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New flexible and inexact Golub-Kahan algorithms for inverse problems

Malena Sabaté Landman, Silvia Gazzola

TL;DR

This paper addresses large-scale linear inverse problems $A x_{\text{true}}+n=b$ where regularization is essential due to ill-posedness and noise. It develops a unified flexible and inexact Krylov framework based on new Golub-Kahan factorizations (FGK and iGK) that compute regularized solutions by projecting onto adaptively generated subspaces with iteration-dependent preconditioners, enabling general data-fidelity terms including $\ell_p$-norm fits. The authors introduce new solvers APD, ADP, and PAD that combine approximation, differentiation and projection in novel orders, together with classical DAP, DPA, PDA methods, and provide inexactness estimates to guide restarts. Numerical experiments in image deblurring and parallel-beam computed tomography demonstrate competitive performance and accuracy gains, highlighting the practical impact for imaging inverse problems. The work paves the way for including explicit regularization and hybrid formulations within the same flexible/inexact Krylov framework.

Abstract

This paper introduces a new class of algorithms for solving large-scale linear inverse problems based on new flexible and inexact Golub-Kahan factorizations. The proposed methods iteratively compute regularized solutions by approximating a solution to (re)weighted least squares problems via projection onto adaptively generated subspaces, where the constraint subspaces for the residuals are (formally) equipped with iteration-dependent preconditioners or inexactness. The new solvers offer a flexible and inexact Krylov subspace alternative to other existing Krylov-based approaches for handling general data fidelity functionals, e.g., those expressed in the $p$-norm. Numerical experiments in imaging applications, such as image deblurring and computed tomography, highlight the effectiveness and competitiveness of the proposed methods with respect to other popular methods.

New flexible and inexact Golub-Kahan algorithms for inverse problems

TL;DR

This paper addresses large-scale linear inverse problems where regularization is essential due to ill-posedness and noise. It develops a unified flexible and inexact Krylov framework based on new Golub-Kahan factorizations (FGK and iGK) that compute regularized solutions by projecting onto adaptively generated subspaces with iteration-dependent preconditioners, enabling general data-fidelity terms including -norm fits. The authors introduce new solvers APD, ADP, and PAD that combine approximation, differentiation and projection in novel orders, together with classical DAP, DPA, PDA methods, and provide inexactness estimates to guide restarts. Numerical experiments in image deblurring and parallel-beam computed tomography demonstrate competitive performance and accuracy gains, highlighting the practical impact for imaging inverse problems. The work paves the way for including explicit regularization and hybrid formulations within the same flexible/inexact Krylov framework.

Abstract

This paper introduces a new class of algorithms for solving large-scale linear inverse problems based on new flexible and inexact Golub-Kahan factorizations. The proposed methods iteratively compute regularized solutions by approximating a solution to (re)weighted least squares problems via projection onto adaptively generated subspaces, where the constraint subspaces for the residuals are (formally) equipped with iteration-dependent preconditioners or inexactness. The new solvers offer a flexible and inexact Krylov subspace alternative to other existing Krylov-based approaches for handling general data fidelity functionals, e.g., those expressed in the -norm. Numerical experiments in imaging applications, such as image deblurring and computed tomography, highlight the effectiveness and competitiveness of the proposed methods with respect to other popular methods.
Paper Structure (14 sections, 1 theorem, 65 equations, 12 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 1 theorem, 65 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Flexible and inexact Golub-Kahan factorizations
  3. Standard, flexible and inexact solvers for general data-fitting problems
  4. Exact Krylov solvers for the exact functional
  5. Mixing and matching differentiation, projection, approximation
  6. DAP, DPA and PDA methods, i.e., 'classical' inexact Krylov solvers
  7. Inexactness estimates
  8. APD, ADP and PAD solvers, i.e., new flexible Krylov solvers
  9. Inexactness estimates
  10. Numerical experiments
  11. Example 1: satellite image deblurring problem
  12. Example 2: Tomography problem
  13. Conclusions and outlook
  14. Basis vectors

Key Result

Proposition 1

Monotonicity properties for flexible MM schemes. Consider the approximate solutions given the new flexible solvers to be $\{{\bf x}_k\}_{k=1,...}$, and assume that ${\bf R}^{-1}_{k+1} = {\bf R}^{-1}({\bf x}_{k-1})$. Then, for $f({\bf x})$ defined in eq:MM_f, we can guarantee monotonicity in the obje for ${\boldsymbol{\nu}} = {\bf r}_k-{\bf r}_{k-1} = {\bf U}_k \left( {\bf y}_k - \right)$.

Figures (12)

  • Figure 1: Data for the first example, representing an image deblurring test problem. From left to right: true solution, point spread function for the blur, noisy blurred image, and noise. The latter is displayed using the square root of the absolute values of each element.
  • Figure 2: Example 1. Relative error norm values versus iteration number for a variety of solvers, with specific emphasis on the effect of restarting (second row). Note that the label 'exact' corresponds to preconditioned LSQR using the weights evaluated in the exact solution.
  • Figure 3: Example 1. Top row: Best reconstructions for a variety of algorithms. Bottom row: Square root of the absolute values of the error vectors (all operations performed entry-wise).
  • Figure 4: Example 1. Top row: Residuals associated to the best reconstructions for a variety of algorithms. Bottom row: residual error vectors, i.e. ${\bf A}{\bf x}-{\bf b}-{\bf r}_\text{true}$. All of the images are displaying the square root of the absolute values of each of the entries.
  • Figure 5: Data for the second example, representing a computed tomography test problem. Left: true solution. Center: noisy measurements (a.k.a. sinogram). Right: square root of the absolute value of the noise, or the residual of the true solution (all operations performed entry-wise).
  • ...and 7 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Proposition 1
  • proof
  • Remark 8