Regularization by inexact Krylov methods with applications to blind deblurring

Silvia Gazzola; Malena Sabate Landman

doi:10.1137/21M1402066

Regularization by inexact Krylov methods with applications to blind deblurring

Silvia Gazzola, Malena Sabate Landman

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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, where both the sharp image and the parameters defining the blur are unkown. When employing a variable projection method jointly with the new inexact solvers in this setting, the latter 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.

Read More: https://epubs.siam.org/doi/abs/10.1137/21M1402066

Original language	English
Pages (from-to)	1528-1552
Number of pages	25
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	42
Issue number	4
Early online date	11 Nov 2021
DOIs	https://doi.org/10.1137/21M1402066
Publication status	Published - 31 Dec 2021

Bibliographical note

Funding:
The work of the first author was partially supported by the EPSRC grant EP/T001593/1.
The work of the second author was supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under grant EP/L015684/1.

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Copyright © 2021 SIAM. The final publication is available at SIAM Journal of Matrix Analysis and Applications via https://doi.org/10.1137/21M1402066
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Fast and Flexible Solvers for Inverse Problems
Gazzola, S. (PI)
Engineering and Physical Sciences Research Council
15/09/19 → 14/09/22
Project: Research council

Cite this

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title = "Regularization by inexact Krylov methods with applications to blind deblurring",

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, where both the sharp image and the parameters defining the blur are unkown. When employing a variable projection method jointly with the new inexact solvers in this setting, the latter 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.Read More: https://epubs.siam.org/doi/abs/10.1137/21M1402066",

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note = "Funding: The work of the first author was partially supported by the EPSRC grant EP/T001593/1. The work of the second author was supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under grant EP/L015684/1. ",

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AB - 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, where both the sharp image and the parameters defining the blur are unkown. When employing a variable projection method jointly with the new inexact solvers in this setting, the latter 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.Read More: https://epubs.siam.org/doi/abs/10.1137/21M1402066

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Regularization by inexact Krylov methods with applications to blind deblurring

Abstract

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Fast and Flexible Solvers for Inverse Problems

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