to separable nonlinear inverse problems arising in blind deblurring, where both the sharp image and the parameters defining the blur are unkown. When employing a variable projection method jointly with the new inexact solvers in this setting, the latter can naturally handle varying inexact blurring parameters while solving the linear deblurring subproblems, allowing for a much reduced number of total iterations and substantial computational savings with respect to their exact counterparts.

This paper is concerned with the regularization of large-scale discrete inverse problems by means of inexact Krylov methods. Specifically, we derive two new inexact Krylov methods that can be efficiently applied to unregularized or Tikhonov-regularized least squares problems, and we study their theoretical properties, including links with their exact counterparts and strategies to monitor the amount of inexactness. We then apply the new methods to separable nonlinear inverse problems arising in blind deblurring. In this setting inexactness stems from the uncertainty in the parameters defining the blur, which may be recovered using a variable projection method leading to an inner-outer iteration scheme (i.e., one cycle of inner iterations is performed to solve one linear deblurring subproblem for any intermediate values of the blurring parameters computed by a nonlinear least squares solver). The new inexact solvers can naturally handle varying inexact blurring parameters while solving the linear deblurring subproblems, allowing for a much reduced number of total iterations and substantial computational savings with respect to their exact counterparts.