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Abstract
This paper presents two new algorithms to compute sparse solutions of largescale linear discrete illposed problems. The proposed approach consists in constructing a sequence of quadratic problems approximating an $\ell_2$$\ell_1$ regularization scheme (with additional smoothing to ensure differentiability at the origin) and partially solving each problem in the sequence using flexible KrylovTikhonov methods. These algorithms are built upon a new solid theoretical justification that guarantees that the sequence of approximate solutions to each problem in the sequence converges to the solution of the considered modified version of the $\ell_2$$\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 wellestablished 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.
Read More: https://epubs.siam.org/doi/abs/10.1137/20M1333948?af=R
Read More: https://epubs.siam.org/doi/abs/10.1137/20M1333948?af=R
Original language  English 

Pages (fromto)  S47S69 
Journal  SIAM Journal on Scientific Computing 
Early online date  11 Feb 2021 
DOIs  
Publication status  Published  31 Dec 2021 
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Fast and Flexible Solvers for Inverse Problems
Engineering and Physical Sciences Research Council
15/09/19 → 14/09/22
Project: Research council