A comparison of parameter choice rules for $$\ell ^p$$-$$\ell ^q$$ minimization

Buccini, Alessandro; Pragliola, Monica; Reichel, Lothar; Sgallari, Fiorella

doi:10.1007/s11565-022-00430-9

Cited by 6 publications

(2 citation statements)

References 37 publications

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“…It is essential that the regularization parameter be chosen properly for the accurate approximation of the desired approximate solution x of (1). A comparison of several techniques for determining has recently been presented in [19]. In this paper, we consider two strategies for determining : for Gaussian noise, we apply the discrepancy principle (DP) [10]; for other kinds of noise, we use generalized crossvalidation (GCV) [9].…”

Section: Numerical Examplesmentioning

confidence: 99%

Limited memory restarted ℓp-ℓq minimization methods using generalized Krylov subspaces

Buccini

Reichel

2023

Adv Comput Math

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Regularization of certain linear discrete ill-posed problems, as well as of certain regression problems, can be formulated as large-scale, possibly nonconvex, minimization problems, whose objective function is the sum of the p th power of the ℓp-norm of a fidelity term and the q th power of the ℓq-norm of a regularization term, with 0 < p,q ≤ 2. We describe new restarted iterative solution methods that require less computer storage and execution time than the methods described by Huang et al. (BIT Numer. Math. 57,351–378, 14). The reduction in computer storage and execution time is achieved by periodic restarts of the method. Computed examples illustrate that restarting does not reduce the quality of the computed solutions.

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Section: Numerical Examplesmentioning

confidence: 99%

Limited memory restarted ℓp-ℓq minimization methods using generalized Krylov subspaces

Buccini

Reichel

2023

Adv Comput Math

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“…For the case of a single scalar regularization parameter µ -that is, when the (TV), (TV 2 ) or (TV 2 2 ) regularizers are used -some effective selection criteria have been proposed in literature; cf. [4,9]. The most popular one is the discrepancy principle (DP) (cf.…”

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Parameter-free restoration of piecewise smooth images

Lanza,

Pragliola,

Sgallari

2023

etna

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We propose a novel strategy for the automatic estimation of the two regularization parameters arising in the image decomposition variational model employed for the restoration task when the underlying corrupting noise is known to be additive white Gaussian. In the model of interest, the target image is decomposed in its piecewise constant and smooth components, with a total variation term penalizing the former and a Tikhonov term acting on the latter. The proposed criterion, which relies on the whiteness property of the noise, extends the residual whiteness principle, originally introduced in the case of a single regularization parameter. The structure of the considered decomposition model allows for an efficient estimation of the pair of unknown parameters, that can be automatically adjusted along the iterations with the alternating direction method of multipliers employed for the numerical solution. The proposed multi-parameter residual whiteness principle is tested on different images with different levels of corruption. The performed tests highlight that the whiteness criterion is particularly effective and robust when moving from a single-parameter to a multi-parameter scenario.

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An Efficient Implementation of the Gauss–Newton Method Via Generalized Krylov Subspaces

Buccini,

de Alba,

Pes

et al. 2023

J Sci Comput

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The solution of nonlinear inverse problems is a challenging task in numerical analysis. In most cases, this kind of problems is solved by iterative procedures that, at each iteration, linearize the problem in a neighborhood of the currently available approximation of the solution. The linearized problem is then solved by a direct or iterative method. Among this class of solution methods, the Gauss–Newton method is one of the most popular ones. We propose an efficient implementation of this method for large-scale problems. Our implementation is based on projecting the nonlinear problem into a sequence of nested subspaces, referred to as Generalized Krylov Subspaces, whose dimension increases with the number of iterations, except for when restarts are carried out. When the computation of the Jacobian matrix is expensive, we combine our iterative method with secant (Broyden) updates to further reduce the computational cost. We show convergence of the proposed solution methods and provide a few numerical examples that illustrate their performance.

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A comparison of parameter choice rules for $$\ell ^p$$-$$\ell ^q$$ minimization

Cited by 6 publications

References 37 publications

Limited memory restarted ℓp-ℓq minimization methods using generalized Krylov subspaces

Limited memory restarted ℓp-ℓq minimization methods using generalized Krylov subspaces

Parameter-free restoration of piecewise smooth images

An Efficient Implementation of the Gauss–Newton Method Via Generalized Krylov Subspaces

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