Iteratively Reweighted FGMRES and FLSQR for Sparse Reconstruction

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 quadratic problems approximating an $\ell_2$-$\ell_1$ regularization scheme (with additional smoothing ensure differentiability at the origin) and partially solving each problem using flexible Krylov--Tikhonov methods. These are built upon solid theoretical justification that guarantees approximate converges solution considered modified version problem. Compared other traditional methods, have advantage building single (flexible) approximation (Krylov) subspace encodes through variable “preconditioning” is expanded as soon defined. Links between solvers well-established based on augmenting Krylov subspaces also established. performance these shown variety numerical examples modeling image deblurring computed tomography.