Automatic parameter setting for Arnoldi–Tikhonov methods

Gazzola, Silvia; Novati, Paolo

doi:10.1016/j.cam.2013.07.023

Cited by 42 publications

(82 citation statements)

References 11 publications

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“…Brezinski et al [3] consider this problem for small to moderately sized problems. More recent treatments are provided by Gazzola and Novati [9] and Lu and Pereverzyev [18]. With x = V y as before, and applying the decompositions (2.6), we get similarly as for (3.1) that (3.8) minimized over R(V ) is equivalent to the reduced minimization problem…”

Section: Multi-parameter Tikhonov Regularizationmentioning

confidence: 82%

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A Golub–Kahan-Type Reduction Method for Matrix Pairs

Hochstenbach

Reichel

2015

J Sci Comput

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Section: Multi-parameter Tikhonov Regularizationmentioning

confidence: 82%

“…Methods for determining suitable regularization parameters for this minimization problem are discussed in [2,3,9,18].…”

Section: Multi-parameter Tikhonov Regularizationmentioning

confidence: 99%

A Golub–Kahan-Type Reduction Method for Matrix Pairs

Hochstenbach

Reichel

2015

J Sci Comput

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“…We then compute a rotation matrix Z 2 so that Z 2 c 2:3 = ± c 2:3 e 1 , and let Z be defined as in (6). The matrices H 1+2 Z * and K 1+2 Z * are no longer have their original structure, hence, we need to compute orthonormal P and Q such that P H 1+2 Z * is again upper-Hessenberg and Q K 1+2 Z * is upper-triangular.…”

Section: Subspace Expansion For Multiparameter Tikhonovmentioning

confidence: 99%

“…See for example [1,2,6,14,16,20,20]. In particular, there is no obvious multiparameter extension of the discrepancy principle.…”

Section: A Multiparameter Selection Strategymentioning

confidence: 99%

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Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

Zwaan

Hochstenbach

2016

J Sci Comput

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Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results.This work is part of the research program "Innovative methods for large matrix problems", which is (partly) financed by the Netherlands Organization for Scientific Research (NWO).

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Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

Gazzola

Landman

2020

GAMM-Mitteilungen

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Large-scale linear systems coming from suitable discretizations of linear inverse problems are challenging to solve. Indeed, since they are inherently ill-posed, appropriate regularization should be applied; since they are large-scale, well-established direct regularization methods (such as Tikhonov regularization) cannot often be straightforwardly employed, and iterative linear solvers should be exploited. Moreover, every regularization method crucially depends on the choice of one or more regularization parameters, which should be suitably tuned. The aim of this paper is twofold: (a) survey some well-established regularizing projection methods based on Krylov subspace methods (with a particular emphasis on methods based on the Golub-Kahan bidiagonalization algorithm), and the so-called hybrid approaches (which combine Tikhonov regularization and projection onto Krylov subspaces of increasing dimension); (b) introduce a new principled and adaptive algorithmic approach for regularization similar to specific instances of hybrid methods. In particular, the new strategy provides reliable parameter choice rules by leveraging the framework of bilevel optimization, and the links between Gauss quadrature and Golub-Kahan bidiagonalization. Numerical tests modeling inverse problems in imaging illustrate the performance of existing regularizing Krylov methods, and validate the new algorithms. K E Y W O R D S hybrid methods, imaging problems, Krylov subspace methods, large-scale linear inverse problems, regularization parameter choice rules, Tikhonov regularization 1 INTRODUCTION This paper considers linear, large-scale, discrete ill-posed problems of the form Ax true + e = b true + e = b , (1) where the matrix A ∈ R m×n represents a forward mapping that is ill-conditioned with ill-determined rank (ie, the singular values of A decay and cluster at zero without an evident gap between two consecutive ones), x true ∈ R n is the desired solution, and e ∈ R m is some unknown noise that affects the data b ∈ R m. Systems like (1) typically stem from the discretization of first-kind Fredholm integral equations, and model inverse problems arising in a variety of applications, such This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Automatic parameter setting for Arnoldi–Tikhonov methods

Cited by 42 publications

References 11 publications

A Golub–Kahan-Type Reduction Method for Matrix Pairs

A Golub–Kahan-Type Reduction Method for Matrix Pairs

Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

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