GCV for Tikhonov regularization via global Golub–Kahan decomposition

Fenu, Caterina; Reichel, Lothar; Rodriguez, Giuseppe

doi:10.1002/nla.2034

Cited by 54 publications

(47 citation statements)

References 34 publications

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“…Both methods bring along some downsides, such as, just to name a few, the existence of a corner in the L-curve and the feasibility of the computations in the GCV, especially when dealing with large-scale problems; see [7] for a more extensive discussion and non-quadratic regularizers. A plethora of literature has been focused on developing strategies to overcome the downsides of these classical methods [2,8,12,17,18].…”

mentioning

confidence: 99%

Residual whiteness principle for parameter-free image restoration

Lanza¹,

Pragliola²,

Sgallari³

2020

etna

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Selecting the regularization parameter in the image restoration variational framework is of crucial importance, since it can highly influence the quality of the final restoration. In this paper, we propose a parameter-free approach for automatically selecting the regularization parameter when the blur is space-invariant and known and the noise is additive white Gaussian with unknown standard deviation, based on the so-called residual whiteness principle. More precisely, the regularization parameter is required to minimize the residual whiteness function, namely the normalized auto-correlation of the residual image of the restoration. The proposed method can be applied to a wide class of variational models, such as those including in their formulation regularizers of Tikhonov and Total Variation type. For non-quadratic regularizers, the residual whiteness principle is nested in an iterative optimization scheme based on the alternating direction method of multipliers. The effectiveness of the proposed approach is verified by solving some test examples and performing a comparison with other parameter estimation state-of-the-art strategies, such as the discrepancy principle.

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confidence: 99%

Residual whiteness principle for parameter-free image restoration

Lanza¹,

Pragliola²,

Sgallari³

2020

etna

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“…Substituting x = V ℓ y, y ∈ R ℓ , into (5) and determining an approximate solution by a Galerkin method gives the equation…”

Section: Krylov Subspace Methods For Nonnegative Tikhonov Regularizationmentioning

confidence: 99%

“…We will use the discrepancy principle in the computed examples reported in Section 4. However, the solution methods discussed in this paper also can be applied in conjunction with other techniques for determining a suitable value of µ > 0, including the L-curve criterion and generalized cross validation; see [5,4,6,7] for discussions and comparisons of many methods for determining a suitable value of the regularization parameter. Due to the fact that many singular values of the matrix A cluster at the origin, the least-squares problem (1) may be numerically rank-deficient.…”

Section: Introductionmentioning

confidence: 99%

Modulus-based iterative methods for constrained Tikhonov regularization

Bai

Buccini

Hayami

et al. 2017

Journal of Computational and Applied Mathematics

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Tikhonov regularization is one of the most popular methods for the solution of linear discrete ill-posed problems. In many applications the desired solution is known to lie in the nonnegative cone. It is then natural to require that the approximate solution determined by Tikhonov regularization also lies in this cone. The present paper describes two iterative methods, that employ modulus-based iterative methods, to compute approximate solutions in the nonnegative cone of large-scale Tikhonov regularization problems. The first method considered consists of two steps: first the given linear discrete ill-posed problem is reduced to a small problem by a Krylov subspace method, and then the reduced Tikhonov regularization problems so obtained is solved. The second method described explores the structure of certain image restoration problems. Computed examples illustrate the performances of these methods.

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“…The choice of the regularization parameter is important and many methods have been proposed in the literature; see, e.g., [10,13,15,17] and references therein. We will apply the discrepancy principle in the present paper.…”

Section: The Discrepancy Principle and Zero-findersmentioning

confidence: 99%

“…The regularization parameter µ > 0 determines the sensitivity of x µ to the error e in b, and how close x µ is to the desired vector x true . Many methods for determining a suitable value of the regularization parameter, including the discrepancy principle, the L-curve criterion, and generalized cross validation, require repeated solution of (4) for several values of µ; see, e.g., [10,13,15,17]. When the matrices A and L are of small to moderate size, one therefore often first computes the generalized singular value decomposition (GSVD) of the matrix pair {A, L}.…”

Section: Introductionmentioning

confidence: 99%

Generalized singular value decomposition with iterated Tikhonov regularization

Buccini

Pasha

Reichel

2020

Journal of Computational and Applied Mathematics

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Linear discrete ill-posed problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods try to reduce the sensitivity by replacing the given problem by a nearby one, whose solution is less affected by perturbations. This paper describes how generalized singular value decomposition can be combined with iterated Tikhonov regularization and illustrates that the method so obtained determines approximate solutions of higher quality than the more commonly used approach of pairing generalized singular value decomposition with (standard) Tikhonov regularization. The regularization parameter is determined with the aid of the discrepancy principle. This requires the application of a zero-finder. Several zero-finders are compared.

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GCV for Tikhonov regularization via global Golub–Kahan decomposition

Cited by 54 publications

References 34 publications

Residual whiteness principle for parameter-free image restoration

Residual whiteness principle for parameter-free image restoration

Modulus-based iterative methods for constrained Tikhonov regularization

Generalized singular value decomposition with iterated Tikhonov regularization

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