New flexible and inexact Golub-Kahan algorithms for inverse problems

TL;DR

This paper presents innovative flexible and inexact Golub-Kahan algorithms for large-scale inverse problems, enabling efficient regularized solutions with adaptive preconditioning and inexact computations, demonstrated through imaging applications.

Contribution

It introduces a new class of flexible and inexact Golub-Kahan algorithms that improve upon existing Krylov subspace methods for inverse problems.

Findings

Effective in image deblurring and tomography

Competitive performance compared to existing methods

Handles general data fidelity functionals

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.