Iteratively Reweighted FGMRES and FLSQR for Sparse Reconstruction

Silvia Gazzola, James G. Nagy, Malena Sabate Landman

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8 Citations (SciVal)
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Abstract

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.

Original languageEnglish
Pages (from-to)S47-S69
JournalSIAM Journal on Scientific Computing
Early online date11 Feb 2021
DOIs
Publication statusPublished - 31 Dec 2021

Bibliographical note

Funding:
The work of the first author was partially funded by EPSRC under grant EP/T001593/1.
The work of the second author was partially supported by the U.S. National Science
Foundation under grant DMS -1819042.
The work of the third author was supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa) under project EP/L015684/1.

Keywords

  • Augmented Krylov methods
  • Flexible Krylov methods
  • Imaging problems
  • Inverse problems
  • Krylov methods
  • Sparse reconstruction

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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