Regularization by inexact Krylov methods with applications to blind deblurring

TL;DR

This paper introduces two new inexact Krylov methods for large-scale inverse problems, demonstrating their effectiveness in blind deblurring by reducing computational effort through handling uncertain parameters efficiently.

Contribution

The paper develops and analyzes novel inexact Krylov methods tailored for inverse problems, especially in blind deblurring, with strategies to manage inexactness and improve computational efficiency.

Findings

New inexact Krylov methods are effective for large-scale inverse problems.

The methods significantly reduce computational costs in blind deblurring.

Theoretical analysis links inexact and exact Krylov methods.

Abstract

This paper is concerned with the regularization of large-scale discrete inverse problems by means of inexact Krylov methods. Specifically, we derive two new inexact Krylov methods that can be efficiently applied to unregularized or Tikhonov-regularized least squares problems, and we study their theoretical properties, including links with their exact counterparts and strategies to monitor the amount of inexactness. We then apply the new methods to separable nonlinear inverse problems arising in blind deblurring. In this setting inexactness stems from the uncertainty in the parameters defining the blur, which may be recovered using a variable projection method leading to an inner-outer iteration scheme (i.e., one cycle of inner iterations is performed to solve one linear deblurring subproblem for any intermediate values of the blurring parameters computed by a nonlinear least squares solver). The new inexact solvers can naturally handle varying inexact blurring parameters while solving the linear deblurring subproblems, allowing for a much reduced number of total iterations and substantial computational savings with respect to their exact counterparts.