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Iterative Refinement and Flexible Iteratively Reweighed Solvers for Linear Inverse Problems with Sparse Solutions

Lucas Onisk, Malena Sabaté Landman

TL;DR

The paper introduces a restartable framework that blends iteratively reweighted norm (IRN) schemes with iterative refinement and flexible Krylov subspaces to solve large-scale linear inverse problems with sparse solutions. By interpreting IRN as iterative refinement and employing iteration-dependent preconditioning, the authors develop IR-FGMRES, IR-FLSQR and corrected variants CIR-FGMRES, CIR-FLSQR that support restarts and automatic parameter selection via the discrepancy principle. Theoretical results establish monotone decrease of a smoothed objective and convergence properties under mild conditions, while numerical experiments on 1D deblurring, a star-cluster image, and oversampled CT demonstrate memory efficiency and competitive accuracy compared with existing Krylov solvers. The proposed methods are particularly impactful for high-noise, memory-constrained settings common in imaging inverse problems, offering robust performance and scalable, automated regularization control.

Abstract

This paper presents a new algorithmic framework for computing sparse solutions to large-scale linear discrete ill-posed problems. The approach is motivated by recent perspectives on iteratively reweighted norm schemes, viewed through the lens of iterative refinement. This framework leverages the efficiency and fast convergence of flexible Krylov methods while achieving higher accuracy through suitable restarts. Additionally, we demonstrate that the proposed methods outperform other flexible Krylov approaches in memory-limited scenarios. Relevant convergence theory is discussed, and the performance of the proposed algorithms is illustrated through a range of numerical examples, including image deblurring and computed tomography.

Iterative Refinement and Flexible Iteratively Reweighed Solvers for Linear Inverse Problems with Sparse Solutions

TL;DR

The paper introduces a restartable framework that blends iteratively reweighted norm (IRN) schemes with iterative refinement and flexible Krylov subspaces to solve large-scale linear inverse problems with sparse solutions. By interpreting IRN as iterative refinement and employing iteration-dependent preconditioning, the authors develop IR-FGMRES, IR-FLSQR and corrected variants CIR-FGMRES, CIR-FLSQR that support restarts and automatic parameter selection via the discrepancy principle. Theoretical results establish monotone decrease of a smoothed objective and convergence properties under mild conditions, while numerical experiments on 1D deblurring, a star-cluster image, and oversampled CT demonstrate memory efficiency and competitive accuracy compared with existing Krylov solvers. The proposed methods are particularly impactful for high-noise, memory-constrained settings common in imaging inverse problems, offering robust performance and scalable, automated regularization control.

Abstract

This paper presents a new algorithmic framework for computing sparse solutions to large-scale linear discrete ill-posed problems. The approach is motivated by recent perspectives on iteratively reweighted norm schemes, viewed through the lens of iterative refinement. This framework leverages the efficiency and fast convergence of flexible Krylov methods while achieving higher accuracy through suitable restarts. Additionally, we demonstrate that the proposed methods outperform other flexible Krylov approaches in memory-limited scenarios. Relevant convergence theory is discussed, and the performance of the proposed algorithms is illustrated through a range of numerical examples, including image deblurring and computed tomography.

Paper Structure

This paper contains 18 sections, 4 theorems, 34 equations, 9 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. IRN schemes for $\ell_p$ regularization
  3. Background on IRN
  4. Krylov methods for $\ell_p$ regularization
  5. Flexible Krylov methods for $\ell_p$ regularization
  6. IRN as an iterative refinement scheme
  7. Background on iterative refinement
  8. IRN schemes as iterative refinement
  9. Combining IRW flexible Krylov methods and iterative refinement
  10. Iteratively Refined and (restarted) flexible Krylov methods
  11. Corrected iteratively refined flexible IRN
  12. Criteria for the choice of regularization parameters
  13. Theoretical considerations
  14. Numerical experiments
  15. One dimensional deblurring problem
  16. ...and 3 more sections

Key Result

Proposition 5.1

Let $\{\mathbf{x}_k\}_k$ be the sequence of approximate solutions computed by either IR-FGMRES, IF-FLSQR, CIR-FGMRES, or CIR-FLSQR (in Algorithms alg:FIRKS, alg:CFIRKS_A and alg:CFIRKS_GKB, respectively). Assume that, for all $\mathbf{W}_k=\widetilde{W}(\mathbf{L}\mathbf{x}_k)$ the following holds:

Figures (9)

  • Figure 1: Setting for the 1D signal restoration example Spectra.
  • Figure 2: Selection of basis vectors for the search spaces corresponding to the new methods: IR-FGMRES and IR-FLSQR; in comparison with the standard GMRES and LSQR for the test problem Spectra. Underlaid, in black discontinuos lines, the true solution.
  • Figure 3: Different error norm histories across the iterations for the test problem Spectra. In the top row, comparisons between methods based on the (flexible) Arnoldi method; on the bottom row, comparisons between methods based on the (flexible) Golub-Kahan method. In the second and third column, the discrepancy principle is used at each iteration when possible. Note that FISTA and SpaRSA require a user-specified regularization parameter.
  • Figure 4: True image and measurements for the test problem star cluster. Note that the color bars are in different scales. This is because the measurements have lost contrast due to the blurring, and it would be hard to visualize $\mathbf{b}$ otherwise.
  • Figure 5: Different error norm histories for the test problem star cluster.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Proposition 5.1
  • proof
  • Corollary 5.2
  • proof
  • Proposition 5.3
  • proof
  • Proposition 5.4
  • proof