Iterative Methods for Ill-Posed Problems

Bakushinsky, A. B.; Kokurin, M. Yu.; Smirnova, Alexandra

doi:10.1515/9783110250657

Cited by 40 publications

(44 citation statements)

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“…However, as far as regularization is concerned, it should be mentioned that simple constraints such as non‐negativity or box constraints, which sometimes have a dramatic effect in enhancing the quality of the reconstructions, cannot be straightforwardly incorporated when adopting the basic Krylov subspace methods discussed in this paper (although some approaches based on flexible Krylov methods are available; see [82]). Similarly, the methods discussed in this paper cannot be straightforwardly applied to nonlinear ill‐posed problems; see, eg, [83,84]. Extending the new class of algorithms to handle these situations can be the focus of future research.…”

Section: Discussionmentioning

confidence: 99%

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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Section: Discussionmentioning

confidence: 99%

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

Gazzola

Landman

2020

GAMM-Mitteilungen

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“…Я. И. ВЕДЕЛЬ, С. В. ДЕНИСОВ, В. В. СЕМЁНОВ Следуя А. Б. Бакушинскому [1], назовем задачу (3) аппроксимацией Тихонова-Браудера двухуровневой задачи (2) 1 . Из результатов [11] следует существование и единственность решения x ε ∈ C задачи (3) для любого ε > 0.…”

Section: постановка задачиunclassified

“…Ключевые слова: двухуровневая задача, вариационное неравенство, задача о равновесии, двухэтапный проксимальный метод, итеративная регуляризация, сильная сходимость. Введение В оптимизации, теории некорректных задач распространен следующий прием решения задач с неединственным решением [1][2][3]. Задаче ставится в соответствие семейство возмущенных задач, однозначно и корректно разрешимых.…”

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Algorithm for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problems

Vedel

Денисов

Semenov

2020

JNAM

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In this paper, we consider bilevel problem: variational inequality problem over the set of solutions the equilibrium problems. To solve this problem, an iterative algorithm is proposed that combines the ideas of a two-stage proximal method and iterative regularization. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz continuous operators, the theorem on strong convergence of sequences generated by the algorithm is proved.

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“…For what concerns the variational inequalities and their approximations, there is an extensive literature. More precisely, existence and approximations of solutions to variational inequalities for various classes of operators in Hilbert and Banach spaces have been considered by Browder [11], Stampacchia [43], Mosco [30,31], Alber [2], Bakushinskii [7], Doktor and Kucera [16], Liskovets [23], Alber and Rjazantseva [3], Rjazantseva [37,38], Liskovets [24,25], McLinden [29], Tossings [46], Gwinner [19], see also Liu [26], Liu and Nashed [27,34], the related references cited in [34], and the monographs by Tikhonov [44,45], Kaplan and Tichartschke [20], Bakushinskii and Goncharskii [8,9], Vasin and Ageev [47] and others. Browder [11] and Stampacchia [43] investigated on the convergence of solutions to variational inequalities when there is no perturbation of the convex set.…”

Section: Introductionmentioning

confidence: 99%

“…Liskovets [24] considered the problem under an assumption of (S)-property. Bakushinskii [7] and Bakushinskii and Goncharskii [8] investigated on the convergence and covergence rate for iterative solutions. Versions of Mosco's perturbation and convergence scheme have been developed in several papers.…”

Section: Introductionmentioning

confidence: 99%

On ill-posedness and stability of tensor variational inequalities: application to an economic equilibrium

Barbagallo

Bianco

2019

J Glob Optim

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The general tensor variational inequalities, recently introduced in Barbagallo et al. (J Nonconvex Anal 19:711-729, 2018), are very useful in order to analyze economic equilibrium models. For this reason, the study of existence and regularity results for such inequalities has an important rule to the light of applications. To this aim, we start to consider some existence and uniqueness theorems for tensor variational inequalities. Then, we investigate on the approximation of solutions to tensor variational inequalities by using suitable perturbed tensor variational inequalities. We establish the convergence of solutions to the regularized tensor variational inequalities to a solution of the original tensor variational inequality making use of the set convergence in Kuratowski's sense. After that, we focus our attention on some stability results. At last, we apply the theoretical results to examine a general oligopolistic market equilibrium problem.

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Iterative Methods for Ill-Posed Problems

Cited by 40 publications

References 0 publications

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

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

Algorithm for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problems

On ill-posedness and stability of tensor variational inequalities: application to an economic equilibrium

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