# Iteratively Reweighted FGMRES and FLSQR for Sparse Reconstruction

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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 $\ell_2$-$\ell_1$ 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 $\ell_2$-$\ell_1$ 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.