Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ<sup>1</sup>-regularization

Flemming, Jens; Gerth, Daniel

doi:10.1515/jiip-2017-0008

Cited by 18 publications

(61 citation statements)

References 17 publications

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“…2016 b ). In special cases such as -regularization, improved results can be obtained, because the effective finite-dimensionality implies that this case is almost well-posed (Grasmair, Scherzer and Haltmeier 2011, Grasmair 2011, Burger, Flemming and Hofmann 2013 a , Flemming, Hofmann and Veselić 2015, Flemming, Hofmann and Veselić 2016, Flemming and Gerth 2017). Converse results have also been recently obtained (Flemming 2017 a , Hohage and Weidling 2017).…”

Section: Variational Regularization Methodsmentioning

confidence: 99%

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Modern regularization methods for inverse problems

2018

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Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this development towards modern nonlinear regularization methods, including their analysis, applications, and issues for future research.In particular we will discuss variational methods and techniques derived from those, since they have attracted particular interest in the last years and link to other fields like image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions, and learning theory.

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Section: Variational Regularization Methodsmentioning

confidence: 99%

“…In special cases such as 1 -regularization improved results can be obtained, due to the effective finite-dimensionality this case is on the borderline to being well-posed (cf. [193,190,69,178,179,176]). Recently also converse results could be obtained (cf.…”

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confidence: 99%

Modern regularization methods for inverse problems

2018

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“…Namely, the canonical basis vectors e (k) do not necessarily belong to the range of A * but to its closure. For the proof of Proposition 4.3 we refer to [11]. Most of the steps are identical or at least similar to the proof of Proposition 5.4 which we will give later.…”

Section: (I)mentioning

confidence: 96%

On ℓ 1 -Regularization Under Continuity of the Forward Operator in Weaker Topologies

Gerth

Hofmann

2018

Trends in Mathematics

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Our focus is on the stable approximate solution of linear operator equations based on noisy data by using 1 -regularization as a sparsity-enforcing version of Tikhonov regularization. We summarize recent results on situations where the sparsity of the solution slightly fails. In particular, we show how the recently established theory for weak*-to-weak continuous linear forward operators can be extended to the case of weak*-to-weak* continuity. This might be of interest when the image space is non-reflexive. We discuss existence, stability and convergence of regularized solutions. For injective operators, we will formulate convergence rates by exploiting variational source conditions. The typical rate function obtained under an ill-posed operator is strictly concave and the degree of failure of the solution sparsity has an impact on its behavior. Linear convergence rates just occur in the two borderline cases of proper sparsity, where the solutions belong to 0 , and of well-posedness. For an exemplary operator, we demonstrate that the technical properties used in our theory can be verified in practice. In the last section, we briefly mention the difficult case of oversmoothing regularization where x † does not belong to 1 .

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“…Variational source conditions can be obtained from classical and general source conditions for linear problems in Hilbert spaces (see [4, Section 3.2], [19], [5,Chapter 13]), as well as from problem specific calculations for certain nonlinear problems and Banach space settings (see [3,7,11,14,18]). Details on variational source conditions can be found in, e. g., [2,5,9,16].…”

Section: Usually One Usesmentioning

confidence: 99%

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

Flemming

2018

Journal of Inverse and Ill-Posed Problems

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We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

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Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ¹-regularization

Cited by 18 publications

References 17 publications

Modern regularization methods for inverse problems

Modern regularization methods for inverse problems

On ℓ 1 -Regularization Under Continuity of the Forward Operator in Weaker Topologies

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

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Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ1-regularization

Cited by 18 publications

References 17 publications

Modern regularization methods for inverse problems

Modern regularization methods for inverse problems

On ℓ 1 -Regularization Under Continuity of the Forward Operator in Weaker Topologies

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

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Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ¹-regularization